abstract sample in quantitative research

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Abstract Writing: A Step-by-Step Guide With Tips & Examples

Table of Contents

Introduction

Abstracts of research papers have always played an essential role in describing your research concisely and clearly to researchers and editors of journals, enticing them to continue reading. However, with the widespread availability of scientific databases, the need to write a convincing abstract is more crucial now than during the time of paper-bound manuscripts.

Abstracts serve to "sell" your research and can be compared with your "executive outline" of a resume or, rather, a formal summary of the critical aspects of your work. Also, it can be the "gist" of your study. Since most educational research is done online, it's a sign that you have a shorter time for impressing your readers, and have more competition from other abstracts that are available to be read.

The APCI (Academic Publishing and Conferences International) articulates 12 issues or points considered during the final approval process for conferences & journals and emphasises the importance of writing an abstract that checks all these boxes (12 points). Since it's the only opportunity you have to captivate your readers, you must invest time and effort in creating an abstract that accurately reflects the critical points of your research.

With that in mind, let’s head over to understand and discover the core concept and guidelines to create a substantial abstract. Also, learn how to organise the ideas or plots into an effective abstract that will be awe-inspiring to the readers you want to reach.

What is Abstract? Definition and Overview

The word "Abstract' is derived from Latin abstractus meaning "drawn off." This etymological meaning also applies to art movements as well as music, like abstract expressionism. In this context, it refers to the revealing of the artist's intention.

Based on this, you can determine the meaning of an abstract: A condensed research summary. It must be self-contained and independent of the body of the research. However, it should outline the subject, the strategies used to study the problem, and the methods implemented to attain the outcomes. The specific elements of the study differ based on the area of study; however, together, it must be a succinct summary of the entire research paper.

Abstracts are typically written at the end of the paper, even though it serves as a prologue. In general, the abstract must be in a position to:

• Describe the paper.
• Identify the problem or the issue at hand.
• Explain to the reader the research process, the results you came up with, and what conclusion you've reached using these results.
• Include keywords to guide your strategy and the content.

Furthermore, the abstract you submit should not reflect upon any of  the following elements:

• Examine, analyse or defend the paper or your opinion.
• What you want to study, achieve or discover.
• Be redundant or irrelevant.

After reading an abstract, your audience should understand the reason - what the research was about in the first place, what the study has revealed and how it can be utilised or can be used to benefit others. You can understand the importance of abstract by knowing the fact that the abstract is the most frequently read portion of any research paper. In simpler terms, it should contain all the main points of the research paper.

What is the Purpose of an Abstract?

Abstracts are typically an essential requirement for research papers; however, it's not an obligation to preserve traditional reasons without any purpose. Abstracts allow readers to scan the text to determine whether it is relevant to their research or studies. The abstract allows other researchers to decide if your research paper can provide them with some additional information. A good abstract paves the interest of the audience to pore through your entire paper to find the content or context they're searching for.

Abstract writing is essential for indexing, as well. The Digital Repository of academic papers makes use of abstracts to index the entire content of academic research papers. Like meta descriptions in the regular Google outcomes, abstracts must include keywords that help researchers locate what they seek.

Types of Abstract

Informative and Descriptive are two kinds of abstracts often used in scientific writing.

A descriptive abstract gives readers an outline of the author's main points in their study. The reader can determine if they want to stick to the research work, based on their interest in the topic. An abstract that is descriptive is similar to the contents table of books, however, the format of an abstract depicts complete sentences encapsulated in one paragraph. It is unfortunate that the abstract can't be used as a substitute for reading a piece of writing because it's just an overview, which omits readers from getting an entire view. Also, it cannot be a way to fill in the gaps the reader may have after reading this kind of abstract since it does not contain crucial information needed to evaluate the article.

To conclude, a descriptive abstract is:

• A simple summary of the task, just summarises the work, but some researchers think it is much more of an outline
• Typically, the length is approximately 100 words. It is too short when compared to an informative abstract.
• A brief explanation but doesn't provide the reader with the complete information they need;
• An overview that omits conclusions and results

An informative abstract is a comprehensive outline of the research. There are times when people rely on the abstract as an information source. And the reason is why it is crucial to provide entire data of particular research. A well-written, informative abstract could be a good substitute for the remainder of the paper on its own.

A well-written abstract typically follows a particular style. The author begins by providing the identifying information, backed by citations and other identifiers of the papers. Then, the major elements are summarised to make the reader aware of the study. It is followed by the methodology and all-important findings from the study. The conclusion then presents study results and ends the abstract with a comprehensive summary.

In a nutshell, an informative abstract:

• Has a length that can vary, based on the subject, but is not longer than 300 words.
• Contains all the content-like methods and intentions
• Offers evidence and possible recommendations.

Informative Abstracts are more frequent than descriptive abstracts because of their extensive content and linkage to the topic specifically. You should select different types of abstracts to papers based on their length: informative abstracts for extended and more complex abstracts and descriptive ones for simpler and shorter research papers.

What are the Characteristics of a Good Abstract?

• A good abstract clearly defines the goals and purposes of the study.
• It should clearly describe the research methodology with a primary focus on data gathering, processing, and subsequent analysis.
• A good abstract should provide specific research findings.
• It presents the principal conclusions of the systematic study.
• It should be concise, clear, and relevant to the field of study.
• A well-designed abstract should be unifying and coherent.
• It is easy to grasp and free of technical jargon.
• It is written impartially and objectively.

What are the various sections of an ideal Abstract?

By now, you must have gained some concrete idea of the essential elements that your abstract needs to convey . Accordingly, the information is broken down into six key sections of the abstract, which include:

An Introduction or Background

Research methodology, objectives and goals, limitations.

Let's go over them in detail.

The introduction, also known as background, is the most concise part of your abstract. Ideally, it comprises a couple of sentences. Some researchers only write one sentence to introduce their abstract. The idea behind this is to guide readers through the key factors that led to your study.

It's understandable that this information might seem difficult to explain in a couple of sentences. For example, think about the following two questions like the background of your study:

• What is currently available about the subject with respect to the paper being discussed?
• What isn't understood about this issue? (This is the subject of your research)

While writing the abstract’s introduction, make sure that it is not lengthy. Because if it crosses the word limit, it may eat up the words meant to be used for providing other key information.

Research methodology is where you describe the theories and techniques you used in your research. It is recommended that you describe what you have done and the method you used to get your thorough investigation results. Certainly, it is the second-longest paragraph in the abstract.

In the research methodology section, it is essential to mention the kind of research you conducted; for instance, qualitative research or quantitative research (this will guide your research methodology too) . If you've conducted quantitative research, your abstract should contain information like the sample size, data collection method, sampling techniques, and duration of the study. Likewise, your abstract should reflect observational data, opinions, questionnaires (especially the non-numerical data) if you work on qualitative research.

The research objectives and goals speak about what you intend to accomplish with your research. The majority of research projects focus on the long-term effects of a project, and the goals focus on the immediate, short-term outcomes of the research. It is possible to summarise both in just multiple sentences.

In stating your objectives and goals, you give readers a picture of the scope of the study, its depth and the direction your research ultimately follows. Your readers can evaluate the results of your research against the goals and stated objectives to determine if you have achieved the goal of your research.

In the end, your readers are more attracted by the results you've obtained through your study. Therefore, you must take the time to explain each relevant result and explain how they impact your research. The results section exists as the longest in your abstract, and nothing should diminish its reach or quality.

One of the most important things you should adhere to is to spell out details and figures on the results of your research.

Instead of making a vague assertion such as, "We noticed that response rates varied greatly between respondents with high incomes and those with low incomes", Try these: "The response rate was higher for high-income respondents than those with lower incomes (59 30 percent vs. 30 percent in both cases; P<0.01)."

You're likely to encounter certain obstacles during your research. It could have been during data collection or even during conducting the sample . Whatever the issue, it's essential to inform your readers about them and their effects on the research.

Research limitations offer an opportunity to suggest further and deep research. If, for instance, you were forced to change for convenient sampling and snowball samples because of difficulties in reaching well-suited research participants, then you should mention this reason when you write your research abstract. In addition, a lack of prior studies on the subject could hinder your research.

Your conclusion should include the same number of sentences to wrap the abstract as the introduction. The majority of researchers offer an idea of the consequences of their research in this case.

Your conclusion should include three essential components:

• A significant take-home message.
• Corresponding important findings.
• The Interpretation.

Even though the conclusion of your abstract needs to be brief, it can have an enormous influence on the way that readers view your research. Therefore, make use of this section to reinforce the central message from your research. Be sure that your statements reflect the actual results and the methods you used to conduct your research.

Good Abstract Examples

Abstract example #1.

Children’s consumption behavior in response to food product placements in movies.

The abstract:

"Almost all research into the effects of brand placements on children has focused on the brand's attitudes or behavior intentions. Based on the significant differences between attitudes and behavioral intentions on one hand and actual behavior on the other hand, this study examines the impact of placements by brands on children's eating habits. Children aged 6-14 years old were shown an excerpt from the popular film Alvin and the Chipmunks and were shown places for the item Cheese Balls. Three different versions were developed with no placements, one with moderately frequent placements and the third with the highest frequency of placement. The results revealed that exposure to high-frequency places had a profound effect on snack consumption, however, there was no impact on consumer attitudes towards brands or products. The effects were not dependent on the age of the children. These findings are of major importance to researchers studying consumer behavior as well as nutrition experts as well as policy regulators."

Abstract Example #2

Social comparisons on social media: The impact of Facebook on young women’s body image concerns and mood. The abstract:

"The research conducted in this study investigated the effects of Facebook use on women's moods and body image if the effects are different from an internet-based fashion journal and if the appearance comparison tendencies moderate one or more of these effects. Participants who were female ( N = 112) were randomly allocated to spend 10 minutes exploring their Facebook account or a magazine's website or an appearance neutral control website prior to completing state assessments of body dissatisfaction, mood, and differences in appearance (weight-related and facial hair, face, and skin). Participants also completed a test of the tendency to compare appearances. The participants who used Facebook were reported to be more depressed than those who stayed on the control site. In addition, women who have the tendency to compare appearances reported more facial, hair and skin-related issues following Facebook exposure than when they were exposed to the control site. Due to its popularity it is imperative to conduct more research to understand the effect that Facebook affects the way people view themselves."

Abstract Example #3

The Relationship Between Cell Phone Use and Academic Performance in a Sample of U.S. College Students

"The cellphone is always present on campuses of colleges and is often utilised in situations in which learning takes place. The study examined the connection between the use of cell phones and the actual grades point average (GPA) after adjusting for predictors that are known to be a factor. In the end 536 students in the undergraduate program from 82 self-reported majors of an enormous, public institution were studied. Hierarchical analysis ( R 2 = .449) showed that use of mobile phones is significantly ( p < .001) and negative (b equal to -.164) connected to the actual college GPA, after taking into account factors such as demographics, self-efficacy in self-regulated learning, self-efficacy to improve academic performance, and the actual high school GPA that were all important predictors ( p < .05). Therefore, after adjusting for other known predictors increasing cell phone usage was associated with lower academic performance. While more research is required to determine the mechanisms behind these results, they suggest the need to educate teachers and students to the possible academic risks that are associated with high-frequency mobile phone usage."

Quick tips on writing a good abstract

There exists a common dilemma among early age researchers whether to write the abstract at first or last? However, it's recommended to compose your abstract when you've completed the research since you'll have all the information to give to your readers. You can, however, write a draft at the beginning of your research and add in any gaps later.

If you find abstract writing a herculean task, here are the few tips to help you with it:

1. Always develop a framework to support your abstract

Before writing, ensure you create a clear outline for your abstract. Divide it into sections and draw the primary and supporting elements in each one. You can include keywords and a few sentences that convey the essence of your message.

2. Review Other Abstracts

Abstracts are among the most frequently used research documents, and thousands of them were written in the past. Therefore, prior to writing yours, take a look at some examples from other abstracts. There are plenty of examples of abstracts for dissertations in the dissertation and thesis databases.

3. Avoid Jargon To the Maximum

When you write your abstract, focus on simplicity over formality. You should  write in simple language, and avoid excessive filler words or ambiguous sentences. Keep in mind that your abstract must be readable to those who aren't acquainted with your subject.

4. Focus on Your Research

It's a given fact that the abstract you write should be about your research and the findings you've made. It is not the right time to mention secondary and primary data sources unless it's absolutely required.

Conclusion: How to Structure an Interesting Abstract?

Abstracts are a short outline of your essay. However, it's among the most important, if not the most important. The process of writing an abstract is not straightforward. A few early-age researchers tend to begin by writing it, thinking they are doing it to "tease" the next step (the document itself). However, it is better to treat it as a spoiler.

The simple, concise style of the abstract lends itself to a well-written and well-investigated study. If your research paper doesn't provide definitive results, or the goal of your research is questioned, so will the abstract. Thus, only write your abstract after witnessing your findings and put your findings in the context of a larger scenario.

The process of writing an abstract can be daunting, but with these guidelines, you will succeed. The most efficient method of writing an excellent abstract is to centre the primary points of your abstract, including the research question and goals methods, as well as key results.

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Writing an Abstract for Your Research Paper

Definition and Purpose of Abstracts

An abstract is a short summary of your (published or unpublished) research paper, usually about a paragraph (c. 6-7 sentences, 150-250 words) long. A well-written abstract serves multiple purposes:

• an abstract lets readers get the gist or essence of your paper or article quickly, in order to decide whether to read the full paper;
• an abstract prepares readers to follow the detailed information, analyses, and arguments in your full paper;
• and, later, an abstract helps readers remember key points from your paper.

It’s also worth remembering that search engines and bibliographic databases use abstracts, as well as the title, to identify key terms for indexing your published paper. So what you include in your abstract and in your title are crucial for helping other researchers find your paper or article.

If you are writing an abstract for a course paper, your professor may give you specific guidelines for what to include and how to organize your abstract. Similarly, academic journals often have specific requirements for abstracts. So in addition to following the advice on this page, you should be sure to look for and follow any guidelines from the course or journal you’re writing for.

The Contents of an Abstract

Abstracts contain most of the following kinds of information in brief form. The body of your paper will, of course, develop and explain these ideas much more fully. As you will see in the samples below, the proportion of your abstract that you devote to each kind of information—and the sequence of that information—will vary, depending on the nature and genre of the paper that you are summarizing in your abstract. And in some cases, some of this information is implied, rather than stated explicitly. The Publication Manual of the American Psychological Association , which is widely used in the social sciences, gives specific guidelines for what to include in the abstract for different kinds of papers—for empirical studies, literature reviews or meta-analyses, theoretical papers, methodological papers, and case studies.

Here are the typical kinds of information found in most abstracts:

• the context or background information for your research; the general topic under study; the specific topic of your research
• the central questions or statement of the problem your research addresses
• what’s already known about this question, what previous research has done or shown
• the main reason(s) , the exigency, the rationale , the goals for your research—Why is it important to address these questions? Are you, for example, examining a new topic? Why is that topic worth examining? Are you filling a gap in previous research? Applying new methods to take a fresh look at existing ideas or data? Resolving a dispute within the literature in your field? . . .
• your research and/or analytical methods
• your main findings , results , or arguments
• the significance or implications of your findings or arguments.

Your abstract should be intelligible on its own, without a reader’s having to read your entire paper. And in an abstract, you usually do not cite references—most of your abstract will describe what you have studied in your research and what you have found and what you argue in your paper. In the body of your paper, you will cite the specific literature that informs your research.

When to Write Your Abstract

Although you might be tempted to write your abstract first because it will appear as the very first part of your paper, it’s a good idea to wait to write your abstract until after you’ve drafted your full paper, so that you know what you’re summarizing.

What follows are some sample abstracts in published papers or articles, all written by faculty at UW-Madison who come from a variety of disciplines. We have annotated these samples to help you see the work that these authors are doing within their abstracts.

Choosing Verb Tenses within Your Abstract

The social science sample (Sample 1) below uses the present tense to describe general facts and interpretations that have been and are currently true, including the prevailing explanation for the social phenomenon under study. That abstract also uses the present tense to describe the methods, the findings, the arguments, and the implications of the findings from their new research study. The authors use the past tense to describe previous research.

The humanities sample (Sample 2) below uses the past tense to describe completed events in the past (the texts created in the pulp fiction industry in the 1970s and 80s) and uses the present tense to describe what is happening in those texts, to explain the significance or meaning of those texts, and to describe the arguments presented in the article.

The science samples (Samples 3 and 4) below use the past tense to describe what previous research studies have done and the research the authors have conducted, the methods they have followed, and what they have found. In their rationale or justification for their research (what remains to be done), they use the present tense. They also use the present tense to introduce their study (in Sample 3, “Here we report . . .”) and to explain the significance of their study (In Sample 3, This reprogramming . . . “provides a scalable cell source for. . .”).

Sample Abstract 1

From the social sciences.

Reporting new findings about the reasons for increasing economic homogamy among spouses

Gonalons-Pons, Pilar, and Christine R. Schwartz. “Trends in Economic Homogamy: Changes in Assortative Mating or the Division of Labor in Marriage?” Demography , vol. 54, no. 3, 2017, pp. 985-1005.

Sample Abstract 2

From the humanities.

Analyzing underground pulp fiction publications in Tanzania, this article makes an argument about the cultural significance of those publications

Emily Callaci. “Street Textuality: Socialism, Masculinity, and Urban Belonging in Tanzania’s Pulp Fiction Publishing Industry, 1975-1985.” Comparative Studies in Society and History , vol. 59, no. 1, 2017, pp. 183-210.

Sample Abstract/Summary 3

From the sciences.

Reporting a new method for reprogramming adult mouse fibroblasts into induced cardiac progenitor cells

Lalit, Pratik A., Max R. Salick, Daryl O. Nelson, Jayne M. Squirrell, Christina M. Shafer, Neel G. Patel, Imaan Saeed, Eric G. Schmuck, Yogananda S. Markandeya, Rachel Wong, Martin R. Lea, Kevin W. Eliceiri, Timothy A. Hacker, Wendy C. Crone, Michael Kyba, Daniel J. Garry, Ron Stewart, James A. Thomson, Karen M. Downs, Gary E. Lyons, and Timothy J. Kamp. “Lineage Reprogramming of Fibroblasts into Proliferative Induced Cardiac Progenitor Cells by Defined Factors.” Cell Stem Cell , vol. 18, 2016, pp. 354-367.

Sample Abstract 4, a Structured Abstract

Reporting results about the effectiveness of antibiotic therapy in managing acute bacterial sinusitis, from a rigorously controlled study

Note: This journal requires authors to organize their abstract into four specific sections, with strict word limits. Because the headings for this structured abstract are self-explanatory, we have chosen not to add annotations to this sample abstract.

Wald, Ellen R., David Nash, and Jens Eickhoff. “Effectiveness of Amoxicillin/Clavulanate Potassium in the Treatment of Acute Bacterial Sinusitis in Children.” Pediatrics , vol. 124, no. 1, 2009, pp. 9-15.

“OBJECTIVE: The role of antibiotic therapy in managing acute bacterial sinusitis (ABS) in children is controversial. The purpose of this study was to determine the effectiveness of high-dose amoxicillin/potassium clavulanate in the treatment of children diagnosed with ABS.

METHODS : This was a randomized, double-blind, placebo-controlled study. Children 1 to 10 years of age with a clinical presentation compatible with ABS were eligible for participation. Patients were stratified according to age (<6 or ≥6 years) and clinical severity and randomly assigned to receive either amoxicillin (90 mg/kg) with potassium clavulanate (6.4 mg/kg) or placebo. A symptom survey was performed on days 0, 1, 2, 3, 5, 7, 10, 20, and 30. Patients were examined on day 14. Children’s conditions were rated as cured, improved, or failed according to scoring rules.

RESULTS: Two thousand one hundred thirty-five children with respiratory complaints were screened for enrollment; 139 (6.5%) had ABS. Fifty-eight patients were enrolled, and 56 were randomly assigned. The mean age was 6630 months. Fifty (89%) patients presented with persistent symptoms, and 6 (11%) presented with nonpersistent symptoms. In 24 (43%) children, the illness was classified as mild, whereas in the remaining 32 (57%) children it was severe. Of the 28 children who received the antibiotic, 14 (50%) were cured, 4 (14%) were improved, 4(14%) experienced treatment failure, and 6 (21%) withdrew. Of the 28children who received placebo, 4 (14%) were cured, 5 (18%) improved, and 19 (68%) experienced treatment failure. Children receiving the antibiotic were more likely to be cured (50% vs 14%) and less likely to have treatment failure (14% vs 68%) than children receiving the placebo.

CONCLUSIONS : ABS is a common complication of viral upper respiratory infections. Amoxicillin/potassium clavulanate results in significantly more cures and fewer failures than placebo, according to parental report of time to resolution.” (9)

Some Excellent Advice about Writing Abstracts for Basic Science Research Papers, by Professor Adriano Aguzzi from the Institute of Neuropathology at the University of Zurich:

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• How to Write An Abstract For Research Papers: Tips & Examples

In many ways, an abstract is like a trailer of a movie or the synopsis of your favorite book. Its job is to whet the reader’s appetite by sharing important information about your work. After reading a well-written abstract, one should have enough interest to explore the full research thesis. 

So how do you write an interesting abstract that captures the core of your study? First, you need to understand your research objectives and match them with the key results of your study. In this article, we will share some tips for writing an effective abstract, plus samples you can learn from. 

What is an Abstract in Research Writing?

In simple terms, an abstract is a concise write-up that gives an overview of your systematic investigation. According to Grammarly, it is a self-contained summary of a larger work, and it serves as a preview of the bigger document. 

It usually appears at the beginning of your thesis or research paper and helps the reader to have an overview of your work without going into great detail. This means that when someone reads your abstract, it should give them a clear idea of the purpose of your systematic investigation, your problem statement, key results, and any gaps requiring further investigation. 

So how long should your abstract be to capture all of these details? The reality is you don’t need a lot of words to capture key pieces of information in your abstract. Typically, 6–7 sentences made up of 150–250 words should be just right. 

Read: Writing Research Proposals: Tips, Examples & Mistakes

What are the Characteristics of a Good Abstract? 

• A good abstract clearly states the aims and objectives of the research.
• It outlines the research methodology for data gathering , processing and analysis. 
• A good abstract summarizes specific research results.
• It states the key conclusions of the systematic investigation.
• It is brief yet straight to the point. 
• A good abstract is unified and coherent. 
• It is easy to understand and devoid of technical jargon. 
• It is written in an unbiased and objective manner. 

What is the Purpose of an Abstract? 

Every abstract has two major purposes. First, it communicates the relevance of your systematic investigation to readers. After reading your abstract, people can determine how relevant your study is to their primary or secondary research purpose. 

The second purpose of an abstract is to communicate your key findings to those who don’t have time to read the whole paper. Research papers typically run into tens of pages so it takes time to read and digest them. To help readers grasp the core ideas in a systematic investigation, it pays to have a well-written abstract that outlines important information concerning your study. 

In all, your abstract should accurately outline the most important information in your research. Many times, it determines whether people would go ahead to read your dissertation. Abstracts are often indexed along with keywords on academic databases, so they make your thesis easily findable.

Learn About: How to Write a Problem Statement for your Research

What are the Sections of an Abstract?

You already know the key pieces of information that your abstract should communicate. These details are broken into six important sections of the abstract which are: 

• The Introduction or Background
• Research Methodology
• Aims and Objectives 
• Limitations

Let’s discuss them in detail. 

• The Introduction or Background 

The introduction or background is the shortest part of your abstract and usually consists of 2–3 sentences. In fact, some researchers write a single sentence as the introduction of their abstract. The whole idea here is to take the reader through the important events leading to your research. 

Understandably, this information may appear difficult to convey in a few sentences. To help out, consider answering these two questions in the background to your study : 

• What is already known about the subject, related to the paper in question? 
• What is not known about the subject (this is the focus of your study)? 

As much as possible, ensure that your abstract’s introduction doesn’t eat into the word count for the other key information. 

• Research Methodology 

This is the section where you spell out any theories and methods adopted for your study. Ideally, you should cover what has been done and how you went about it to achieve the results of your systematic investigation. It is usually the second-longest section in the abstract. 

In the research methodology section, you should also state the type of research you embarked on; that is, qualitative research or quantitative research —this will inform your research methods too. If you’ve conducted quantitative research, your abstract should contain information like the sample size, data collection methods , sampling technique, and duration of your experiment. 

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In the end, readers are most interested in the results you’ve achieved with your study. This means you should take time to outline every relevant outcome and show how they affect your research population . Typically, the results section should be the longest one in your abstract and nothing should compromise its range and quality. 

An important thing you should do here is spelled out facts and figures about research outcomes. Instead of a vague statement like, “we noticed that response rates differed greatly between high-income and low-income respondents”, try this: “The response rate was higher in high-income respondents than in their low-income counterparts (59% vs 30%, respectively; P

• Conclusion 

Like the introduction, your conclusion should contain a few sentences that wrap up your abstract. Most researchers express a theoretical opinion about the implications of their study, here. 

Your conclusion should contain three important elements: 

• The primary take-home message
• The additional findings of importance
• The perspective 

Although the conclusion of your abstract should be short, it has a great impact on how readers perceive your study. So, take advantage of this section to reiterate the core message in your systematic investigation. Also, make sure any statements here reflect the true outcomes and methods of your research. 

• Limitations 

Chances are you must have faced certain challenges in the course of your research—it could be at the data collection phase or during sampling . Whatever these challenges are, it pays to let your readers know about them, and the impact they had on your study. 

For example, if you had to switch to convenience sampling or snowball sampling due to difficulties in contacting well-suited research participants, you should include this in your abstract. Also, a lack of previous studies in the research area could pose a limitation on your study. Research limitations provide an opportunity to make suggestions for further research. 

Research aims and objectives speak to what you want to achieve with your study. Typically, research aims focus on a project’s long-term outcomes while the objectives focus on the immediate, short-term outcome of the investigation. You may summarize both using a single paragraph comprising a few sentences.

Stating your aims and objectives will give readers a clear idea of the scope, depth, and direction that your research will ultimately take. Readers would measure your research outcomes against stated aims and objectives to know if you achieved the purpose of your study. 

Use For Free: Research Form Templates

Abstract Writing Styles and General Guidelines 

Now that you know the different sections plus information that your abstract should contain, let’s look at how to write an abstract for your research paper.

A common question that comes up is, should I write my abstract first or last? It’s best to write your abstract after you’ve finished working on the research because you have full information to present to your readers. However, you can always create a draft at the beginning of your systematic investigation and fill in the gaps later.  

Does writing an abstract seem like a herculean task? Here are a few tips to help out. 

1. Always create a framework for your abstract 

Before you start writing, take time to develop a detailed outline for your abstract. Break it into sections and sketch the main and supporting points for each section. You can list keywords plus 1–2 sentences that capture your core messaging. 

2. Read Other Abstracts 

Abstracts are one of the most common research documents, and thousands of them have been written in time past. So, before writing yours, try to study a couple of samples from others. You can find lots of dissertation abstract examples in thesis and dissertation databases.

3. Steer Clear of Jargon As Much As Possible 

While writing your abstract, emphasize clarity over style. This means you should communicate in simple terms and avoid unnecessary filler words and ambiguous sentences. Remember, your abstract should be understandable to readers who are not familiar with your topic. 

4. Focus on Your Research

It goes without saying that your abstract should be solely focused on your research and what you’ve discovered. It’s not the time to cite primary and secondary data sources unless this is absolutely necessary. 

This doesn’t mean you should ignore the scholarly background of your work. You might include a sentence or two summarizing the scholarly background to show the relevance of your work to a broader debate, but there’s no need to mention specific publications. 

Going further, here are some abstract writing guidelines from the University of Bergen: 

• An abstract briefly explains the salient aspects of the content. 
• Abstracts should be accurate and succinct, self-contained, and readable.  
• The abstract should paraphrase and summarise rather than quote from the paper.
• Abstracts should relate only to the paper to be presented/assessed.

Types of Abstracts with Examples 

According to the University of Adelaide, there are two major types of abstracts written for research purposes. First, we have informative abstracts and descriptive abstracts. 

1. Informative Abstract  

An informative abstract is the more common type of abstract written for academic research. It highlights the most important aspects of your systematic investigation without going into unnecessary or irrelevant details that the reader might not find useful. 

The length varies according to discipline, but an informative abstract is rarely more than 10% of the length of the entire work. In the case of longer work, it may be much less.

In any informative abstract, you’d touch on information like the purpose, method, scope, results, and conclusion of your study. By now, you’re thinking, “this is the type of abstract we’ve been discussing all along”, and you wouldn’t be far from the truth. 

Advantages of Informative Abstracts

• These abstracts save time for both the researcher and the readers. 
• It’s easy to refer to these abstracts as secondary research sources. 

Disadvantages of Informative Abstracts

• These types of abstracts lack personality.

Example of an Informative Abstract

• Sample Informative Abstract Based on Experimental Work From Colorado State University
• Sample Informative Abstract Based on Non-experimental Work From Colorado State University

2. Descriptive Abstract 

A descriptive abstract reads like a synopsis and focuses on enticing the reader with interesting information. They don’t care as much for data and details, and instead read more like overviews that don’t give too much away. 

You’d find descriptive abstracts in artistic criticism pieces and entertainment research as opposed to scientific investigations. This type of abstract makes no judgments about the work, nor does it provide results or conclusions of the research. They are usually written in 100 words or less. 

Advantages of Descriptive Abstracts

• It gives a very brief overview of the research paper. 
• It is easier to write descriptive abstracts compared to informational abstracts. 

Disadvantages of Descriptive Abstracts

• They are suitable for scientific research. 
• Descriptive abstracts might omit relevant information that deepens your knowledge of the systematic investigation.

Example of Descriptive Abstracts 

• Sample Descriptive Abstract From Colorado State University

FAQs About Writing Abstracts in Research Papers

1. How Long Should an Abstract Be?

A typical abstract should be about six sentences long or less than 150 words. Most universities have specific word count requirements that fall within 150–300 words. 

2. How Do You Start an Abstract Sentence?

There are several ways to start your abstract. Consider the following methods: 

• State a problem or uncertainty
• Make a general statement with the present research action.
• State the purpose or objective of your research
• State a real-world phenomena or a standard practice.

3. Should you cite in an abstract?

While you can refer to information from specific research papers, there’s no need to cite sources in your abstract. Your abstract should focus on your original research, not on the work of others. 

4. What should not be included in an abstract?

An abstract shouldn’t have numeric references, bibliographies, sections, or even footnotes. 

5. Which tense is used in writing an abstract?

An abstract should be written in the third-person present tense. Use the simple past tense when describing your methodology and specific findings from your study. 

Writing an abstract might appear challenging but with these steps, you should get it right. The easiest approach to writing a good abstract is centering it on key information including your research problem and objectives, methodology, and key results.

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• How to Write an Abstract | Steps & Examples

How to Write an Abstract | Steps & Examples

Published on 1 March 2019 by Shona McCombes . Revised on 10 October 2022 by Eoghan Ryan.

An abstract is a short summary of a longer work (such as a dissertation or research paper ). The abstract concisely reports the aims and outcomes of your research, so that readers know exactly what your paper is about.

Although the structure may vary slightly depending on your discipline, your abstract should describe the purpose of your work, the methods you’ve used, and the conclusions you’ve drawn.

One common way to structure your abstract is to use the IMRaD structure. This stands for:

• Introduction

Abstracts are usually around 100–300 words, but there’s often a strict word limit, so make sure to check the relevant requirements.

In a dissertation or thesis , include the abstract on a separate page, after the title page and acknowledgements but before the table of contents .

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Table of contents

Abstract example, when to write an abstract, step 1: introduction, step 2: methods, step 3: results, step 4: discussion, tips for writing an abstract, frequently asked questions about abstracts.

Hover over the different parts of the abstract to see how it is constructed.

This paper examines the role of silent movies as a mode of shared experience in the UK during the early twentieth century. At this time, high immigration rates resulted in a significant percentage of non-English-speaking citizens. These immigrants faced numerous economic and social obstacles, including exclusion from public entertainment and modes of discourse (newspapers, theater, radio).

Incorporating evidence from reviews, personal correspondence, and diaries, this study demonstrates that silent films were an affordable and inclusive source of entertainment. It argues for the accessible economic and representational nature of early cinema. These concerns are particularly evident in the low price of admission and in the democratic nature of the actors’ exaggerated gestures, which allowed the plots and action to be easily grasped by a diverse audience despite language barriers.

Keywords: silent movies, immigration, public discourse, entertainment, early cinema, language barriers.

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You will almost always have to include an abstract when:

• Completing a thesis or dissertation
• Submitting a research paper to an academic journal
• Writing a book proposal
• Applying for research grants

It’s easiest to write your abstract last, because it’s a summary of the work you’ve already done. Your abstract should:

• Be a self-contained text, not an excerpt from your paper
• Be fully understandable on its own
• Reflect the structure of your larger work

Start by clearly defining the purpose of your research. What practical or theoretical problem does the research respond to, or what research question did you aim to answer?

You can include some brief context on the social or academic relevance of your topic, but don’t go into detailed background information. If your abstract uses specialised terms that would be unfamiliar to the average academic reader or that have various different meanings, give a concise definition.

After identifying the problem, state the objective of your research. Use verbs like “investigate,” “test,” “analyse,” or “evaluate” to describe exactly what you set out to do.

This part of the abstract can be written in the present or past simple tense  but should never refer to the future, as the research is already complete.

• This study will investigate the relationship between coffee consumption and productivity.
• This study investigates the relationship between coffee consumption and productivity.

Next, indicate the research methods that you used to answer your question. This part should be a straightforward description of what you did in one or two sentences. It is usually written in the past simple tense, as it refers to completed actions.

• Structured interviews will be conducted with 25 participants.
• Structured interviews were conducted with 25 participants.

Don’t evaluate validity or obstacles here — the goal is not to give an account of the methodology’s strengths and weaknesses, but to give the reader a quick insight into the overall approach and procedures you used.

Next, summarise the main research results . This part of the abstract can be in the present or past simple tense.

• Our analysis has shown a strong correlation between coffee consumption and productivity.
• Our analysis shows a strong correlation between coffee consumption and productivity.
• Our analysis showed a strong correlation between coffee consumption and productivity.

Depending on how long and complex your research is, you may not be able to include all results here. Try to highlight only the most important findings that will allow the reader to understand your conclusions.

Finally, you should discuss the main conclusions of your research : what is your answer to the problem or question? The reader should finish with a clear understanding of the central point that your research has proved or argued. Conclusions are usually written in the present simple tense.

• We concluded that coffee consumption increases productivity.
• We conclude that coffee consumption increases productivity.

If there are important limitations to your research (for example, related to your sample size or methods), you should mention them briefly in the abstract. This allows the reader to accurately assess the credibility and generalisability of your research.

If your aim was to solve a practical problem, your discussion might include recommendations for implementation. If relevant, you can briefly make suggestions for further research.

If your paper will be published, you might have to add a list of keywords at the end of the abstract. These keywords should reference the most important elements of the research to help potential readers find your paper during their own literature searches.

Be aware that some publication manuals, such as APA Style , have specific formatting requirements for these keywords.

It can be a real challenge to condense your whole work into just a couple of hundred words, but the abstract will be the first (and sometimes only) part that people read, so it’s important to get it right. These strategies can help you get started.

Read other abstracts

The best way to learn the conventions of writing an abstract in your discipline is to read other people’s. You probably already read lots of journal article abstracts while conducting your literature review —try using them as a framework for structure and style.

You can also find lots of dissertation abstract examples in thesis and dissertation databases .

Reverse outline

Not all abstracts will contain precisely the same elements. For longer works, you can write your abstract through a process of reverse outlining.

For each chapter or section, list keywords and draft one to two sentences that summarise the central point or argument. This will give you a framework of your abstract’s structure. Next, revise the sentences to make connections and show how the argument develops.

Write clearly and concisely

A good abstract is short but impactful, so make sure every word counts. Each sentence should clearly communicate one main point.

To keep your abstract or summary short and clear:

• Avoid passive sentences: Passive constructions are often unnecessarily long. You can easily make them shorter and clearer by using the active voice.
• Avoid long sentences: Substitute longer expressions for concise expressions or single words (e.g., “In order to” for “To”).
• Avoid obscure jargon: The abstract should be understandable to readers who are not familiar with your topic.
• Avoid repetition and filler words: Replace nouns with pronouns when possible and eliminate unnecessary words.
• Avoid detailed descriptions: An abstract is not expected to provide detailed definitions, background information, or discussions of other scholars’ work. Instead, include this information in the body of your thesis or paper.

If you’re struggling to edit down to the required length, you can get help from expert editors with Scribbr’s professional proofreading services .

Check your formatting

If you are writing a thesis or dissertation or submitting to a journal, there are often specific formatting requirements for the abstract—make sure to check the guidelines and format your work correctly. For APA research papers you can follow the APA abstract format .

Checklist: Abstract

The word count is within the required length, or a maximum of one page.

The abstract appears after the title page and acknowledgements and before the table of contents .

I have clearly stated my research problem and objectives.

I have briefly described my methodology .

I have summarized the most important results .

I have stated my main conclusions .

I have mentioned any important limitations and recommendations.

The abstract can be understood by someone without prior knowledge of the topic.

You've written a great abstract! Use the other checklists to continue improving your thesis or dissertation.

An abstract is a concise summary of an academic text (such as a journal article or dissertation ). It serves two main purposes:

• To help potential readers determine the relevance of your paper for their own research.
• To communicate your key findings to those who don’t have time to read the whole paper.

Abstracts are often indexed along with keywords on academic databases, so they make your work more easily findable. Since the abstract is the first thing any reader sees, it’s important that it clearly and accurately summarises the contents of your paper.

An abstract for a thesis or dissertation is usually around 150–300 words. There’s often a strict word limit, so make sure to check your university’s requirements.

The abstract is the very last thing you write. You should only write it after your research is complete, so that you can accurately summarize the entirety of your thesis or paper.

Avoid citing sources in your abstract . There are two reasons for this:

• The abstract should focus on your original research, not on the work of others.
• The abstract should be self-contained and fully understandable without reference to other sources.

There are some circumstances where you might need to mention other sources in an abstract: for example, if your research responds directly to another study or focuses on the work of a single theorist. In general, though, don’t include citations unless absolutely necessary.

The abstract appears on its own page, after the title page and acknowledgements but before the table of contents .

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What this handout is about

This handout provides definitions and examples of the two main types of abstracts: descriptive and informative. It also provides guidelines for constructing an abstract and general tips for you to keep in mind when drafting. Finally, it includes a few examples of abstracts broken down into their component parts.

What is an abstract?

An abstract is a self-contained, short, and powerful statement that describes a larger work. Components vary according to discipline. An abstract of a social science or scientific work may contain the scope, purpose, results, and contents of the work. An abstract of a humanities work may contain the thesis, background, and conclusion of the larger work. An abstract is not a review, nor does it evaluate the work being abstracted. While it contains key words found in the larger work, the abstract is an original document rather than an excerpted passage.

Why write an abstract?

You may write an abstract for various reasons. The two most important are selection and indexing. Abstracts allow readers who may be interested in a longer work to quickly decide whether it is worth their time to read it. Also, many online databases use abstracts to index larger works. Therefore, abstracts should contain keywords and phrases that allow for easy searching.

Say you are beginning a research project on how Brazilian newspapers helped Brazil’s ultra-liberal president Luiz Ignácio da Silva wrest power from the traditional, conservative power base. A good first place to start your research is to search Dissertation Abstracts International for all dissertations that deal with the interaction between newspapers and politics. “Newspapers and politics” returned 569 hits. A more selective search of “newspapers and Brazil” returned 22 hits. That is still a fair number of dissertations. Titles can sometimes help winnow the field, but many titles are not very descriptive. For example, one dissertation is titled “Rhetoric and Riot in Rio de Janeiro.” It is unclear from the title what this dissertation has to do with newspapers in Brazil. One option would be to download or order the entire dissertation on the chance that it might speak specifically to the topic. A better option is to read the abstract. In this case, the abstract reveals the main focus of the dissertation:

This dissertation examines the role of newspaper editors in the political turmoil and strife that characterized late First Empire Rio de Janeiro (1827-1831). Newspaper editors and their journals helped change the political culture of late First Empire Rio de Janeiro by involving the people in the discussion of state. This change in political culture is apparent in Emperor Pedro I’s gradual loss of control over the mechanisms of power. As the newspapers became more numerous and powerful, the Emperor lost his legitimacy in the eyes of the people. To explore the role of the newspapers in the political events of the late First Empire, this dissertation analyzes all available newspapers published in Rio de Janeiro from 1827 to 1831. Newspapers and their editors were leading forces in the effort to remove power from the hands of the ruling elite and place it under the control of the people. In the process, newspapers helped change how politics operated in the constitutional monarchy of Brazil.

From this abstract you now know that although the dissertation has nothing to do with modern Brazilian politics, it does cover the role of newspapers in changing traditional mechanisms of power. After reading the abstract, you can make an informed judgment about whether the dissertation would be worthwhile to read.

Besides selection, the other main purpose of the abstract is for indexing. Most article databases in the online catalog of the library enable you to search abstracts. This allows for quick retrieval by users and limits the extraneous items recalled by a “full-text” search. However, for an abstract to be useful in an online retrieval system, it must incorporate the key terms that a potential researcher would use to search. For example, if you search Dissertation Abstracts International using the keywords “France” “revolution” and “politics,” the search engine would search through all the abstracts in the database that included those three words. Without an abstract, the search engine would be forced to search titles, which, as we have seen, may not be fruitful, or else search the full text. It’s likely that a lot more than 60 dissertations have been written with those three words somewhere in the body of the entire work. By incorporating keywords into the abstract, the author emphasizes the central topics of the work and gives prospective readers enough information to make an informed judgment about the applicability of the work.

When do people write abstracts?

• when submitting articles to journals, especially online journals
• when applying for research grants
• when writing a book proposal
• when completing the Ph.D. dissertation or M.A. thesis
• when writing a proposal for a conference paper
• when writing a proposal for a book chapter

Most often, the author of the entire work (or prospective work) writes the abstract. However, there are professional abstracting services that hire writers to draft abstracts of other people’s work. In a work with multiple authors, the first author usually writes the abstract. Undergraduates are sometimes asked to draft abstracts of books/articles for classmates who have not read the larger work.

Types of abstracts

There are two types of abstracts: descriptive and informative. They have different aims, so as a consequence they have different components and styles. There is also a third type called critical, but it is rarely used. If you want to find out more about writing a critique or a review of a work, see the UNC Writing Center handout on writing a literature review . If you are unsure which type of abstract you should write, ask your instructor (if the abstract is for a class) or read other abstracts in your field or in the journal where you are submitting your article.

Descriptive abstracts

A descriptive abstract indicates the type of information found in the work. It makes no judgments about the work, nor does it provide results or conclusions of the research. It does incorporate key words found in the text and may include the purpose, methods, and scope of the research. Essentially, the descriptive abstract describes the work being abstracted. Some people consider it an outline of the work, rather than a summary. Descriptive abstracts are usually very short—100 words or less.

Informative abstracts

The majority of abstracts are informative. While they still do not critique or evaluate a work, they do more than describe it. A good informative abstract acts as a surrogate for the work itself. That is, the writer presents and explains all the main arguments and the important results and evidence in the complete article/paper/book. An informative abstract includes the information that can be found in a descriptive abstract (purpose, methods, scope) but also includes the results and conclusions of the research and the recommendations of the author. The length varies according to discipline, but an informative abstract is rarely more than 10% of the length of the entire work. In the case of a longer work, it may be much less.

Here are examples of a descriptive and an informative abstract of this handout on abstracts . Descriptive abstract:

The two most common abstract types—descriptive and informative—are described and examples of each are provided.

Informative abstract:

Abstracts present the essential elements of a longer work in a short and powerful statement. The purpose of an abstract is to provide prospective readers the opportunity to judge the relevance of the longer work to their projects. Abstracts also include the key terms found in the longer work and the purpose and methods of the research. Authors abstract various longer works, including book proposals, dissertations, and online journal articles. There are two main types of abstracts: descriptive and informative. A descriptive abstract briefly describes the longer work, while an informative abstract presents all the main arguments and important results. This handout provides examples of various types of abstracts and instructions on how to construct one.

Which type should I use?

Your best bet in this case is to ask your instructor or refer to the instructions provided by the publisher. You can also make a guess based on the length allowed; i.e., 100-120 words = descriptive; 250+ words = informative.

How do I write an abstract?

The format of your abstract will depend on the work being abstracted. An abstract of a scientific research paper will contain elements not found in an abstract of a literature article, and vice versa. However, all abstracts share several mandatory components, and there are also some optional parts that you can decide to include or not. When preparing to draft your abstract, keep the following key process elements in mind:

• Reason for writing: What is the importance of the research? Why would a reader be interested in the larger work?
• Problem: What problem does this work attempt to solve? What is the scope of the project? What is the main argument/thesis/claim?
• Methodology: An abstract of a scientific work may include specific models or approaches used in the larger study. Other abstracts may describe the types of evidence used in the research.
• Results: Again, an abstract of a scientific work may include specific data that indicates the results of the project. Other abstracts may discuss the findings in a more general way.
• Implications: What changes should be implemented as a result of the findings of the work? How does this work add to the body of knowledge on the topic?

(This list of elements is adapted with permission from Philip Koopman, “How to Write an Abstract.” )

All abstracts include:

• A full citation of the source, preceding the abstract.
• The most important information first.
• The same type and style of language found in the original, including technical language.
• Key words and phrases that quickly identify the content and focus of the work.
• Clear, concise, and powerful language.

Abstracts may include:

• The thesis of the work, usually in the first sentence.
• Background information that places the work in the larger body of literature.
• The same chronological structure as the original work.

How not to write an abstract:

• Do not refer extensively to other works.
• Do not add information not contained in the original work.
• Do not define terms.

If you are abstracting your own writing

When abstracting your own work, it may be difficult to condense a piece of writing that you have agonized over for weeks (or months, or even years) into a 250-word statement. There are some tricks that you could use to make it easier, however.

Reverse outlining:

This technique is commonly used when you are having trouble organizing your own writing. The process involves writing down the main idea of each paragraph on a separate piece of paper– see our short video . For the purposes of writing an abstract, try grouping the main ideas of each section of the paper into a single sentence. Practice grouping ideas using webbing or color coding .

For a scientific paper, you may have sections titled Purpose, Methods, Results, and Discussion. Each one of these sections will be longer than one paragraph, but each is grouped around a central idea. Use reverse outlining to discover the central idea in each section and then distill these ideas into one statement.

Cut and paste:

To create a first draft of an abstract of your own work, you can read through the entire paper and cut and paste sentences that capture key passages. This technique is useful for social science research with findings that cannot be encapsulated by neat numbers or concrete results. A well-written humanities draft will have a clear and direct thesis statement and informative topic sentences for paragraphs or sections. Isolate these sentences in a separate document and work on revising them into a unified paragraph.

If you are abstracting someone else’s writing

When abstracting something you have not written, you cannot summarize key ideas just by cutting and pasting. Instead, you must determine what a prospective reader would want to know about the work. There are a few techniques that will help you in this process:

Identify key terms:

Search through the entire document for key terms that identify the purpose, scope, and methods of the work. Pay close attention to the Introduction (or Purpose) and the Conclusion (or Discussion). These sections should contain all the main ideas and key terms in the paper. When writing the abstract, be sure to incorporate the key terms.

Highlight key phrases and sentences:

Instead of cutting and pasting the actual words, try highlighting sentences or phrases that appear to be central to the work. Then, in a separate document, rewrite the sentences and phrases in your own words.

Don’t look back:

After reading the entire work, put it aside and write a paragraph about the work without referring to it. In the first draft, you may not remember all the key terms or the results, but you will remember what the main point of the work was. Remember not to include any information you did not get from the work being abstracted.

Revise, revise, revise

No matter what type of abstract you are writing, or whether you are abstracting your own work or someone else’s, the most important step in writing an abstract is to revise early and often. When revising, delete all extraneous words and incorporate meaningful and powerful words. The idea is to be as clear and complete as possible in the shortest possible amount of space. The Word Count feature of Microsoft Word can help you keep track of how long your abstract is and help you hit your target length.

Example 1: Humanities abstract

Kenneth Tait Andrews, “‘Freedom is a constant struggle’: The dynamics and consequences of the Mississippi Civil Rights Movement, 1960-1984” Ph.D. State University of New York at Stony Brook, 1997 DAI-A 59/02, p. 620, Aug 1998

This dissertation examines the impacts of social movements through a multi-layered study of the Mississippi Civil Rights Movement from its peak in the early 1960s through the early 1980s. By examining this historically important case, I clarify the process by which movements transform social structures and the constraints movements face when they try to do so. The time period studied includes the expansion of voting rights and gains in black political power, the desegregation of public schools and the emergence of white-flight academies, and the rise and fall of federal anti-poverty programs. I use two major research strategies: (1) a quantitative analysis of county-level data and (2) three case studies. Data have been collected from archives, interviews, newspapers, and published reports. This dissertation challenges the argument that movements are inconsequential. Some view federal agencies, courts, political parties, or economic elites as the agents driving institutional change, but typically these groups acted in response to the leverage brought to bear by the civil rights movement. The Mississippi movement attempted to forge independent structures for sustaining challenges to local inequities and injustices. By propelling change in an array of local institutions, movement infrastructures had an enduring legacy in Mississippi.

Now let’s break down this abstract into its component parts to see how the author has distilled his entire dissertation into a ~200 word abstract.

What the dissertation does This dissertation examines the impacts of social movements through a multi-layered study of the Mississippi Civil Rights Movement from its peak in the early 1960s through the early 1980s. By examining this historically important case, I clarify the process by which movements transform social structures and the constraints movements face when they try to do so.

How the dissertation does it The time period studied in this dissertation includes the expansion of voting rights and gains in black political power, the desegregation of public schools and the emergence of white-flight academies, and the rise and fall of federal anti-poverty programs. I use two major research strategies: (1) a quantitative analysis of county-level data and (2) three case studies.

What materials are used Data have been collected from archives, interviews, newspapers, and published reports.

Conclusion This dissertation challenges the argument that movements are inconsequential. Some view federal agencies, courts, political parties, or economic elites as the agents driving institutional change, but typically these groups acted in response to movement demands and the leverage brought to bear by the civil rights movement. The Mississippi movement attempted to forge independent structures for sustaining challenges to local inequities and injustices. By propelling change in an array of local institutions, movement infrastructures had an enduring legacy in Mississippi.

Keywords social movements Civil Rights Movement Mississippi voting rights desegregation

Example 2: Science Abstract

Luis Lehner, “Gravitational radiation from black hole spacetimes” Ph.D. University of Pittsburgh, 1998 DAI-B 59/06, p. 2797, Dec 1998

The problem of detecting gravitational radiation is receiving considerable attention with the construction of new detectors in the United States, Europe, and Japan. The theoretical modeling of the wave forms that would be produced in particular systems will expedite the search for and analysis of detected signals. The characteristic formulation of GR is implemented to obtain an algorithm capable of evolving black holes in 3D asymptotically flat spacetimes. Using compactification techniques, future null infinity is included in the evolved region, which enables the unambiguous calculation of the radiation produced by some compact source. A module to calculate the waveforms is constructed and included in the evolution algorithm. This code is shown to be second-order convergent and to handle highly non-linear spacetimes. In particular, we have shown that the code can handle spacetimes whose radiation is equivalent to a galaxy converting its whole mass into gravitational radiation in one second. We further use the characteristic formulation to treat the region close to the singularity in black hole spacetimes. The code carefully excises a region surrounding the singularity and accurately evolves generic black hole spacetimes with apparently unlimited stability.

This science abstract covers much of the same ground as the humanities one, but it asks slightly different questions.

Why do this study The problem of detecting gravitational radiation is receiving considerable attention with the construction of new detectors in the United States, Europe, and Japan. The theoretical modeling of the wave forms that would be produced in particular systems will expedite the search and analysis of the detected signals.

What the study does The characteristic formulation of GR is implemented to obtain an algorithm capable of evolving black holes in 3D asymptotically flat spacetimes. Using compactification techniques, future null infinity is included in the evolved region, which enables the unambiguous calculation of the radiation produced by some compact source. A module to calculate the waveforms is constructed and included in the evolution algorithm.

Results This code is shown to be second-order convergent and to handle highly non-linear spacetimes. In particular, we have shown that the code can handle spacetimes whose radiation is equivalent to a galaxy converting its whole mass into gravitational radiation in one second. We further use the characteristic formulation to treat the region close to the singularity in black hole spacetimes. The code carefully excises a region surrounding the singularity and accurately evolves generic black hole spacetimes with apparently unlimited stability.

Keywords gravitational radiation (GR) spacetimes black holes

Works consulted

We consulted these works while writing this handout. This is not a comprehensive list of resources on the handout’s topic, and we encourage you to do your own research to find additional publications. Please do not use this list as a model for the format of your own reference list, as it may not match the citation style you are using. For guidance on formatting citations, please see the UNC Libraries citation tutorial . We revise these tips periodically and welcome feedback.

Belcher, Wendy Laura. 2009. Writing Your Journal Article in Twelve Weeks: A Guide to Academic Publishing Success. Thousand Oaks, CA: Sage Press.

Koopman, Philip. 1997. “How to Write an Abstract.” Carnegie Mellon University. October 1997. http://users.ece.cmu.edu/~koopman/essays/abstract.html .

Lancaster, F.W. 2003. Indexing And Abstracting in Theory and Practice , 3rd ed. London: Facet Publishing.

You may reproduce it for non-commercial use if you use the entire handout and attribute the source: The Writing Center, University of North Carolina at Chapel Hill

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Organizing Your Social Sciences Research Paper

• 3. The Abstract
• Purpose of Guide
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An abstract summarizes, usually in one paragraph of 300 words or less, the major aspects of the entire paper in a prescribed sequence that includes: 1) the overall purpose of the study and the research problem(s) you investigated; 2) the basic design of the study; 3) major findings or trends found as a result of your analysis; and, 4) a brief summary of your interpretations and conclusions.

Writing an Abstract. The Writing Center. Clarion University, 2009; Writing an Abstract for Your Research Paper. The Writing Center, University of Wisconsin, Madison; Koltay, Tibor. Abstracts and Abstracting: A Genre and Set of Skills for the Twenty-first Century . Oxford, UK: Chandos Publishing, 2010;

Importance of a Good Abstract

Sometimes your professor will ask you to include an abstract, or general summary of your work, with your research paper. The abstract allows you to elaborate upon each major aspect of the paper and helps readers decide whether they want to read the rest of the paper. Therefore, enough key information [e.g., summary results, observations, trends, etc.] must be included to make the abstract useful to someone who may want to examine your work.

How do you know when you have enough information in your abstract? A simple rule-of-thumb is to imagine that you are another researcher doing a similar study. Then ask yourself: if your abstract was the only part of the paper you could access, would you be happy with the amount of information presented there? Does it tell the whole story about your study? If the answer is "no" then the abstract likely needs to be revised.

Farkas, David K. “A Scheme for Understanding and Writing Summaries.” Technical Communication 67 (August 2020): 45-60;  How to Write a Research Abstract. Office of Undergraduate Research. University of Kentucky; Staiger, David L. “What Today’s Students Need to Know about Writing Abstracts.” International Journal of Business Communication January 3 (1966): 29-33; Swales, John M. and Christine B. Feak. Abstracts and the Writing of Abstracts . Ann Arbor, MI: University of Michigan Press, 2009.

Structure and Writing Style

I.  Types of Abstracts

To begin, you need to determine which type of abstract you should include with your paper. There are four general types.

Critical Abstract A critical abstract provides, in addition to describing main findings and information, a judgment or comment about the study’s validity, reliability, or completeness. The researcher evaluates the paper and often compares it with other works on the same subject. Critical abstracts are generally 400-500 words in length due to the additional interpretive commentary. These types of abstracts are used infrequently.

Descriptive Abstract A descriptive abstract indicates the type of information found in the work. It makes no judgments about the work, nor does it provide results or conclusions of the research. It does incorporate key words found in the text and may include the purpose, methods, and scope of the research. Essentially, the descriptive abstract only describes the work being summarized. Some researchers consider it an outline of the work, rather than a summary. Descriptive abstracts are usually very short, 100 words or less. Informative Abstract The majority of abstracts are informative. While they still do not critique or evaluate a work, they do more than describe it. A good informative abstract acts as a surrogate for the work itself. That is, the researcher presents and explains all the main arguments and the important results and evidence in the paper. An informative abstract includes the information that can be found in a descriptive abstract [purpose, methods, scope] but it also includes the results and conclusions of the research and the recommendations of the author. The length varies according to discipline, but an informative abstract is usually no more than 300 words in length.

Highlight Abstract A highlight abstract is specifically written to attract the reader’s attention to the study. No pretense is made of there being either a balanced or complete picture of the paper and, in fact, incomplete and leading remarks may be used to spark the reader’s interest. In that a highlight abstract cannot stand independent of its associated article, it is not a true abstract and, therefore, rarely used in academic writing.

II.  Writing Style

Use the active voice when possible , but note that much of your abstract may require passive sentence constructions. Regardless, write your abstract using concise, but complete, sentences. Get to the point quickly and always use the past tense because you are reporting on a study that has been completed.

Abstracts should be formatted as a single paragraph in a block format and with no paragraph indentations. In most cases, the abstract page immediately follows the title page. Do not number the page. Rules set forth in writing manual vary but, in general, you should center the word "Abstract" at the top of the page with double spacing between the heading and the abstract. The final sentences of an abstract concisely summarize your study’s conclusions, implications, or applications to practice and, if appropriate, can be followed by a statement about the need for additional research revealed from the findings.

Composing Your Abstract

Although it is the first section of your paper, the abstract should be written last since it will summarize the contents of your entire paper. A good strategy to begin composing your abstract is to take whole sentences or key phrases from each section of the paper and put them in a sequence that summarizes the contents. Then revise or add connecting phrases or words to make the narrative flow clearly and smoothly. Note that statistical findings should be reported parenthetically [i.e., written in parentheses].

Before handing in your final paper, check to make sure that the information in the abstract completely agrees with what you have written in the paper. Think of the abstract as a sequential set of complete sentences describing the most crucial information using the fewest necessary words. The abstract SHOULD NOT contain:

• A catchy introductory phrase, provocative quote, or other device to grab the reader's attention,
• Lengthy background or contextual information,
• Redundant phrases, unnecessary adverbs and adjectives, and repetitive information;
• Acronyms or abbreviations,
• References to other literature [say something like, "current research shows that..." or "studies have indicated..."],
• Using ellipticals [i.e., ending with "..."] or incomplete sentences,
• Jargon or terms that may be confusing to the reader,
• Citations to other works, and
• Any sort of image, illustration, figure, or table, or references to them.

Abstract. Writing Center. University of Kansas; Abstract. The Structure, Format, Content, and Style of a Journal-Style Scientific Paper. Department of Biology. Bates College; Abstracts. The Writing Center. University of North Carolina; Borko, Harold and Seymour Chatman. "Criteria for Acceptable Abstracts: A Survey of Abstracters' Instructions." American Documentation 14 (April 1963): 149-160; Abstracts. The Writer’s Handbook. Writing Center. University of Wisconsin, Madison; Hartley, James and Lucy Betts. "Common Weaknesses in Traditional Abstracts in the Social Sciences." Journal of the American Society for Information Science and Technology 60 (October 2009): 2010-2018; Koltay, Tibor. Abstracts and Abstracting: A Genre and Set of Skills for the Twenty-first Century. Oxford, UK: Chandos Publishing, 2010; Procter, Margaret. The Abstract. University College Writing Centre. University of Toronto; Riordan, Laura. “Mastering the Art of Abstracts.” The Journal of the American Osteopathic Association 115 (January 2015 ): 41-47; Writing Report Abstracts. The Writing Lab and The OWL. Purdue University; Writing Abstracts. Writing Tutorial Services, Center for Innovative Teaching and Learning. Indiana University; Koltay, Tibor. Abstracts and Abstracting: A Genre and Set of Skills for the Twenty-First Century . Oxford, UK: 2010; Writing an Abstract for Your Research Paper. The Writing Center, University of Wisconsin, Madison.

Writing Tip

Never Cite Just the Abstract!

Citing to just a journal article's abstract does not confirm for the reader that you have conducted a thorough or reliable review of the literature. If the full-text is not available, go to the USC Libraries main page and enter the title of the article [NOT the title of the journal]. If the Libraries have a subscription to the journal, the article should appear with a link to the full-text or to the journal publisher page where you can get the article. If the article does not appear, try searching Google Scholar using the link on the USC Libraries main page. If you still can't find the article after doing this, contact a librarian or you can request it from our free i nterlibrary loan and document delivery service .

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How to Write an Abstract for a Research Paper | Examples

What is a research paper abstract?

Research paper abstracts summarize your study quickly and succinctly to journal editors and researchers and prompt them to read further. But with the ubiquity of online publication databases, writing a compelling abstract is even more important today than it was in the days of bound paper manuscripts.

Abstracts exist to “sell”  your work, and they could thus be compared to the “executive summary” of a business resume: an official briefing on what is most important about your research. Or the “gist” of your research. With the majority of academic transactions being conducted online, this means that you have even less time to impress readers–and increased competition in terms of other abstracts out there to read.

The APCI (Academic Publishing and Conferences International) notes that there are  12 questions or “points” considered in the selection process  for journals and conferences and stresses the importance of having an abstract that ticks all of these boxes. Because it is often the ONLY chance you have to convince readers to keep reading, it is important that you spend time and energy crafting an abstract that faithfully represents the central parts of your study and captivates your audience.

With that in mind, follow these suggestions when structuring and writing your abstract, and learn how exactly to put these ideas into a solid abstract that will captivate your target readers.

Before Writing Your Abstract

How long should an abstract be.

All abstracts are written with the same essential objective: to give a summary of your study. But there are two basic styles of abstract: descriptive and informative . Here is a brief delineation of the two:

Of the two types of abstracts, informative abstracts are much more common, and they are widely used for submission to journals and conferences. Informative abstracts apply to lengthier and more technical research and are common in the sciences, engineering, and psychology, while descriptive abstracts are more likely used in humanities and social science papers. The best method of determining which abstract type you need to use is to follow the instructions for journal submissions and to read as many other published articles in those journals as possible.

Research Abstract Guidelines and Requirements

As any article about research writing will tell you, authors must always closely follow the specific guidelines and requirements indicated in the Guide for Authors section of their target journal’s website. The same kind of adherence to conventions should be applied to journal publications, for consideration at a conference, and even when completing a class assignment.

Each publisher has particular demands when it comes to formatting and structure. Here are some common questions addressed in the journal guidelines:

• Is there a maximum or minimum word/character length?
• What are the style and formatting requirements?
• What is the appropriate abstract type?
• Are there any specific content or organization rules that apply?

There are of course other rules to consider when composing a research paper abstract. But if you follow the stated rules the first time you submit your manuscript, you can avoid your work being thrown in the “circular file” right off the bat.

Identify Your Target Readership

The main purpose of your abstract is to lead researchers to the full text of your research paper. In scientific journals, abstracts let readers decide whether the research discussed is relevant to their own interests or study. Abstracts also help readers understand your main argument quickly. Consider these questions as you write your abstract:

• Are other academics in your field the main target of your study?
• Will your study perhaps be useful to members of the general public?
• Do your study results include the wider implications presented in the abstract?

Outlining and Writing Your Abstract

What to include in an abstract.

Just as your  research paper title  should cover as much ground as possible in a few short words, your abstract must cover  all  parts of your study in order to fully explain your paper and research. Because it must accomplish this task in the space of only a few hundred words, it is important not to include ambiguous references or phrases that will confuse the reader or mislead them about the content and objectives of your research. Follow these  dos  and  don’ts  when it comes to what kind of writing to include:

• Avoid acronyms or abbreviations since these will need to be explained in order to make sense to the reader, which takes up valuable abstract space. Instead, explain these terms in the Introduction section of the main text.
• Only use references to people or other works if they are well-known. Otherwise, avoid referencing anything outside of your study in the abstract.
• Never include tables, figures, sources, or long quotations in your abstract; you will have plenty of time to present and refer to these in the body of your paper.

Use keywords in your abstract to focus your topic

A vital search tool is the research paper keywords section, which lists the most relevant terms directly underneath the abstract. Think of these keywords as the “tubes” that readers will seek and enter—via queries on databases and search engines—to ultimately land at their destination, which is your paper. Your abstract keywords should thus be words that are commonly used in searches but should also be highly relevant to your work and found in the text of your abstract. Include 5 to 10 important words or short phrases central to your research in both the abstract and the keywords section.

For example, if you are writing a paper on the prevalence of obesity among lower classes that crosses international boundaries, you should include terms like “obesity,” “prevalence,” “international,” “lower classes,” and “cross-cultural.” These are terms that should net a wide array of people interested in your topic of study. Look at our nine rules for choosing keywords for your research paper if you need more input on this.

Research Paper Abstract Structure

As mentioned above, the abstract (especially the informative abstract) acts as a surrogate or synopsis of your research paper, doing almost as much work as the thousands of words that follow it in the body of the main text. In the hard sciences and most social sciences, the abstract includes the following sections and organizational schema.

Each section is quite compact—only a single sentence or two, although there is room for expansion if one element or statement is particularly interesting or compelling. As the abstract is almost always one long paragraph, the individual sections should naturally merge into one another to create a holistic effect. Use the following as a checklist to ensure that you have included all of the necessary content in your abstract.

1) Identify your purpose and motivation

So your research is about rabies in Brazilian squirrels. Why is this important? You should start your abstract by explaining why people should care about this study—why is it significant to your field and perhaps to the wider world? And what is the exact purpose of your study; what are you trying to achieve? Start by answering the following questions:

• What made you decide to do this study or project?
• Why is this study important to your field or to the lay reader?
• Why should someone read your entire article?

In summary, the first section of your abstract should include the importance of the research and its impact on related research fields or on the wider scientific domain.

2) Explain the research problem you are addressing

Stating the research problem that your study addresses is the corollary to why your specific study is important and necessary. For instance, even if the issue of “rabies in Brazilian squirrels” is important, what is the problem—the “missing piece of the puzzle”—that your study helps resolve?

You can combine the problem with the motivation section, but from a perspective of organization and clarity, it is best to separate the two. Here are some precise questions to address:

• What is your research trying to better understand or what problem is it trying to solve?
• What is the scope of your study—does it try to explain something general or specific?
• What is your central claim or argument?

3) Discuss your research approach

Your specific study approach is detailed in the Methods and Materials section .  You have already established the importance of the research, your motivation for studying this issue, and the specific problem your paper addresses. Now you need to discuss  how  you solved or made progress on this problem—how you conducted your research. If your study includes your own work or that of your team, describe that here. If in your paper you reviewed the work of others, explain this here. Did you use analytic models? A simulation? A double-blind study? A case study? You are basically showing the reader the internal engine of your research machine and how it functioned in the study. Be sure to:

• Detail your research—include methods/type of the study, your variables, and the extent of the work
• Briefly present evidence to support your claim
• Highlight your most important sources

4) Briefly summarize your results

Here you will give an overview of the outcome of your study. Avoid using too many vague qualitative terms (e.g, “very,” “small,” or “tremendous”) and try to use at least some quantitative terms (i.e., percentages, figures, numbers). Save your qualitative language for the conclusion statement. Answer questions like these:

• What did your study yield in concrete terms (e.g., trends, figures, correlation between phenomena)?
• How did your results compare to your hypothesis? Was the study successful?
• Where there any highly unexpected outcomes or were they all largely predicted?

5) State your conclusion

In the last section of your abstract, you will give a statement about the implications and  limitations of the study . Be sure to connect this statement closely to your results and not the area of study in general. Are the results of this study going to shake up the scientific world? Will they impact how people see “Brazilian squirrels”? Or are the implications minor? Try not to boast about your study or present its impact as  too  far-reaching, as researchers and journals will tend to be skeptical of bold claims in scientific papers. Answer one of these questions:

• What are the exact effects of these results on my field? On the wider world?
• What other kind of study would yield further solutions to problems?
• What other information is needed to expand knowledge in this area?

After Completing the First Draft of Your Abstract

Revise your abstract.

The abstract, like any piece of academic writing, should be revised before being considered complete. Check it for  grammatical and spelling errors  and make sure it is formatted properly.

Get feedback from a peer

Getting a fresh set of eyes to review your abstract is a great way to find out whether you’ve summarized your research well. Find a reader who understands research papers but is not an expert in this field or is not affiliated with your study. Ask your reader to summarize what your study is about (including all key points of each section). This should tell you if you have communicated your key points clearly.

In addition to research peers, consider consulting with a professor or even a specialist or generalist writing center consultant about your abstract. Use any resource that helps you see your work from another perspective.

Consider getting professional editing and proofreading

While peer feedback is quite important to ensure the effectiveness of your abstract content, it may be a good idea to find an academic editor  to fix mistakes in grammar, spelling, mechanics, style, or formatting. The presence of basic errors in the abstract may not affect your content, but it might dissuade someone from reading your entire study. Wordvice provides English editing services that both correct objective errors and enhance the readability and impact of your work.

Additional Abstract Rules and Guidelines

Write your abstract after completing your paper.

Although the abstract goes at the beginning of your manuscript, it does not merely introduce your research topic (that is the job of the title), but rather summarizes your entire paper. Writing the abstract last will ensure that it is complete and consistent with the findings and statements in your paper.

Keep your content in the correct order

Both questions and answers should be organized in a standard and familiar way to make the content easier for readers to absorb. Ideally, it should mimic the overall format of your essay and the classic “introduction,” “body,” and “conclusion” form, even if the parts are not neatly divided as such.

Write the abstract from scratch

Because the abstract is a self-contained piece of writing viewed separately from the body of the paper, you should write it separately as well. Never copy and paste direct quotes from the paper and avoid paraphrasing sentences in the paper. Using new vocabulary and phrases will keep your abstract interesting and free of redundancies while conserving space.

Don’t include too many details in the abstract

Again, the density of your abstract makes it incompatible with including specific points other than possibly names or locations. You can make references to terms, but do not explain or define them in the abstract. Try to strike a balance between being specific to your study and presenting a relatively broad overview of your work.

Wordvice Resources

If you think your abstract is fine now but you need input on abstract writing or require English editing services (including paper editing ), then head over to the Wordvice academic resources page, where you will find many more articles, for example on writing the Results , Methods , and Discussion sections of your manuscript, on choosing a title for your paper , or on how to finalize your journal submission with a strong cover letter .    

The Dissertation Abstract: 101

How to write a clear & concise abstract (with examples).

By:   Madeline Fink (MSc) Reviewed By: Derek Jansen (MBA)   | June 2020

So, you’ve (finally) finished your thesis or dissertation or thesis. Now it’s time to write up your abstract (sometimes also called the executive summary). If you’re here, chances are you’re not quite sure what you need to cover in this section, or how to go about writing it. Fear not – we’ll explain it all in plain language , step by step , with clear examples .

Overview: The Dissertation/Thesis Abstract

• What exactly is a dissertation (or thesis) abstract
• What’s the purpose and function of the abstract
• Why is the abstract so important
• How to write a high-quality dissertation abstract
• Example/sample of a quality abstract
• Quick tips to write a high-quality dissertation abstract

What is an abstract?

Simply put, the abstract in a dissertation or thesis is a short (but well structured) summary that outlines the most important points of your research (i.e. the key takeaways). The abstract is usually 1 paragraph or about 300-500 words long (about one page), but but this can vary between universities.

A quick note regarding terminology – strictly speaking, an abstract and an executive summary are two different things when it comes to academic publications. Typically, an abstract only states what the research will be about, but doesn’t explore the findings – whereas an executive summary covers both . However, in the context of a dissertation or thesis, the abstract usually covers both, providing a summary of the full project.

In terms of content, a good dissertation abstract usually covers the following points:

• The purpose of the research (what’s it about and why’s that important)
• The methodology (how you carried out the research)
• The key research findings (what answers you found)
• The implications of these findings (what these answers mean)

We’ll explain each of these in more detail a little later in this post. Buckle up.

What’s the purpose of the abstract?

A dissertation abstract has two main functions:

The first purpose is to  inform potential readers  of the main idea of your research without them having to read your entire piece of work. Specifically, it needs to communicate what your research is about (what were you trying to find out) and what your findings were . When readers are deciding whether to read your dissertation or thesis, the abstract is the first part they’ll consider. 

The second purpose of the abstract is to  inform search engines and dissertation databases  as they index your dissertation or thesis. The keywords and phrases in your abstract (as well as your keyword list) will often be used by these search engines to categorize your work and make it accessible to users. 

Simply put, your abstract is your shopfront display window – it’s what passers-by (both human and digital) will look at before deciding to step inside. 

Why’s it so important?

The short answer – because most people don’t have time to read your full dissertation or thesis! Time is money, after all…

If you think back to when you undertook your literature review , you’ll quickly realise just how important abstracts are! Researchers reviewing the literature on any given topic face a mountain of reading, so they need to optimise their approach. A good dissertation abstract gives the reader a “TLDR” version of your work – it helps them decide whether to continue to read it in its entirety. So, your abstract, as your shopfront display window, needs to “sell” your research to time-poor readers.

You might be thinking, “but I don’t plan to publish my dissertation”. Even so, you still need to provide an impactful abstract for your markers. Your ability to concisely summarise your work is one of the things they’re assessing, so it’s vital to invest time and effort into crafting an enticing shop window.  

A good abstract also has an added purpose for grad students . As a freshly minted graduate, your dissertation or thesis is often your most significant professional accomplishment and highlights where your unique expertise lies. Potential employers who want to know about this expertise are likely to only read the abstract (as opposed to reading your entire document) – so it needs to be good!

Think about it this way – if your thesis or dissertation were a book, then the abstract would be the blurb on the back cover. For better or worse, readers will absolutely judge your book by its cover .

How to write your abstract

As we touched on earlier, your abstract should cover four important aspects of your research: the purpose , methodology , findings , and implications . Therefore, the structure of your dissertation or thesis abstract needs to reflect these four essentials, in the same order.  Let’s take a closer look at each of them, step by step:

Step 1: Describe the purpose and value of your research

Here you need to concisely explain the purpose and value of your research. In other words, you need to explain what your research set out to discover and why that’s important. When stating the purpose of research, you need to clearly discuss the following:

• What were your research aims and research questions ?
• Why were these aims and questions important?

It’s essential to make this section extremely clear, concise and convincing . As the opening section, this is where you’ll “hook” your reader (marker) in and get them interested in your project. If you don’t put in the effort here, you’ll likely lose their interest.

Step 2: Briefly outline your study’s methodology

In this part of your abstract, you need to very briefly explain how you went about answering your research questions . In other words, what research design and methodology you adopted in your research. Some important questions to address here include:

• Did you take a qualitative or quantitative approach ?
• Who/what did your sample consist of?
• How did you collect your data?
• How did you analyse your data?

Simply put, this section needs to address the “ how ” of your research. It doesn’t need to be lengthy (this is just a summary, after all), but it should clearly address the four questions above.

Need a helping hand?

Step 3: Present your key findings

Next, you need to briefly highlight the key findings . Your research likely produced a wealth of data and findings, so there may be a temptation to ramble here. However, this section is just about the key findings – in other words, the answers to the original questions that you set out to address.

Again, brevity and clarity are important here. You need to concisely present the most important findings for your reader.

Step 4: Describe the implications of your research

Have you ever found yourself reading through a large report, struggling to figure out what all the findings mean in terms of the bigger picture? Well, that’s the purpose of the implications section – to highlight the “so what?” of your research. 

In this part of your abstract, you should address the following questions:

• What is the impact of your research findings on the industry /field investigated? In other words, what’s the impact on the “real world”. 
• What is the impact of your findings on the existing body of knowledge ? For example, do they support the existing research?
• What might your findings mean for future research conducted on your topic?

If you include these four essential ingredients in your dissertation abstract, you’ll be on headed in a good direction.

Example: Dissertation/thesis abstract

Here is an example of an abstract from a master’s thesis, with the purpose , methods , findings , and implications colour coded.

The U.S. citizenship application process is a legal and symbolic journey shaped by many cultural processes. This research project aims to bring to light the experiences of immigrants and citizenship applicants living in Dallas, Texas, to promote a better understanding of Dallas’ increasingly diverse population. Additionally, the purpose of this project is to provide insights to a specific client, the office of Dallas Welcoming Communities and Immigrant Affairs, about Dallas’ lawful permanent residents who are eligible for citizenship and their reasons for pursuing citizenship status . The data for this project was collected through observation at various citizenship workshops and community events, as well as through semi-structured interviews with 14 U.S. citizenship applicants . Reasons for applying for U.S. citizenship discussed in this project include a desire for membership in U.S. society, access to better educational and economic opportunities, improved ease of travel and the desire to vote. Barriers to the citizenship process discussed in this project include the amount of time one must dedicate to the application, lack of clear knowledge about the process and the financial cost of the application. Other themes include the effects of capital on applicant’s experience with the citizenship process, symbolic meanings of citizenship, transnationalism and ideas of deserving and undeserving surrounding the issues of residency and U.S. citizenship. These findings indicate the need for educational resources and mentorship for Dallas-area residents applying for U.S. citizenship, as well as a need for local government programs that foster a sense of community among citizenship applicants and their neighbours.

Practical tips for writing your abstract

When crafting the abstract for your dissertation or thesis, the most powerful technique you can use is to try and put yourself in the shoes of a potential reader. Assume the reader is not an expert in the field, but is interested in the research area. In other words, write for the intelligent layman, not for the seasoned topic expert. 

Start by trying to answer the question “why should I read this dissertation?”

Remember the WWHS.

Make sure you include the  what , why ,  how , and  so what  of your research in your abstract:

• What you studied (who and where are included in this part)
• Why the topic was important
• How you designed your study (i.e. your research methodology)
• So what were the big findings and implications of your research

Keep it simple.

Use terminology appropriate to your field of study, but don’t overload your abstract with big words and jargon that cloud the meaning and make your writing difficult to digest. A good abstract should appeal to all levels of potential readers and should be a (relatively) easy read. Remember, you need to write for the intelligent layman.

Be specific.

When writing your abstract, clearly outline your most important findings and insights and don’t worry about “giving away” too much about your research – there’s no need to withhold information. This is the one way your abstract is not like a blurb on the back of a book – the reader should be able to clearly understand the key takeaways of your thesis or dissertation after reading the abstract. Of course, if they then want more detail, they need to step into the restaurant and try out the menu.

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A Practical Guide to Writing Quantitative and Qualitative Research Questions and Hypotheses in Scholarly Articles

Edward barroga.

1 Department of General Education, Graduate School of Nursing Science, St. Luke’s International University, Tokyo, Japan.

Glafera Janet Matanguihan

2 Department of Biological Sciences, Messiah University, Mechanicsburg, PA, USA.

The development of research questions and the subsequent hypotheses are prerequisites to defining the main research purpose and specific objectives of a study. Consequently, these objectives determine the study design and research outcome. The development of research questions is a process based on knowledge of current trends, cutting-edge studies, and technological advances in the research field. Excellent research questions are focused and require a comprehensive literature search and in-depth understanding of the problem being investigated. Initially, research questions may be written as descriptive questions which could be developed into inferential questions. These questions must be specific and concise to provide a clear foundation for developing hypotheses. Hypotheses are more formal predictions about the research outcomes. These specify the possible results that may or may not be expected regarding the relationship between groups. Thus, research questions and hypotheses clarify the main purpose and specific objectives of the study, which in turn dictate the design of the study, its direction, and outcome. Studies developed from good research questions and hypotheses will have trustworthy outcomes with wide-ranging social and health implications.

INTRODUCTION

Scientific research is usually initiated by posing evidenced-based research questions which are then explicitly restated as hypotheses. 1 , 2 The hypotheses provide directions to guide the study, solutions, explanations, and expected results. 3 , 4 Both research questions and hypotheses are essentially formulated based on conventional theories and real-world processes, which allow the inception of novel studies and the ethical testing of ideas. 5 , 6

It is crucial to have knowledge of both quantitative and qualitative research 2 as both types of research involve writing research questions and hypotheses. 7 However, these crucial elements of research are sometimes overlooked; if not overlooked, then framed without the forethought and meticulous attention it needs. Planning and careful consideration are needed when developing quantitative or qualitative research, particularly when conceptualizing research questions and hypotheses. 4

There is a continuing need to support researchers in the creation of innovative research questions and hypotheses, as well as for journal articles that carefully review these elements. 1 When research questions and hypotheses are not carefully thought of, unethical studies and poor outcomes usually ensue. Carefully formulated research questions and hypotheses define well-founded objectives, which in turn determine the appropriate design, course, and outcome of the study. This article then aims to discuss in detail the various aspects of crafting research questions and hypotheses, with the goal of guiding researchers as they develop their own. Examples from the authors and peer-reviewed scientific articles in the healthcare field are provided to illustrate key points.

DEFINITIONS AND RELATIONSHIP OF RESEARCH QUESTIONS AND HYPOTHESES

A research question is what a study aims to answer after data analysis and interpretation. The answer is written in length in the discussion section of the paper. Thus, the research question gives a preview of the different parts and variables of the study meant to address the problem posed in the research question. 1 An excellent research question clarifies the research writing while facilitating understanding of the research topic, objective, scope, and limitations of the study. 5

On the other hand, a research hypothesis is an educated statement of an expected outcome. This statement is based on background research and current knowledge. 8 , 9 The research hypothesis makes a specific prediction about a new phenomenon 10 or a formal statement on the expected relationship between an independent variable and a dependent variable. 3 , 11 It provides a tentative answer to the research question to be tested or explored. 4

Hypotheses employ reasoning to predict a theory-based outcome. 10 These can also be developed from theories by focusing on components of theories that have not yet been observed. 10 The validity of hypotheses is often based on the testability of the prediction made in a reproducible experiment. 8

Conversely, hypotheses can also be rephrased as research questions. Several hypotheses based on existing theories and knowledge may be needed to answer a research question. Developing ethical research questions and hypotheses creates a research design that has logical relationships among variables. These relationships serve as a solid foundation for the conduct of the study. 4 , 11 Haphazardly constructed research questions can result in poorly formulated hypotheses and improper study designs, leading to unreliable results. Thus, the formulations of relevant research questions and verifiable hypotheses are crucial when beginning research. 12

CHARACTERISTICS OF GOOD RESEARCH QUESTIONS AND HYPOTHESES

Excellent research questions are specific and focused. These integrate collective data and observations to confirm or refute the subsequent hypotheses. Well-constructed hypotheses are based on previous reports and verify the research context. These are realistic, in-depth, sufficiently complex, and reproducible. More importantly, these hypotheses can be addressed and tested. 13

There are several characteristics of well-developed hypotheses. Good hypotheses are 1) empirically testable 7 , 10 , 11 , 13 ; 2) backed by preliminary evidence 9 ; 3) testable by ethical research 7 , 9 ; 4) based on original ideas 9 ; 5) have evidenced-based logical reasoning 10 ; and 6) can be predicted. 11 Good hypotheses can infer ethical and positive implications, indicating the presence of a relationship or effect relevant to the research theme. 7 , 11 These are initially developed from a general theory and branch into specific hypotheses by deductive reasoning. In the absence of a theory to base the hypotheses, inductive reasoning based on specific observations or findings form more general hypotheses. 10

TYPES OF RESEARCH QUESTIONS AND HYPOTHESES

Research questions and hypotheses are developed according to the type of research, which can be broadly classified into quantitative and qualitative research. We provide a summary of the types of research questions and hypotheses under quantitative and qualitative research categories in Table 1 .

Research questions in quantitative research

In quantitative research, research questions inquire about the relationships among variables being investigated and are usually framed at the start of the study. These are precise and typically linked to the subject population, dependent and independent variables, and research design. 1 Research questions may also attempt to describe the behavior of a population in relation to one or more variables, or describe the characteristics of variables to be measured ( descriptive research questions ). 1 , 5 , 14 These questions may also aim to discover differences between groups within the context of an outcome variable ( comparative research questions ), 1 , 5 , 14 or elucidate trends and interactions among variables ( relationship research questions ). 1 , 5 We provide examples of descriptive, comparative, and relationship research questions in quantitative research in Table 2 .

Hypotheses in quantitative research

In quantitative research, hypotheses predict the expected relationships among variables. 15 Relationships among variables that can be predicted include 1) between a single dependent variable and a single independent variable ( simple hypothesis ) or 2) between two or more independent and dependent variables ( complex hypothesis ). 4 , 11 Hypotheses may also specify the expected direction to be followed and imply an intellectual commitment to a particular outcome ( directional hypothesis ) 4 . On the other hand, hypotheses may not predict the exact direction and are used in the absence of a theory, or when findings contradict previous studies ( non-directional hypothesis ). 4 In addition, hypotheses can 1) define interdependency between variables ( associative hypothesis ), 4 2) propose an effect on the dependent variable from manipulation of the independent variable ( causal hypothesis ), 4 3) state a negative relationship between two variables ( null hypothesis ), 4 , 11 , 15 4) replace the working hypothesis if rejected ( alternative hypothesis ), 15 explain the relationship of phenomena to possibly generate a theory ( working hypothesis ), 11 5) involve quantifiable variables that can be tested statistically ( statistical hypothesis ), 11 6) or express a relationship whose interlinks can be verified logically ( logical hypothesis ). 11 We provide examples of simple, complex, directional, non-directional, associative, causal, null, alternative, working, statistical, and logical hypotheses in quantitative research, as well as the definition of quantitative hypothesis-testing research in Table 3 .

Research questions in qualitative research

Unlike research questions in quantitative research, research questions in qualitative research are usually continuously reviewed and reformulated. The central question and associated subquestions are stated more than the hypotheses. 15 The central question broadly explores a complex set of factors surrounding the central phenomenon, aiming to present the varied perspectives of participants. 15

There are varied goals for which qualitative research questions are developed. These questions can function in several ways, such as to 1) identify and describe existing conditions ( contextual research question s); 2) describe a phenomenon ( descriptive research questions ); 3) assess the effectiveness of existing methods, protocols, theories, or procedures ( evaluation research questions ); 4) examine a phenomenon or analyze the reasons or relationships between subjects or phenomena ( explanatory research questions ); or 5) focus on unknown aspects of a particular topic ( exploratory research questions ). 5 In addition, some qualitative research questions provide new ideas for the development of theories and actions ( generative research questions ) or advance specific ideologies of a position ( ideological research questions ). 1 Other qualitative research questions may build on a body of existing literature and become working guidelines ( ethnographic research questions ). Research questions may also be broadly stated without specific reference to the existing literature or a typology of questions ( phenomenological research questions ), may be directed towards generating a theory of some process ( grounded theory questions ), or may address a description of the case and the emerging themes ( qualitative case study questions ). 15 We provide examples of contextual, descriptive, evaluation, explanatory, exploratory, generative, ideological, ethnographic, phenomenological, grounded theory, and qualitative case study research questions in qualitative research in Table 4 , and the definition of qualitative hypothesis-generating research in Table 5 .

Qualitative studies usually pose at least one central research question and several subquestions starting with How or What . These research questions use exploratory verbs such as explore or describe . These also focus on one central phenomenon of interest, and may mention the participants and research site. 15

Hypotheses in qualitative research

Hypotheses in qualitative research are stated in the form of a clear statement concerning the problem to be investigated. Unlike in quantitative research where hypotheses are usually developed to be tested, qualitative research can lead to both hypothesis-testing and hypothesis-generating outcomes. 2 When studies require both quantitative and qualitative research questions, this suggests an integrative process between both research methods wherein a single mixed-methods research question can be developed. 1

FRAMEWORKS FOR DEVELOPING RESEARCH QUESTIONS AND HYPOTHESES

Research questions followed by hypotheses should be developed before the start of the study. 1 , 12 , 14 It is crucial to develop feasible research questions on a topic that is interesting to both the researcher and the scientific community. This can be achieved by a meticulous review of previous and current studies to establish a novel topic. Specific areas are subsequently focused on to generate ethical research questions. The relevance of the research questions is evaluated in terms of clarity of the resulting data, specificity of the methodology, objectivity of the outcome, depth of the research, and impact of the study. 1 , 5 These aspects constitute the FINER criteria (i.e., Feasible, Interesting, Novel, Ethical, and Relevant). 1 Clarity and effectiveness are achieved if research questions meet the FINER criteria. In addition to the FINER criteria, Ratan et al. described focus, complexity, novelty, feasibility, and measurability for evaluating the effectiveness of research questions. 14

The PICOT and PEO frameworks are also used when developing research questions. 1 The following elements are addressed in these frameworks, PICOT: P-population/patients/problem, I-intervention or indicator being studied, C-comparison group, O-outcome of interest, and T-timeframe of the study; PEO: P-population being studied, E-exposure to preexisting conditions, and O-outcome of interest. 1 Research questions are also considered good if these meet the “FINERMAPS” framework: Feasible, Interesting, Novel, Ethical, Relevant, Manageable, Appropriate, Potential value/publishable, and Systematic. 14

As we indicated earlier, research questions and hypotheses that are not carefully formulated result in unethical studies or poor outcomes. To illustrate this, we provide some examples of ambiguous research question and hypotheses that result in unclear and weak research objectives in quantitative research ( Table 6 ) 16 and qualitative research ( Table 7 ) 17 , and how to transform these ambiguous research question(s) and hypothesis(es) into clear and good statements.

a These statements were composed for comparison and illustrative purposes only.

b These statements are direct quotes from Higashihara and Horiuchi. 16

a This statement is a direct quote from Shimoda et al. 17

The other statements were composed for comparison and illustrative purposes only.

CONSTRUCTING RESEARCH QUESTIONS AND HYPOTHESES

To construct effective research questions and hypotheses, it is very important to 1) clarify the background and 2) identify the research problem at the outset of the research, within a specific timeframe. 9 Then, 3) review or conduct preliminary research to collect all available knowledge about the possible research questions by studying theories and previous studies. 18 Afterwards, 4) construct research questions to investigate the research problem. Identify variables to be accessed from the research questions 4 and make operational definitions of constructs from the research problem and questions. Thereafter, 5) construct specific deductive or inductive predictions in the form of hypotheses. 4 Finally, 6) state the study aims . This general flow for constructing effective research questions and hypotheses prior to conducting research is shown in Fig. 1 .

Research questions are used more frequently in qualitative research than objectives or hypotheses. 3 These questions seek to discover, understand, explore or describe experiences by asking “What” or “How.” The questions are open-ended to elicit a description rather than to relate variables or compare groups. The questions are continually reviewed, reformulated, and changed during the qualitative study. 3 Research questions are also used more frequently in survey projects than hypotheses in experiments in quantitative research to compare variables and their relationships.

Hypotheses are constructed based on the variables identified and as an if-then statement, following the template, ‘If a specific action is taken, then a certain outcome is expected.’ At this stage, some ideas regarding expectations from the research to be conducted must be drawn. 18 Then, the variables to be manipulated (independent) and influenced (dependent) are defined. 4 Thereafter, the hypothesis is stated and refined, and reproducible data tailored to the hypothesis are identified, collected, and analyzed. 4 The hypotheses must be testable and specific, 18 and should describe the variables and their relationships, the specific group being studied, and the predicted research outcome. 18 Hypotheses construction involves a testable proposition to be deduced from theory, and independent and dependent variables to be separated and measured separately. 3 Therefore, good hypotheses must be based on good research questions constructed at the start of a study or trial. 12

In summary, research questions are constructed after establishing the background of the study. Hypotheses are then developed based on the research questions. Thus, it is crucial to have excellent research questions to generate superior hypotheses. In turn, these would determine the research objectives and the design of the study, and ultimately, the outcome of the research. 12 Algorithms for building research questions and hypotheses are shown in Fig. 2 for quantitative research and in Fig. 3 for qualitative research.

EXAMPLES OF RESEARCH QUESTIONS FROM PUBLISHED ARTICLES

• EXAMPLE 1. Descriptive research question (quantitative research)
• - Presents research variables to be assessed (distinct phenotypes and subphenotypes)
• “BACKGROUND: Since COVID-19 was identified, its clinical and biological heterogeneity has been recognized. Identifying COVID-19 phenotypes might help guide basic, clinical, and translational research efforts.
• RESEARCH QUESTION: Does the clinical spectrum of patients with COVID-19 contain distinct phenotypes and subphenotypes? ” 19
• EXAMPLE 2. Relationship research question (quantitative research)
• - Shows interactions between dependent variable (static postural control) and independent variable (peripheral visual field loss)
• “Background: Integration of visual, vestibular, and proprioceptive sensations contributes to postural control. People with peripheral visual field loss have serious postural instability. However, the directional specificity of postural stability and sensory reweighting caused by gradual peripheral visual field loss remain unclear.
• Research question: What are the effects of peripheral visual field loss on static postural control ?” 20
• EXAMPLE 3. Comparative research question (quantitative research)
• - Clarifies the difference among groups with an outcome variable (patients enrolled in COMPERA with moderate PH or severe PH in COPD) and another group without the outcome variable (patients with idiopathic pulmonary arterial hypertension (IPAH))
• “BACKGROUND: Pulmonary hypertension (PH) in COPD is a poorly investigated clinical condition.
• RESEARCH QUESTION: Which factors determine the outcome of PH in COPD?
• STUDY DESIGN AND METHODS: We analyzed the characteristics and outcome of patients enrolled in the Comparative, Prospective Registry of Newly Initiated Therapies for Pulmonary Hypertension (COMPERA) with moderate or severe PH in COPD as defined during the 6th PH World Symposium who received medical therapy for PH and compared them with patients with idiopathic pulmonary arterial hypertension (IPAH) .” 21
• EXAMPLE 4. Exploratory research question (qualitative research)
• - Explores areas that have not been fully investigated (perspectives of families and children who receive care in clinic-based child obesity treatment) to have a deeper understanding of the research problem
• “Problem: Interventions for children with obesity lead to only modest improvements in BMI and long-term outcomes, and data are limited on the perspectives of families of children with obesity in clinic-based treatment. This scoping review seeks to answer the question: What is known about the perspectives of families and children who receive care in clinic-based child obesity treatment? This review aims to explore the scope of perspectives reported by families of children with obesity who have received individualized outpatient clinic-based obesity treatment.” 22
• EXAMPLE 5. Relationship research question (quantitative research)
• - Defines interactions between dependent variable (use of ankle strategies) and independent variable (changes in muscle tone)
• “Background: To maintain an upright standing posture against external disturbances, the human body mainly employs two types of postural control strategies: “ankle strategy” and “hip strategy.” While it has been reported that the magnitude of the disturbance alters the use of postural control strategies, it has not been elucidated how the level of muscle tone, one of the crucial parameters of bodily function, determines the use of each strategy. We have previously confirmed using forward dynamics simulations of human musculoskeletal models that an increased muscle tone promotes the use of ankle strategies. The objective of the present study was to experimentally evaluate a hypothesis: an increased muscle tone promotes the use of ankle strategies. Research question: Do changes in the muscle tone affect the use of ankle strategies ?” 23

EXAMPLES OF HYPOTHESES IN PUBLISHED ARTICLES

• EXAMPLE 1. Working hypothesis (quantitative research)
• - A hypothesis that is initially accepted for further research to produce a feasible theory
• “As fever may have benefit in shortening the duration of viral illness, it is plausible to hypothesize that the antipyretic efficacy of ibuprofen may be hindering the benefits of a fever response when taken during the early stages of COVID-19 illness .” 24
• “In conclusion, it is plausible to hypothesize that the antipyretic efficacy of ibuprofen may be hindering the benefits of a fever response . The difference in perceived safety of these agents in COVID-19 illness could be related to the more potent efficacy to reduce fever with ibuprofen compared to acetaminophen. Compelling data on the benefit of fever warrant further research and review to determine when to treat or withhold ibuprofen for early stage fever for COVID-19 and other related viral illnesses .” 24
• EXAMPLE 2. Exploratory hypothesis (qualitative research)
• - Explores particular areas deeper to clarify subjective experience and develop a formal hypothesis potentially testable in a future quantitative approach
• “We hypothesized that when thinking about a past experience of help-seeking, a self distancing prompt would cause increased help-seeking intentions and more favorable help-seeking outcome expectations .” 25
• “Conclusion
• Although a priori hypotheses were not supported, further research is warranted as results indicate the potential for using self-distancing approaches to increasing help-seeking among some people with depressive symptomatology.” 25
• EXAMPLE 3. Hypothesis-generating research to establish a framework for hypothesis testing (qualitative research)
• “We hypothesize that compassionate care is beneficial for patients (better outcomes), healthcare systems and payers (lower costs), and healthcare providers (lower burnout). ” 26
• Compassionomics is the branch of knowledge and scientific study of the effects of compassionate healthcare. Our main hypotheses are that compassionate healthcare is beneficial for (1) patients, by improving clinical outcomes, (2) healthcare systems and payers, by supporting financial sustainability, and (3) HCPs, by lowering burnout and promoting resilience and well-being. The purpose of this paper is to establish a scientific framework for testing the hypotheses above . If these hypotheses are confirmed through rigorous research, compassionomics will belong in the science of evidence-based medicine, with major implications for all healthcare domains.” 26
• EXAMPLE 4. Statistical hypothesis (quantitative research)
• - An assumption is made about the relationship among several population characteristics ( gender differences in sociodemographic and clinical characteristics of adults with ADHD ). Validity is tested by statistical experiment or analysis ( chi-square test, Students t-test, and logistic regression analysis)
• “Our research investigated gender differences in sociodemographic and clinical characteristics of adults with ADHD in a Japanese clinical sample. Due to unique Japanese cultural ideals and expectations of women's behavior that are in opposition to ADHD symptoms, we hypothesized that women with ADHD experience more difficulties and present more dysfunctions than men . We tested the following hypotheses: first, women with ADHD have more comorbidities than men with ADHD; second, women with ADHD experience more social hardships than men, such as having less full-time employment and being more likely to be divorced.” 27
• “Statistical Analysis
• ( text omitted ) Between-gender comparisons were made using the chi-squared test for categorical variables and Students t-test for continuous variables…( text omitted ). A logistic regression analysis was performed for employment status, marital status, and comorbidity to evaluate the independent effects of gender on these dependent variables.” 27

EXAMPLES OF HYPOTHESIS AS WRITTEN IN PUBLISHED ARTICLES IN RELATION TO OTHER PARTS

• EXAMPLE 1. Background, hypotheses, and aims are provided
• “Pregnant women need skilled care during pregnancy and childbirth, but that skilled care is often delayed in some countries …( text omitted ). The focused antenatal care (FANC) model of WHO recommends that nurses provide information or counseling to all pregnant women …( text omitted ). Job aids are visual support materials that provide the right kind of information using graphics and words in a simple and yet effective manner. When nurses are not highly trained or have many work details to attend to, these job aids can serve as a content reminder for the nurses and can be used for educating their patients (Jennings, Yebadokpo, Affo, & Agbogbe, 2010) ( text omitted ). Importantly, additional evidence is needed to confirm how job aids can further improve the quality of ANC counseling by health workers in maternal care …( text omitted )” 28
• “ This has led us to hypothesize that the quality of ANC counseling would be better if supported by job aids. Consequently, a better quality of ANC counseling is expected to produce higher levels of awareness concerning the danger signs of pregnancy and a more favorable impression of the caring behavior of nurses .” 28
• “This study aimed to examine the differences in the responses of pregnant women to a job aid-supported intervention during ANC visit in terms of 1) their understanding of the danger signs of pregnancy and 2) their impression of the caring behaviors of nurses to pregnant women in rural Tanzania.” 28
• EXAMPLE 2. Background, hypotheses, and aims are provided
• “We conducted a two-arm randomized controlled trial (RCT) to evaluate and compare changes in salivary cortisol and oxytocin levels of first-time pregnant women between experimental and control groups. The women in the experimental group touched and held an infant for 30 min (experimental intervention protocol), whereas those in the control group watched a DVD movie of an infant (control intervention protocol). The primary outcome was salivary cortisol level and the secondary outcome was salivary oxytocin level.” 29
• “ We hypothesize that at 30 min after touching and holding an infant, the salivary cortisol level will significantly decrease and the salivary oxytocin level will increase in the experimental group compared with the control group .” 29
• EXAMPLE 3. Background, aim, and hypothesis are provided
• “In countries where the maternal mortality ratio remains high, antenatal education to increase Birth Preparedness and Complication Readiness (BPCR) is considered one of the top priorities [1]. BPCR includes birth plans during the antenatal period, such as the birthplace, birth attendant, transportation, health facility for complications, expenses, and birth materials, as well as family coordination to achieve such birth plans. In Tanzania, although increasing, only about half of all pregnant women attend an antenatal clinic more than four times [4]. Moreover, the information provided during antenatal care (ANC) is insufficient. In the resource-poor settings, antenatal group education is a potential approach because of the limited time for individual counseling at antenatal clinics.” 30
• “This study aimed to evaluate an antenatal group education program among pregnant women and their families with respect to birth-preparedness and maternal and infant outcomes in rural villages of Tanzania.” 30
• “ The study hypothesis was if Tanzanian pregnant women and their families received a family-oriented antenatal group education, they would (1) have a higher level of BPCR, (2) attend antenatal clinic four or more times, (3) give birth in a health facility, (4) have less complications of women at birth, and (5) have less complications and deaths of infants than those who did not receive the education .” 30

Research questions and hypotheses are crucial components to any type of research, whether quantitative or qualitative. These questions should be developed at the very beginning of the study. Excellent research questions lead to superior hypotheses, which, like a compass, set the direction of research, and can often determine the successful conduct of the study. Many research studies have floundered because the development of research questions and subsequent hypotheses was not given the thought and meticulous attention needed. The development of research questions and hypotheses is an iterative process based on extensive knowledge of the literature and insightful grasp of the knowledge gap. Focused, concise, and specific research questions provide a strong foundation for constructing hypotheses which serve as formal predictions about the research outcomes. Research questions and hypotheses are crucial elements of research that should not be overlooked. They should be carefully thought of and constructed when planning research. This avoids unethical studies and poor outcomes by defining well-founded objectives that determine the design, course, and outcome of the study.

Disclosure: The authors have no potential conflicts of interest to disclose.

Author Contributions:

• Conceptualization: Barroga E, Matanguihan GJ.
• Methodology: Barroga E, Matanguihan GJ.
• Writing - original draft: Barroga E, Matanguihan GJ.
• Writing - review & editing: Barroga E, Matanguihan GJ.

Writing Quantitative Research Studies

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• Adyya Gupta 3 &
• Karen G. Peres 4  

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Summarizing quantitative data and its effective presentation and discussion can be challenging for students and researchers. This chapter provides a framework for adequately reporting findings from quantitative analysis in a research study for those contemplating to write a research paper. The rationale underpinning the reporting methods to maintain the credibility and integrity of quantitative studies is outlined. Commonly used terminologies in empirical studies are defined and discussed with suitable examples. Key elements that build consistency between different sections (background, methods, results, and the discussion) of a research study using quantitative methods in a journal article are explicated. Specifically, recommended standard guidelines for randomized controlled trials and observational studies for reporting and discussion of findings from quantitative studies are elaborated. Key aspects of methodology that include describing the study population, sampling strategy, data collection methods, measurements/variables, and statistical analysis which informs the quality of a study from the reviewer’s perspective are described. Effective use of references in the methods section to strengthen the rationale behind specific statistical techniques and choice of measures has been highlighted with examples. Identifying ways in which data can be most succinctly and effectively summarized in tables and graphs according to their suitability and purpose of information is also detailed in this chapter. Strategies to present and discuss the quantitative findings in a structured discussion section are also provided. Overall, the chapter provides the readers with a comprehensive set of tools to identify key strategies to be considered when reporting quantitative research.

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Bhaumik S, Arora M, Singh A, Sargent JD. Impact of entertainment media smoking on adolescent smoking behaviours. Cochrane Database Syst Rev. 2015;6:1–12. https://doi.org/10.1002/14651858.CD011720 .

Article   Google Scholar  

Dickersin K, Manheimer E, Wieland S, Robinson KA, Lefebvre C, McDonald S. Development of the Cochrane Collaboration’s CENTRAL register of controlled clinical trials. Eval Health Prof. 2002;25(1):38–64.

Google Scholar  

Docherty M, Smith R. The case for structuring the discussion of scientific papers: much the same as that for structuring abstracts. Br Med J. 1999;318(7193):1224–5.

Greenland S, Pearl J, Robins JM. Causal diagrams for epidemiologic research. Epidemiology. 1999;10(1):37–48.

Horton R. The rhetoric of research. Br Med J. 1995;310(6985):985–7.

Kool B, Ziersch A, Robinson P, Wolfenden L, Lowe JB. The ‘Seven deadly sins’ of rejected papers. Aust N Z J Public Health. 2016;40(1):3–4.

Mannocci A, Saulle R, Colamesta V, D’Aguanno S, Giraldi G, Maffongelli E, et al. What is the impact of reporting guidelines on public health journals in Europe? The case of STROBE, CONSORT and PRISMA. J Public Health. 2015;37(4):737–40.

Rothwell PM. External validity of randomised controlled trials: “to whom do the results of this trial apply?”. Lancet. 2005;365(9453):82–93.

Schulz KF, Altman DG, Moher D. CONSORT 2010 statement: updated guidelines for reporting parallel group randomised trials. PLoS Med. 2010;7(3):e1000251.

Szklo M. Quality of scientific articles. Rev Saude Publica. 2006;40 Spec no:30–5.

Vandenbroucke JP, von Elm E, Altman DG, Gotzsche PC, Mulrow CD, Pocock SJ, et al. Strengthening the reporting of observational studies in epidemiology (STROBE): explanation and elaboration. PLoS Med. 2007;4(10):e297.

Weiss NS, Koepsell TD, Psaty BM. Generalizability of the results of randomized trials. Arch Intern Med. 2008;168(2):133–5.

Singh A, Gupta A, Peres MA, Watt RG, Tsakos G, Mathur MR. Association between tooth loss and hypertension among a primarily rural middle aged and older Indian adult population. J Public Health Dent. 2016;76:198–205.

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Adyya Gupta

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Singh, A., Gupta, A., Peres, K.G. (2019). Writing Quantitative Research Studies. In: Liamputtong, P. (eds) Handbook of Research Methods in Health Social Sciences. Springer, Singapore. https://doi.org/10.1007/978-981-10-5251-4_117

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Article Contents

1 introduction, 2 basic ingredients: rigidity of the exact inclusion and linearized estimate, 3 proof of proposition 1.3, 4 proof of theorem 1.1, 5 a |$3\times 3$| counter-example, acknowledgments.

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Quantitative Rigidity of Differential Inclusions in Two Dimensions

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Xavier Lamy, Andrew Lorent, Guanying Peng, Quantitative Rigidity of Differential Inclusions in Two Dimensions, International Mathematics Research Notices , Volume 2024, Issue 8, April 2024, Pages 6325–6349, https://doi.org/10.1093/imrn/rnad108

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In 1850, Liouville [ 22 ] proved that, given a domain |$\Omega \subset{{\mathbb{R}}}^3$|⁠ , any smooth map |$u\colon \Omega \to{{\mathbb{R}}}^3$| satisfying the differential inclusion |$Du(x)\in{{\mathbb{R}}}_+ O(n)$| for all |$x\in \Omega $| must be either affine or a Möbius transform. A corollary to Liouville’s Theorem is that a |$C^3$| function whose gradient belongs everywhere to |$SO(n)$| is an affine mapping. This phenomenon of being able to globally control a map satisfying a certain differential inclusion |$Du\in K$| is known as “rigidity”. Questions about the stability of differential inclusions under weak convergence and approximate rigidity statements, raised by Tartar in [ 30 , 31 ], are intimately linked with phenomena of compensated compactness and have been extremely influential in the development of weak convergence methods in partial differential equation (PDE).

A number of works have extended the above result ( 1 ) to cover various larger classes of matrices than |$K=SO (n)$|⁠ . Chaudhuri and Müller [ 8 ] and later De Lellis and Székelyhidi [ 9 ] considered a set of the form |$K = SO (n) A\cup SO (n) B$| where |$A$| and |$B$| are strongly incompatible in the sense of Matos [ 24 ]. Faraco and Zhong [ 13 ] proved an analogous quantitative rigidity result with |$K=\mathfrak m\cdot SO(n)$| where |$\mathfrak m \subset (0,+\infty )$| is compact. There the infimum in the left-hand side of ( 1 ) also needs to include the gradients of Möbius transforms, and the integral is over a smaller subset |$\Omega ^{\prime}\subset \subset \Omega $|⁠ . Similar results for maps defined on the sphere have been obtained recently by Luckhaus and Zemas [ 23 ]. Best constants for ( 1 ) are investigated by Lewicka and Müller in [ 21 ].

Our main result is an optimal generalization of the quantitative rigidity estimate of [ 14 ] in the context of compact connected submanifolds |$K\subset \mathbb{R}^{2\times 2}$| without boundary.

  Theorem 1.1. Let |$K\subset{{\mathbb{R}}}^{2\times 2}$| be a smooth, compact, and connected |$1$| -manifold without boundary. Assume that K has no rank-one connections, and satisfies the stronger property of being elliptic in the sense that there exists |$C_*>0$| such that $$\begin{align} |M-M^{\prime}|^2\leq C_* \det(M-M^{\prime})\qquad\forall M,M^{\prime}\in K. \end{align}$$ (2) Then, for any |$u\in H^1(B_1;{{\mathbb{R}}}^2)$|⁠ , we have $$\begin{align} \inf_{M\in K}\int_{B_{1/2}}| Du -M |^2\, \textrm{d}x \leq C \int_{B_1} \operatorname{dist}^2(Du, K)\, \textrm{d}x, \end{align}$$ (3) for some constant |$C=C(K)>0$|⁠ .

  Remark 1.2. A covering argument as in [ 13 ] shows that the estimate ( 3 ) in the balls |$B_{1/2}\subset B_1$| automatically improves to $$\begin{align*} \inf_{M\in K}\int_{\Omega^{\prime}}| Du -M |^2\, \textrm{d}x \leq C \int_{\Omega} \operatorname{dist}^2(Du, K)\, \textrm{d}x,\qquad\forall u\in H^1(\Omega;{{\mathbb{R}}}^2), \end{align*}$$ for any bounded domain |$\Omega \subset{{\mathbb{R}}}^2$| and open subset |$\Omega ^{\prime}\subset \subset \Omega \subset{{\mathbb{R}}}^2$|⁠ , and a constant |$C=C(K,\Omega ^{\prime},\Omega )>0$|⁠ .

This result is optimal among compact connected submanifolds |$K\subset \mathbb R^{2\times 2}$| without boundary for the following reasons:

First, it is classical that the no-rank-one-connections assumption is necessary for the rigidity of the exact differential inclusion (see e.g., [ 18 , 25 ]).

Second, ellipticity is necessary for the validity of the linearized version of ( 3 ) because non-ellipticity would imply that the tangent space |$T_{M}K$| has a rank-one connection for some |$M\in K$|⁠ , and by Remark 2.3 the linearized version of ( 3 ) is implied by ( 3 ).

Third, the two previous conditions (no rank-one connections and ellipticity) imply that the connected submanifold without boundary |$K\subset \mathbb{R}^{2\times 2}$| must be of dimension 1 [ 33 , Corollaries 3.5 and 3.6].

Moreover, we provide in Section 5 an example showing that the two-dimensional setting is also optimal: there exists an elliptic |$1$| -submanifold |$K\subset{{\mathbb{R}}}^{3\times 3}$| without rank-one connection but which contains a so-called |$\mathcal T_4$| configuration, a well-known obstruction to compactness of sequences |$\lbrace u_k\rbrace \subset H^1$| satisfying |$\operatorname{dist}(Du_k,K)\to 0$| in |$L^2$| [ 7 ], and therefore to any type of quantitative rigidity estimate.

One of our motivations for studying differential inclusion into general submanifolds |$K\subset{{\mathbb{R}}}^{2\times 2}$| is our previous work [ 20 ] where we obtained a rigidity result for a non-elliptic differential inclusion related to the so-called Aviles–Giga functional, and pointed out the nice consequences that a corresponding quantitative rigidity estimate would have. Theorem 1.1 is not valid for non-elliptic differential inclusions, but the ideas in the present work should be relevant to attain that goal.

While the statements of the quantitative rigidity results of [ 8 , 13 , 14 ] are elementary, their proofs are not. Their starting point, in addition to rigidity of the exact differential inclusion, is a linearized version of ( 1 ) for the differential inclusion |$Du\in T_{M_0}K$| into a tangent space |$T_{M_0}K$|⁠ . For |$K=SO(n)$| and |$M_0=I$|⁠ , this is Korn’s inequality. A natural linearization procedure then provides a quantitative rigidity estimate, but in terms of the |$L^\infty $| norm of |$\operatorname{dist}(Du,K)$|⁠ , rather than |$L^2$|⁠ . Strengthening the |$L^2$| bound on |$\operatorname{dist}(Du,K)$| into an |$L^\infty $| bound constitutes therefore the main difficulty. A key idea, introduced in [ 14 ], is to use the regularity of an elliptic PDE satisfied by solutions of the exact differential inclusion: for |$K=SO(n)$|⁠ , the exact differential inclusion |$Du\in SO\left (n\right )$| implies that the coordinate functions |$u_k$| are harmonic. For |$K\subset \mathbb{R}_{+}SO\left (n\right )$|⁠ , the coordinate functions satisfy the |$(n-2)$| -Laplace equation |$\operatorname{div} (\left |\nabla u_k\right |^{n-2} \nabla u_k )=0$|⁠ . Such PDE follows from the universal identity |$\operatorname{div}\operatorname{cof}(Du)=0$| (where |$\operatorname{cof}$| denotes the cofactor matrix), together with identities satisfied by matrices in the specific set |$K$|⁠ . It is satisfied by solutions of the exact differential inclusion, and for a general map |$u$| the error from solving that PDE can be controlled in terms of the right-hand side of ( 1 ). This allows to reduce the proof of ( 1 ) to maps solving that PDE. Elliptic regularity then provides, via a compactness argument, a uniform bound on |$\operatorname{dist}(Du,K)$| and the linearization can be performed.

Following this scheme, the main ingredient to prove Theorem 1.1 is to embed |$K$| into the graph of a uniformly monotone vector field: this will be enough to turn the identity |$\operatorname{div}\operatorname{cof}(Du)=0$| into a quasilinear elliptic equation for the exact differential inclusion |$Du\in K$|⁠ .

  Proposition 1.3. Let |$K$| be as in Theorem 1.1 . There exist |$G_1,G_2\colon{{\mathbb{R}}}^2\to{{\mathbb{R}}}^2$| smooth, globally Lipschitz vector fields such that $$\begin{align*} K \subset \left\lbrace \left(\begin{array}{c} A \\ iG_1(A) \end{array}\right)\colon A\in{{\mathbb{R}}}^2 \right\rbrace \cap \left\lbrace \left(\begin{array}{c} -iG_2(B) \\ B \end{array}\right)\colon B\in{{\mathbb{R}}}^2 \right\rbrace, \end{align*}$$ and |$G_1,G_2$| are uniformly monotone, that is, $$\begin{align*} (G_j(X)-G_j(X^{\prime}))\cdot (X-X^{\prime}) \geq \lambda |X-X^{\prime}|^2\qquad\forall X,X^{\prime}\in{{\mathbb{R}}}^2, \end{align*}$$ for some constant |$\lambda>0$| depending only on |$K$|⁠ .

Proposition 1.3 relies on remarkable properties of elliptic subsets of |${{\mathbb{R}}}^{2\times 2}$| with respect to the decomposition into conformal and anticonformal parts, discovered in [ 33 ] and exploited in a striking manner in [ 5 , 11 , 12 ] (see also [ 18 ]). It is also related to the classical link between two-dimensional elliptic PDEs of second-order and complex Beltrami equations, see for example, the introduction of [ 6 ], and [ 1–4 ] for recent developments on nonlinear Beltrami equations. The proof of Proposition 1.3 is in fact extremely close to [ 3 , Theorem 5], and the main point of Proposition 1.3 is to emphasize that the second-order elliptic equations satisfied by the real and imaginary part of solutions to a nonlinear Beltrami equation (as in [ 3 , Theorem 5]) are, in our setting, associated to embeddings of |$K$| into graphs: this fact is crucially used in Step 2 of Theorem 1.1 ’s proof, and we find it convenient to state it independently (although it could also be retrieved from the proof of [ 3 , Theorem 5] applied to the nonlinear Beltrami equation associated to |$K$|⁠ ). Then the proof of Theorem 1.1 follows the scheme outlined above.

The article is organized as follows. In Section 2 , we establish the two basic prerequisites to Theorem 1, rigidity for the exact differential inclusion and the linearized estimate. In Section 3 , we give the proof of Proposition 1.3 . In Section 4 , we gather these ingredients to prove Theorem 1.1 . In Section 5 , we describe the counterexample in |${{\mathbb{R}}}^{3\times 3}$|⁠ .

In this section, let |$K\subset{{\mathbb{R}}}^{2\times 2}$| be as in Theorem 1.1 . We prove the following two Lemmas.

If |$u\in H^1(B_1;{{\mathbb{R}}}^2)$| is such that |$Du\in K$| a.e., then |$Du\equiv M$| for some |$M\in K$|⁠ .

  Lemma 2.2. For all |$M\in K$| and |$u\in H^1(B_1;{{\mathbb{R}}}^2)$| we have $$\begin{align} \inf_{X\in T_M K} \int_{ B_{1}}|Du-X|^2\, \textrm{d}x \leq C \int_{B_1} \operatorname{dist}^2(Du,T_M K)\, \textrm{d}x \end{align}$$ (4) for some constant |$C=C(K)>0$|⁠ , where |$T_M K$| denotes the linear tangent space to |$K$| at |$M$|⁠ .

  Remark 2.3. The linearized estimate ( 4 ), or rather its weaker interior version $$\begin{align} \inf_{X\in T_M K} \int_{ B_{1/2}}|Du-X|^2\, \textrm{d}x \leq C \int_{B_1} \operatorname{dist}^2(Du,T_M K)\, \textrm{d}x, \end{align}$$ (5) is a necessary condition for ( 3 ) to be valid. Assume indeed that ( 3 ) is verified, fix |$u\in C^1(\overline B_1;{{\mathbb{R}}}^2)$|⁠ , and apply ( 3 ) to |$v_{\epsilon }(x)=Mx +{\epsilon } u(x)$| for |${\epsilon }\ll 1$|⁠ . There exists |$M_{\epsilon }\in K$| such that $$\begin{align*} \int_{B_{1/2}}|M-M_{\epsilon} -{\epsilon} Du|^2\, \textrm{d}x &\leq C \int_{B_1}\operatorname{dist}^2(M +{\epsilon} Du,K)\, \textrm{d}x \\ &\leq C {\epsilon}^2\int_{B_1} \operatorname{dist}^2(Du,T_M K)\, \textrm{d}x + o({\epsilon}^2). \end{align*}$$ Hence, letting |$X_{\epsilon }={\epsilon }^{-1}(M-M_{\epsilon })$|⁠ , we have $$\begin{align*} \int_{B_{1/2}}|X_{\epsilon} -Du|^2\, \textrm{d}x \leq C \int_{B_1}\operatorname{dist}^2(Du,T_M K)\, \textrm{d}x + o(1). \end{align*}$$ In particular, |$X_{\epsilon }$| is bounded, and extracting a converging subsequence, we obtain |$X\in T_M K$| showing the validity of ( 5 ) for |$u\in C^1(\overline B_1;{{\mathbb{R}}}^2)$|⁠ , and then by density for |$u\in H^1(B_1;{{\mathbb{R}}}^2)$|⁠ .

Let |$\ell =|K|$| and |$\gamma \colon{{\mathbb{R}}}/\ell Z\to K$| be an arc-length parametrization of |$K$|⁠ . The ellipticity assumption ( 2 ) ensures that |$\gamma ^{\prime}(t)$| is invertible for all |$t\in{{\mathbb{R}}}$|⁠ . Let |$u\in H^1(B_1;{{\mathbb{R}}}^2)$| such that |$Du\in K$| a.e., then |$u$| is smooth by [ 32 ], and since |$B_1$| is simply connected there exists a smooth lifting |$\theta \colon B_1\to{{\mathbb{R}}}$| such that |$Du=\gamma (\theta )$|⁠ . Using that |$\operatorname{div} \operatorname{cof}(Du)=0$|⁠ , where |$\operatorname{cof}$| denotes the cofactor matrix, we find |$\operatorname{cof} (\gamma ^{\prime}(\theta ))\nabla \theta =0$|⁠ , hence |$\nabla \theta =0$| since |$\operatorname{cof}(\gamma ^{\prime}(\theta ))$| is invertible. Therefore, |$Du$| is constant.

Moreover, if |$u\in H^1(B_1;{{\mathbb{R}}}^2)$| satisfies |$P_MDu=0$| a.e., then |$Du=\lambda X_0$| for some |$\lambda \in L^2(B_1;{{\mathbb{R}}})$|⁠ , and the distributional identity |$0=\operatorname{div} \operatorname{cof}(Du)=\operatorname{cof}(X_0)\nabla \lambda $| implies that |$\lambda $| is constant, hence |$Du\equiv X$| for some |$X\in T_M K $|⁠ .

The fact that ( 4 ) follows from ( 6 ) is general and already mentioned in [ 28 , page 74], here we provided details for the reader’s convenience. More precisely, if |$P_M D$| is replaced by a more general differential operator, the validity of ( 4 ) is equivalent to the null space of that differential operator being finite-dimensional [ 28 , Theorem 8.15 and Remark 4], and a compactness argument similar to the one given here shows that the last term in the right-hand side of ( 6 ) can be dropped if |$u$| is orthogonal to that finite-dimensional null space.

The function |$H$| admits a smooth extension |$H\colon{{\mathbb{C}}}\to{{\mathbb{C}}}$|⁠ , which is |$k$| -Lipschitz for some (possibly larger) |$0\leq k<1$|⁠ .

We first fix, thanks to Kirszbraun’s theorem, a |$k$| -Lipschitz extension |$\widehat H\colon{{\mathbb{C}}}\to{{\mathbb{C}}}$|⁠ . In the rest of the proof, we modify |$\widehat H$| to make it smooth while still agreeing with |$H$| on |$p_+( K )$|⁠ , at the cost of slightly increasing its Lipschitz constant.

The map |$F\colon z\mapsto \bar z +H(z)$| is a smooth diffeomorphism from |${{\mathbb{C}}}$| onto |${{\mathbb{C}}}$|⁠ . Moreover, |$F$| is |$(1+k)$| -Lipschitz and |$F^{-1}$| is |$(1-k)^{-1}$| -Lipschitz.

  Proof of Lemma 3.2. For any |$w\in{{\mathbb{C}}}$|⁠ , the equation $$\begin{align*} w=\bar z +H(z) \quad\Leftrightarrow\quad z =\bar w - \bar H(z), \end{align*}$$ admits a unique solution |$z\in{{\mathbb{C}}}$| thanks to the fixed point theorem, since |$z\mapsto \bar w-\bar H(z)$| is |$k$| -Lipschitz and |$0\leq k<1$|⁠ . This shows that |$F$| is bijective. The inequalities $$\begin{align*} (1-k)|z-z^{\prime}| \leq |F(z)-F(z^{\prime})| \leq (1+k) |z-z^{\prime}|, \end{align*}$$ follow directly from the fact that |$H$| is |$k$| -Lipschitz and imply the announced Lipschitz constants of |$F$| and |$F^{-1}$|⁠ . The inverse |$F^{-1}$| is smooth thanks to the Inverse Function Theorem, since |$DF(z)\colon h\mapsto \bar h +DH(z)h$| is invertible for all |$z\in{{\mathbb{C}}}$| because |$\|DH\| <1$|⁠ .

  Proof of Proposition 1.3 completed. Then, identifying |${{\mathbb{C}}}$| with |${{\mathbb{R}}}^2$|⁠ , we define $$\begin{align*} G_1(A)=\overline{F^{-1}(A)} -H(F^{-1}(A)), \end{align*}$$ so that a short calculation shows that $$\begin{align*} [z,H(z)]=\left(\begin{array}{c} A \\ iG_1(A)\end{array}\right)\qquad\textrm{for}\ A=F(z). \end{align*}$$ The map |$G_1$| is smooth and globally Lipschitz with Lipschitz constant |$\Lambda =(1+k)/(1-k)$|⁠ . Moreover, for all |$A=F(z)$|⁠ , |$A^{\prime}=F(z^{\prime})$|⁠ , we have $$\begin{align*} (G_1(A)-G_1(A^{\prime}))\cdot (A-A^{\prime}) & =\det \left(\begin{array}{c} A-A^{\prime} \\ i(G_{1}(A)-G_{1}(A^{\prime}))\end{array}\right)\\ & =\det ([z-z^{\prime},H(z)-H(z^{\prime})] )\\ &=|z-z^{\prime}|^2-|H(z)-H(z^{\prime})|^2 \\ &\geq (1-k^2)|z-z^{\prime}|^2. \end{align*}$$ Since |$F$| is |$(1+k)$| -Lipschitz, this implies $$\begin{align*} (G_{1}(A)-G_{1}(A^{\prime}))\cdot (A-A^{\prime}) \geq \lambda |A-A^{\prime}|^2, \end{align*}$$ for |$\lambda =(1-k^2)/(1+k)^{2}= (1-k)/(1+k)>0$|⁠ . This concludes the proof of Proposition 1.3 .

Step 4. We finally combine Step 3 with the linearized estimate of Lemma 2.2 , to obtain our main estimate ( 3 ). The basic idea, as in [ 13 , 14 ], is to linearize |$\operatorname{dist}^2(\cdot , K )$| around |$M_0\in K $| such that |$|Du-M_0|$| is uniformly small. When doing so, ( 3 ) formally turns into the linearized estimate ( 4 ) of Lemma 2.2 , and it remains to control the error terms. Due to the modification of |$u$| arising from the translation |$X\in T_M K $| in the left-hand side of the linearized estimate ( 4 ), it is not directly obvious that the error terms are negligible. In [ 14 ], this problem is absent because their equivalent of ( 9 ) comes with an explicit |$\rho ({\epsilon })=C{\epsilon }^{\frac 14}$|⁠ . In [ 13 ], it is taken care of via a topological degree argument [ 13 , Proposition 4.7] (see also [ 27 ]), which allows to avoid the translation. While a similar degree argument could be used to iteratively improve the estimate here, we present a simpler alternative method relying on elementary estimates.

We fix |$\delta _0>0$| depending only on |$ K $|⁠ , such that the nearest-point projection |$\Pi _ K $| onto |$ K $| is uniquely defined and smooth in the neighborhood |$\mathcal N_{2\delta _0}( K )$|⁠ . We first choose |${\epsilon }_0$| small enough that |$\rho ({\epsilon }_0)\leq \delta _0$|⁠ , so thanks to ( 9 ) the projection |$\Pi _ K (Du)$| is well-defined.

In this section, we prove that the two-dimensional setting of Theorem 1.1 is optimal in the following sense: a connected |$1$| -submanifold of |${{\mathbb{R}}}^{3\times 3}$| that has no rank-one connection and is elliptic may not satisfy Šverák’s compactness result [ 32 ], and even less a quantitative rigidity estimate.

There exists a smooth, compact, and connected |$1$| -submanifold |$ K \subset{{\mathbb{R}}}^{ 3\times 3}$| without boundary, which is elliptic and has no rank-one connection, but contains a |$\mathcal T_4$| configuration.

|$\rho \left (\theta +2\pi \right )=\rho \left (\theta \right )+2\pi $| for |$\theta \in{{\mathbb{R}}}$|⁠ ,

|$\rho \left (\hat \theta _k\right )=\hat \theta _k$| for |$\hat \theta _k =\theta _a +(k-1)\frac \pi 2$|⁠ , |$k=1,2,3,4$|⁠ .

Notation. With the |$M_{i_1i_2,j_1j_2}$| minor we mean the determinant of the |$2\times 2$| submatrix corresponding to the rows |$i_1,i_2$| and columns |$j_1,j_2$|⁠ .

If |$a>0$| is such that |$\theta _a\notin \frac{\pi }{48}{{\mathbb{Z}}}$|⁠ , the curve |$ K _a$| is elliptic, that is, |$\textrm{Rank}\:\Gamma _a^{\prime}(\theta )> 1$| for all |$\theta \in{{\mathbb{R}}}$|⁠ .

  Proof. The derivative |$\Gamma _a^{\prime}$| is given by $$\begin{align*} \Gamma_a^{\prime}(\theta):=\left(\begin{array}{ccc} -r_a \sin\left(\theta\right) & -8\cos\left(8\theta-8\theta_a\right) & 6\rho^{\prime}(\theta)\cos\left(6\rho\left(\theta\right)-6\theta_a\right) \\ 6\cos\left(6\theta-6\theta_a\right) & r_a \cos\left(\theta\right) & 8\rho^{\prime}(\theta)\cos\left(8\rho\left(\theta\right)-8\theta_a\right) \\ -r_a \sin\left(\theta\right) & 8\cos\left(8\theta-8\theta_a\right) & 6\rho^{\prime}(\theta)\cos\left(6\rho\left(\theta\right)-6\theta_a\right) \end{array}\right). \end{align*}$$ Assume |$\textrm{Rank}\:\Gamma _a^{\prime}(\theta )\leq 1$| for some |$\theta \in{{\mathbb{R}}}$|⁠ . Then calculating the |$M_{12,12}$| minor we have $$\begin{align*} -r_a^2 \sin\left(\theta\right)\cos\left(\theta\right) +48 \cos\left(8\theta-8\theta_a\right)\cos\left(6\theta-6\theta_a\right)=0 \end{align*}$$ and calculating the |$M_{23,12}$| minor we have $$\begin{align*} 48 \cos\left(8\theta-8\theta_a\right)\cos\left(6\theta-6\theta_a\right)+ r_a^2 \sin\left(\theta\right)\cos\left(\theta\right) =0. \end{align*}$$ Adding and subtracting these two equations, we obtain that $$\begin{align*} \sin\left(\theta\right)\cos\left(\theta\right)=0\ \textrm{and}\ \cos\left(8\theta-8\theta_a\right)\cos\left(6\theta-6\theta_a\right)=0. \end{align*}$$ The first equality implies |$\theta \in \frac \pi 2{{\mathbb{Z}}}$|⁠ , and then the second equality becomes $$\begin{align*} \cos(8\theta_a)\cos(6\theta_a)=0, \end{align*}$$ which is impossible for |$\theta _a\notin \frac{\pi }{48}{{\mathbb{Z}}}$|⁠ .

|$\rho \left (\hat \theta _k\right )=\hat \theta _k$| for |$\theta _k =\hat \theta _a +(k-1)\frac \pi 2$|⁠ , |$k=1,2,3,4$|⁠ ,

For any |$\theta , \theta ^{\prime}\in [0,2\pi )\cap \frac{\pi }{24}\mathbb{Z}$|⁠ , we have |$\rho (\theta )-\rho (\theta ^{\prime})\notin \frac \pi{12}{{\mathbb{Z}}}$|⁠ ,

|$\sup _{\theta \in \mathbb{R}}\left |\rho (\theta )-\theta \right |<{\epsilon }$|⁠ .

  Proof. Let |$\delta =\frac 12 \operatorname{dist}(\theta _a,\frac \pi{24}{{\mathbb{Z}}}) = \frac 12 \operatorname{dist}(\lbrace \hat \theta _k\rbrace ,\frac \pi{24}{{\mathbb{Z}}})>0$| and fix a smooth function |$\varphi $| such that |$\operatorname{supp}\varphi \subset (-\delta ,\delta )$|⁠ , |$0\leq \varphi \leq \varphi (0)=1$| and |$|\varphi ^{\prime}|\leq 2/\delta $|⁠ . Define |$\rho $| on |$[0,2\pi )$| by setting $$\begin{align*} \rho(\theta)=\theta +\sum_{j=1}^{47}t_j\,\varphi\left(\theta-j\frac{\pi}{24}\right)\qquad\ \textrm{for}\ \theta\in [0,2\pi), \end{align*}$$ where |$t_1,\ldots ,t_{47}\in (-\eta ,\eta )$| are to be fixed later and |$\eta =\min ({\epsilon },\delta /2)$|⁠ . The choice of |$\delta>0$| ensures that |$\rho (\hat \theta _k)=\hat \theta _k$|⁠ , also since |$|t_j|<{\epsilon }$| we have |$|\rho -id|<{\epsilon }$|⁠ , and finally since |$|t_j|<\delta /2$| we have |$\rho ^{\prime}>0$| on |$[0,2\pi )$|⁠ . Moreover, the function |$\rho -id$| is identically zero near |$0$| and |$2\pi $|⁠ , so it can be extended to a smooth |$2\pi $| periodic function, thus yielding a smooth monotonic extension |$\rho \colon{{\mathbb{R}}}\to{{\mathbb{R}}}$| satisfying |$\rho (\theta +2\pi )=\rho (\theta )+2\pi $| for |$\theta \in{{\mathbb{R}}}$|⁠ . It remains to argue that we can pick |$t_1,\ldots ,t_{47}\in [0,\eta )$| to ensure that the third condition in Lemma 5.3 is satisfied. Denote |$t_0=0$|⁠ . By induction, we may for each |$j=1,\ldots ,47$| choose |$t_j\in [0,\eta )$| to ensure that $$\begin{align*} \rho(j\pi/24)-\rho(\ell\pi/24)=t_j-t_\ell + (j-\ell)\pi/24 \notin \frac\pi{12}{{\mathbb{Z}}} \end{align*}$$ for all |$\ell \in \lbrace 0,\ldots ,j-1\rbrace $|⁠ . This is possible because at each step there is only a discrete set of values of |$t_j$| to avoid.

If |$a>0$| is such that |$\theta _a\notin \frac{\pi }{48}{{\mathbb{Z}}}$|⁠ , |${\epsilon }>0$| is small enough and |$\rho $| is as in Lemma 5.3 , then the curve |$ K _a\subset{{\mathbb{R}}}^{3\times 3}$| does not contain rank-one connections.

Step 2. We have |$\theta -\theta ^{\prime}\in \frac{\pi }{4}\mathbb{Z}$|⁠ .

Step 3. We have |$\theta -\theta ^{\prime}\in \pi \mathbb{Z}$|⁠ .

This work was supported by the Agence Nationale De La Re cherche project [ANR-22-CE40-0006-01 to X.L.]; the Simons foundation [426900 A.L.]; and the National Science Foundation [DMS-2206291 to G.P.].

We thank D. Faraco and A. Guerra for illuminating comments. We also thank the anonymous referees for numerous constructive suggestions to improve the writing of the manuscript.

Communicated by Prof. Michael Struwe

Astala K. A. Clop D. Faraco J. Jääskeläinen “ Manifolds of quasiconformal mappings and the nonlinear Beltrami equation .” JAMA 139 , no. 1 ( 2019 ): 207 – 38 . http://10.1007/s11854-019-0059-x .

Google Scholar

Astala K. A. Clop D. Faraco J. Jääskeläinen A. Koski “ Nonlinear Beltrami operators, Schauder estimates and bounds for the Jacobian .” Ann. Inst. Henri Poincaré, Anal. Non Linéaire 34 , no. 6 ( 2017 ): 1543 – 59 .

Astala K. A. Clop D. Faraco J. Jääskeläinen A. Koski “ Improved Hölder regularity for strongly elliptic PDEs .” J. Math. Pures Appl. (9) 140 ( 2020 ): 230 – 58 . http://10.1016/j.matpur.2020.06.005 .

Astala K. A. Clop D. Faraco J. Jääskeläinen L. J. Székelyhidi “ Uniqueness of normalized homeomorphic solutions to nonlinear Beltrami equations .” Internat. Math. Res. Notices 2012 , no. 18 ( 2012 ): 4101 – 19 . http://10.1093/imrn/rnr178 .

Astala K. D. Faraco “ Quasiregular mappings and young measures .” Proc. R. Soc. Edinb. A Math. 132 , no. 5 ( 2002 ): 1045 – 56 . http://10.1017/S0308210500002006 .

Astala K. Iwaniec T. Martin G. Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane . Princeton Math. Ser. , vol. 48 . Princeton, NJ : Princeton University Press , 2009 , http://10.1515/9781400830114 .

Google Preview

Bhattacharya K. N. B. Firoozye R. D. James R. V. Kohn “ Restrictions on microstructure .” Proc. R. Soc. Edinb. A Math. 124 , no. 5 ( 1994 ): 843 – 78 . http://10.1017/S0308210500022381 .

Chaudhuri N. S. Müller “ Rigidity estimate for two incompatible wells .” Calc. Var. Partial Differential Equations 19 , no. 4 ( 2004 ): 379 – 90 . http://10.1007/s00526-003-0220-2 .

De Lellis C. L. Székelyhidi “ Simple proof of two-well rigidity .” C. R. Math. Acad. Sci. Paris 343 , no. 5 ( 2006 ): 367 – 70 . http://10.1016/j.crma.2006.07.008 .

Evans L. C. Partial Differential Equations , 2nd ed. Graduate Studies in Mathematics , vol. 19 . American Mathematical Society , Providence, RI , 2010 .

Faraco D. “ Tartar conjecture and Beltrami operators .” Mich. Math. J. 52 , no. 1 ( 2004 ): 83 – 104 .

Faraco D. Székelyhidi L. Tartar’s conjecture and localization of the quasiconvex hull in |${\mathbb{R}}^{2\times 2}$| . Acta Math. 200 , no. 2 ( 2008 ): 279 – 305 , http://10.1007/s11511-008-0028-1 .

Faraco D. X. Zhong “ Geometric rigidity of conformal matrices .” Ann. Scuola Norm. Super. Pisa. Class. Sci. Ser. V 4 , no. 4 ( 2005 ): 557 – 85 .

Friesecke G. R. D. James S. Müller “ A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity .” Comm. Pure Appl. Math. 55 , no. 11 ( 2002 ): 1461 – 506 . http://10.1002/cpa.10048 .

Friesecke G. R. D. James S. Müller “ A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence .” Arch. Rational Mech. Anal. 180 , no. 2 ( 2006 ): 183 – 236 . http://10.1007/s00205-005-0400-7 .

Gilbarg D. Trudinger N. S. Elliptic Partial Differential Equations of Second Order . Classics in Mathematics . Springer , Berlin , 2001 . Reprint of the 1998 edition , http://10.1007/978-3-642-61798-0 .

John F. “ Rotation and strain .” Comm. Pure Appl. Math. 14 , no. 1 ( 1961 ): 391 – 413 . http://10.1002/cpa.3160140316 .

Kirchheim B. Rigidity and Geometry of Microstructures . MIS-MPG lecture note 16/2003 , 2003 .

Kohn R. “ New integral estimates for deformations in terms of their nonlinear strains .” Arch. Rational Mech. Anal. 78 , no. 2 ( 1982 ): 131 – 72 . http://10.1007/BF00250837 .

Lamy X. A. Lorent G. Peng “ Rigidity of a non-elliptic differential inclusion related to the Aviles–Giga conjecture .” Arch. Rational Mech. Anal. 238 , no. 1 ( 2020 ): 383 – 413 . http://10.1007/s00205-020-01545-z .

Lewicka M. S. Müller “ On the optimal constants in Korn’s and geometric rigidity estimates in bounded and unbounded domains, under Neumann boundary conditions .” Indiana Univ. Math. J. 65 , no. 2 ( 2016 ): 377 – 97 . http://10.1512/iumj.2016.65.5797 .

Liouville J. Theoreme Sur l’equation |$d{x}^2+d{y}^2+d{z}^2=\lambda \left (d{\alpha }^2+d{\beta }^2+d{\gamma }^2\right )$| . J. Math. Pures Appl. 1 , no. 15 ( 1850 ).

Luckhaus S. Zemas K. “ Rigidity estimates for isometric and conformal maps from |${\mathbb{S}}^{n-}$| to |${\mathbb{R}}^n$| .” Invent. Math. ( 2022 ), 230 , 375 , 461 , http://10.1007/s00222-022-01128-7 .

Matos J. P. “ Young measures and the absence of fine microstructures in a class of phase transitions .” European J. Appl. Math. 3 , no. 1 ( 1992 ): 31 – 54 . http://10.1017/S095679250000067X .

Müller S. “ Variational models for microstructure and phase transitions .” In Calculus of Variations and Geometric Evolution Problems. Lectures Given at the 2nd Session of the Centro Internazionale Matematico Estivo (CIME), Cetraro, Italy, June 15–22, 1996 . Berlin : Springer , 1999 , pp. 85 – 210 , http://10.1007/BFb0092670 .

Resetnjak J. “ Liouville’s conformal mapping theorem under minimal regularity hypotheses .” Sibirsk. Mat. Z. 8 ( 1967 ): 835 – 40 .

Reshetnyak Y. G. Stability Theorems in Geometry and Analysis . Translated from the Russian by N. S. Dairbekov and V. N. Dyatlov. Revised and updated translation, rev. and updated transl. ed . Math. Appl., Dordr. , vol. 304 . Dordrecht : Kluwer Academic Publishers , 1994 , http://10.1007/978-94-015-8360-2 .

Smith K. T. “ Formulas to represent functions by their derivatives .” Math. Ann. 188 ( 1970 ): 53 – 77 . http://10.1007/BF01435415 .

Székelyhidi L. j. “ Rank-one convex hulls in |${\mathbb{R}}^{2\times 2}$| .” Calc. Var. Partial Differ. Equ. 22 , no. 3 ( 2005 ): 253 – 81 , http://10.1007/s00526-004-0272-y .

Tartar L. “ Compensated compactness and applications to partial differential equations .” Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., 39, Pitman, Boston, Mass.-London, 1979 . 136 – 212 , 1979 .

Tartar L. “ The compensated compactness method applied to systems of conservation laws. Systems of nonlinear partial differential equations .” Proc. NATO Adv. Study Inst., Oxford/U.K. 1982, NATO ASI Ser. Ser., C 111 ( 1983 ): 263 – 85 1983 .

Šverák V. “ On Tartar’s conjecture .” Ann. Inst. Henri Poincaré. Anal. Nonlin. 10 , no. 4 ( 1993 ): 405 – 12 . http://10.1016/s0294-1449(16)30208-6 .

Zhang K. “ On connected subsets of |${M}^{2\times 2}$| without rank-one connections .” Proc. R. Soc. Edinb. A Math. 127 , no. 1 ( 1997 ): 207 – 16 . http://10.1017/S0308210500023593 .

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