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Frequency Polygons

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Frequency Polygons in Statistics

Frequency Polygons in Statistics: A frequency polygon is a type of line graph where the frequencies of classes are plotted against their midpoints. This graphical representation closely resembles a histogram and is typically used for comparing data sets or showing cumulative frequency distributions. It uses a line graph to represent quantitative data.

Frequency polygons are one of the great methods to represent statistical data so that it can be read easily. In statistics, we deal with lots of data, and reading it quickly is necessary for solving statistical problems effectively.

Frequency polygons help us to achieve the same result. In this article, we will learn about frequency polygons, their formula, examples, and others in detail.

Table of Content

What is a Frequency Polygon in Statistics?

Frequency polygon graph, cumulative frequency polygon, how to draw frequency polygon, histogram and frequency polygons, frequency polygons examples, frequency polygons are used for – applications.

A visual representation of the frequency distribution of continuous data is the frequency polygon. Karl Pearson, an English statistician, made the initial presentation of it in the late 19th century. Based on the research of earlier statisticians like Francis Galton and Adolphe Quetelet, he put forth the idea. Its value comes from its capacity to graphically portray data, making understanding and analysis easier. They, therefore, serve as an important tool in statistics, assisting researchers in finding patterns and trends in huge data sets.

They are useful for showing data sets with numerous observations or values, which is one of their key features. The polygon represents the data distribution succinctly and clearly by splitting the data into equal intervals and charting the frequency of each container. As they take the shape of points that sit beyond the typical range of the distribution, they are also useful in finding outliers or abnormalities in the data set. Once more, this facilitates identifying and examining odd or unexpected data points.

Frequency Polygon Definition

Frequency polygons are a type of graphical data distribution that aids in recognizing the data by giving it a particular form. Although frequency polygons and histograms are quite similar, they are more effective when comparing two or more sets of data. As a line graph, the graph primarily shows data from the cumulative frequency distribution.

Formula to Find Midpoint of Frequency Polygons

If you want to plot a frequency polygon graph, you must figure out the midpoint or class mark for each of the class intervals. The following is the formula to achieve that:

Class Mark (Midpoint) = (Upper Limit + Lower Limit) / 2

Frequency Polygon Graph is the graphical representation of the data given in the form of class interval and frequency. Let’s consider an example for better understanding:

Example: Plot the graph of the Frequency Polygon for the following data which represents the number of goals scored in a match in a league throughout the season:

For the given data, we can plot the frequency polygon by representing the goals scored on the vertical axis and frequency on the horizontal axis, as follows:

In statistics, the cumulative frequencies of a dataset are shown graphically using cumulative frequency polygons. The dots at the upper-class borders are plotted against the corresponding cumulative frequencies to create the graph. The graph illustrates the visual accumulation of data across time or intervals. The polygon created by the line connecting the points aids in the visualization of the data’s trends and patterns.

It helps analysts grasp the structure and central tendency of cumulative frequency distributions by clearly showing them. Researchers and analysts may get important insights into the distribution of the data and make wise judgments based on the trends seen by employing cumulative frequency polygons.

Following the procedures listed below, construct a histogram before beginning to draw frequency polygons:

Step 1: First, select the class interval and then indicate the values on the axes. Step 2: Label the horizontal axes with the midpoint of each interval. Step 3: Label the vertical axes with the class frequency. Step 4: Mark a point at the height in the centre of each class interval according to the frequency of each class interval. Step 5: Use the line segment to join these spots. Step 6: The representation that was obtained is a frequency polygon .

Read More: Bar Graphs and Histograms

The difference between the frequency polygon and histogram is discussed in the table below:

Example 1: The frequency data below was used to create a frequency polygon.

By first determining the classmark using the equation Classmark = (Upper Limit + Lower Limit) / 2, we may build a frequency polygon without a histogram. Additionally, by combining the previous and next frequencies, we can get the cumulative frequency of each class interval. Class Interval : (59.5 + 49.5/2) = 54.5 (69.5 + 59.5/2) = 64.5 (79.5 + 69.5/2) = 74.5 (89.5 + 79.5/2) = 84.5 (99.5 + 89.5/2) = 94.5 Class Interval Lower Bound Upper Bound Classmark Frequency 49.5-59.5 49.5 59.5 54.5 5 59.5-69.5 59.5 69.5 64.5 10 69.5-79.5 69.5 79.5 74.5 30 79.5-89.5 79.5 89.5 84.5 40 89.5-99.5 89.5 99.5 94.5 15 We note the before and after classmarks as well while plotting the graph. The before in this instance is 44.5, while the after is 104.5. The scores are shown on the x-axis, while the frequency is indicated on the y-axis. Consequently, the frequency polygons graph will seem like follows:

Example 2: Assume that a class of 65 students’ weights are distributed as follows: 15 – 25, 25 – 35, 35 – 45, and 45 – 55. How many grade points would there be for each weight category?

Formula used to get the classmark for a Frequency Polygon Graph is: Classmark = (Upper Limit + Lower Limit) / 2 Hence, Class interval 15-25 = (15 + 25)/2 = 20 Class interval 25-35 = (25 + 35)/2 = 30 Class interval 35-45 = (35 + 45)/2 = 40 Class interval 45-55 = (45 + 55)/2 = 50

Here are some of the key applications of frequency polygons:

• Comparing Distributions : Frequency polygons allow for easy comparison between different datasets. By overlaying multiple polygons on the same graph, you can visually assess differences in shapes, spreads, and central tendencies of the distributions.
• Identifying Skewness and Kurtosis : The shape of a frequency polygon can provide insights into the skewness (asymmetry) and kurtosis (peakedness) of the data distribution. This is useful for understanding the nature of the dataset, whether it is normally distributed, positively skewed, negatively skewed, etc.
• Analyzing Trends : In time series data, frequency polygons can help in visualizing trends over time. This makes it easier to identify patterns such as seasonal variations or cyclic changes.
• Educational Tool : In teaching statistics, frequency polygons serve as a clearer alternative to histograms for some students. They can help in understanding how frequencies change between intervals.
• Cumulative Frequency Analysis : Although not a cumulative frequency graph itself, frequency polygons can be adapted to show cumulative frequencies, providing a visual representation of data accumulation over intervals.

People Also View:

How to Make a Frequency Polygon in Excel? Frequency Distribution – Definition, Types, Table, Graph Types of Polygons

Summary – Frequency Polygons in Statistics

Frequency polygons are graphical tools used in statistics to represent the distribution of data points. They are essentially line graphs that plot class frequencies against class midpoints, making them very similar to histograms but offering a clearer visual comparison between multiple datasets. These graphs are particularly valuable for analyzing trends, comparing distributions, and identifying characteristics such as skewness and kurtosis in the data. By providing a visual summary of data, frequency polygons serve as an effective educational tool to enhance understanding of statistical distributions and are useful in diverse applications from business analytics to academic research.

FAQs on Frequency Polygons in Statistics

What is frequency polygon.

Frequency Polygon is a way of representing continuous data in statistics and is similar to Histogram in statistics.

What Kind of Data is Appropriate for a Frequency Polygon?

Frequency Polygons are suitable for continuous data like height, weight, temperature, or time. They can, however, be used for discrete data that is classified into intervals, such as test scores or income bands.

Why Are Frequency Polygons Used?

Frequency polygon graphs are used to compare a group of data because they are more clear and legible. These graphs are also commonly used to represent the cumulative frequency distribution.

What are the Features of Frequency Polygons?

A frequency polygon graph is a closed dimensional diagram consisting of a line segment connecting the midpoints of the supplied class intervals. The graph can be made either with or without a histogram. The first point lies on the x-axis at y = 0, in the midst of the interval before the first class interval.

What is the Difference Between a Frequency Polygon and a Frequency Curve?

The main distinction between a frequency polygon and a frequency curve is that a frequency polygon is drawn by combining points with a straight line, but a frequency curve is drawn with a smooth hand.

Are Frequency Polygon and Histogram the Same?

No, it is not comparable, however it does have certain characteristics. The main distinction between both is that a histogram is a graphical representation of data made up of contiguous rectangles, whereas a frequency polygon is a curve made up of the midpoints of those rectangles.

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Frequency Polygons

Frequency polygons are a graphical representation of data distribution that helps in understanding the data through a specific shape. Frequency polygons are very similar to histograms but are helpful and useful while comparing two or more data. The graph mainly showcases cumulative frequency distribution data in the form of a line graph. Let us learn about the frequency polygons graph, the steps in creating a graph, and solve a few examples to understand the concept better.

Definition of Frequency Polygons

Frequency Polygons can be defined as a form of a graph that interprets information or data that is widely used in statistics . This visual form of data representation helps in depicting the shape and trend of the data in an organized and systematic manner. Frequency polygons through the shape of the graph depict the number of occurrence of class intervals. This type of graph is usually drawn with a histogram but can be drawn without a histogram as well. While a histogram is a graph with rectangular bars without spaces, a frequency polygon graph is a line graph that represents cumulative frequency distribution data. Frequency polygons look like the image below:

Steps to Construct Frequency Polygons

The curve in a frequency polygon is drawn on an x-axis and y-axis. As a regular graph, the x-axis represents the value in a dataset and the y-axis shows the number of occurrences of each category. While plotting a frequency polygon graph, the most important aspect is the mid-point which is called the class interval or class marks. The frequency polygon curve can be drawn with or without a histogram. For drawing with a histogram, we first draw rectangular bars against the class intervals and join the midpoints of the bars to get the frequency polygons. Here are the steps to drawing a frequency polygon graph without a histogram:

• Step 1: Mark the class intervals for each class on an x-axis while we plot the curve on the y-axis.
• Step 2: Calculate the midpoint of each of the class intervals which is the classmarks. (The formula is mentioned in the next section)
• Step 3: Once the classmarks are obtained, mark them on the x-axis.
• Step 4: Since the height always depicts the frequency, plot the frequency according to each class mark. It should be plotted against the classmark itself and not on the upper or lower limit.
• Step 5: Once the points are marked, join them with a line segment similar to a line graph.
• Step 6: The curve that is obtained by this line segment is the frequency polygon.

Formula to Find the Frequency Polygons Midpoint

While plotting a frequency polygon graph we require to calculate the midpoint or the classmark for each of the class intervals. The formula to do so is:

Class Mark (Midpoint) = (Upper Limit + Lower Limit) / 2

Difference Between Frequency Polygons and Histogram

Even though a frequency polygon graph is similar to a histogram and can be plotted with or without a histogram, the two graphs are yet different from each other. The two graphs have their own unique properties that show the difference visually. The differences are:

Related Topics

Listed below are a few topics that are related to frequency polygons, take a look.

• Frequency Distribution

Examples on Frequency Polygons

Example 1: Construct a frequency polygon without a histogram using the data given below.

To construct a frequency polygon without a histogram we first find the classmark by using the formula Classmark = (Upper Limit + Lower Limit) / 2. And we will find the cumulative frequency of each class interval as well by adding the next frequency and previous frequency together.

Class interval = (59.5 + 49.5)/2 = 54.5, (69.5 + 59.5)/2 = 64.5, (79.5 + 69.5)/2 = 74.5, (89.5 + 79.5)/2 = 84.5, (99.5 + 89.5)/2 = 94.5

While plotting the graph, we also mark the before and after classmark as well. In this case, the before is 44.5 and the after is 104.5. The scores are plotted on the x-axis and the frequency is plotted on the y-axis. Hence, the frequency polygons graph will look like this:

Example 2: In a city, the weekly observations made in a study on the cost of a living index are given in the following table: Draw a frequency polygon for the data below with a histogram.

Solution: To plot a frequency polygon with a histogram, we need to follow these steps to construct a histogram:

• The cost of living index is represented on the x-axis.
• The number of weeks is represented on the y-axis.
• Now rectangular bars of widths equal to the class- size and the length of the bars corresponding to a frequency of the class interval are drawn.

To calculate the midpoint, we use the formula Classmark = (Upper Limit + Lower Limit) / 2

Classmark = (150 + 140)/2 = 145, (160 + 150)/2 = 155 and so on.

While plotting the graph, we also mark the before and after classmark as well. In this case, the before is 135 and the after is 205. ABCDEFGH represents the given data graphically in form of frequency polygon i.e. those are the midpoints. Hence, the frequency polygons graph will look like this:

Example 3: If the weight range for a class of 45 students is distributed by 35 - 45, 45 - 55, 55 - 65, 65 - 75. What would be the class marks for each weight range?

To calculate the classmark for a frequency polygon graph, we use the formula, Classmark = (Upper Limit + Lower Limit) / 2.

Class interval 35 - 45 = (45 + 35)/2 = 40

Class interval 45 - 55 = (55 + 45)/2 = 50

Class interval 55 - 65 = (65 + 55)/2 = 60

Class interval 65 - 75 = (75 + 65)/2 = 70

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Practice Questions on Frequency Polygons

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FAQs on Frequency Polygons

What is frequency polygons.

A frequency polygon is a type of line graph where the class frequency is plotted against the class midpoint and the points are joined by a line segment creating a curve. The curve can be drawn with and without a histogram. A frequency polygon graph helps in depicting the highs and lows of frequency distribution data. To obtain the curve for a frequency polygon, we need to find the classmark or midpoint from the class intervals.

How Do You Construct a Frequency Polygons?

A frequency polygon can be constructed with and without a histogram. The steps to construct a frequency polygon without a histogram are:

• Mark the class intervals for each class on an x-axis while we plot the curve on the y-axis.
• Calculate the midpoint of each of the class intervals which is the classmarks.
• Mark the classmarks on the x-axis.
• Since the height always depicts the frequency, plot the frequency according to each class mark. It should be plotted against the classmark itself and not on the upper or lower limit.
• Once the points are marked, join them with a line segment similar to a line graph.
• The curve that is obtained by this line segment is the frequency polygon.

What is the Difference Between Histogram and Frequency Polygons?

A frequency polygon graph is the improved version of a histogram. A histogram is a bar graph with rectangle-shaped bars depicting the data whereas a frequency polygon is a line graph where a curved line depicts the data. A frequency polygon is more widely used when distributive data needs to be compared since in a histogram the comparison will not be clear.

Why Do We Use Frequency Polygons?

Frequency polygons graphs are used in comparing a set of data as it is clear and more readable. These graphs are also widely used for depicting cumulative frequency distribution.

What are the Characteristics of Frequency Polygons?

A frequency polygon graph is considered as a closed dimensional figure of a line segment joining the midpoints of the given class intervals. The graph can either be drawn with a histogram or without a histogram. The first point is on the x-axis where y = 0 and is placed in the middle of the interval which precedes the first class interval.

What is the Similarity Between Frequency Polygons and Line Graphs?

A frequency polygon is a type of line graph where a line segment curves to join the midpoints of all the class intervals. The shape of the curved line helps in providing accurate data. Both a line graph and frequency polygon graph are widely used when data is required to be compared.

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Statistics: Basic Concepts  - Frequency Polygons

Statistics: basic concepts  -, frequency polygons, statistics: basic concepts frequency polygons.

Statistics: Basic Concepts: Frequency Polygons

Lesson 9: frequency polygons.

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Frequency polygons

What is a frequency polygon.

A frequency polygon shows the overall distribution of a data set. It looks a little bit like a line graph–but the points on the graph can be plotted using data from a histogram or a frequency table . 

Frequency polygons are especially useful for comparing two data sets . In our example, we’ll use the histogram from the last lesson in order to make our frequency polygon. 

How to create a frequency polygon

Here is our histogram from the previous lesson, which shows the age range of members of the orchestra:

• Step 1: Imagine that at the top of each bar, there is a dot located right in the middle. These dots are called midpoints . (They are also sometimes referred to as “class marks.” This is because categories/bins are sometimes called classes).

Each midpoint marked on the graph represents the frequency of each bin or age range.

• Step 2: Once the midpoints have been plotted, the first line segment should connect zero to the first midpoint.

• Step 3: After that, we’ll connect the first midpoint to the second one, and so on.

• Step 4: Remove the bars and copy your value increments onto the x-axis . Since we don't have the bars to represent each bin, make sure you're clear about what values are being shown on the x-axis. In this case, each point represents an age range . 

• Step 5: Now you have your frequency polygon!

Comparing two data sets

Usually a frequency polygon is compared to a different frequency polygon on the same graph. The second frequency polygon comes from another data set . 

For example, if you wanted to compare the age range of the band’s members from 10 years ago with the current band, you could do two things:

• Use frequency tables for your two data sets to plot the points of your frequency polygons.
• Make histograms for each set of data, and then create frequency polygons from your histograms.

You would plot your frequency polygons on the same graph. Let’s use a gray line to represent the data set from 10 years ago, which we can compare with the white line representing the current band’s ages:

On the whole, the trends in the data appear quite similar. The conductor can see that there’s been a dip (or decline) in the 35-40 year age range. 

Characteristics of a frequency polygon

Here’s a quick recap…

• Frequency polygons can be made from a histogram or a frequency table. If you are creating one using a histogram, plot the midpoints at the top of each bar. Then connect the midpoints and remove each bar.
• Generally frequency polygons are used to reflect quantitative data . If you have “bins” like age ranges, these can be plotted on the x-axis. The y-axis is often used to reflect frequency. 
• They’re useful for comparing/contrasting two (or more) data sets reflected on the same graph . Take a look at the overall distribution, and see what conclusions you can draw about the data. 

In the next section of the tutorial, we’ll start diving into some other fundamentals that will help you in statistics.

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Module 2: Descriptive Statistics

Histograms, frequency polygons, and time series graphs, learning outcomes.

• Display data graphically and interpret graphs: stemplots, histograms, and box plots.
• Recognize, describe, and calculate the measures of location of data: quartiles and percentiles.

For most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets. A rule of thumb is to use a histogram when the data set consists of [latex]100[/latex] values or more.

A  histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data.

The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:

• [latex]f[/latex] = frequency
• [latex]n[/latex] = total number of data values (or the sum of the individual frequencies), and
• [latex]RF[/latex] = relative frequency,

then [latex]\displaystyle{R}{F}=\frac{{f}}{{n}}[/latex]

For example, if three students in Mr. Ahab’s English class of [latex]40[/latex] students received from [latex]90[/latex]% to [latex]100[/latex]%, then, [latex]\displaystyle{f}={3},{n}={40}[/latex], and [latex]{R}{F}=\frac{{f}}{{n}}=\frac{{3}}{{40}}={0.075}[/latex]. [latex]7.5[/latex]% of the students received [latex]90–100[/latex]%. [latex]90–100[/latex]% are quantitative measures.

To construct a histogram , first decide how many bars or intervals , also called classes, represent the data. Many histograms consist of five to [latex]15[/latex] bars or classes for clarity. The number of bars needs to be chosen. Choose a starting point for the first interval to be less than the smallest data value. A convenient starting point is a lower value carried out to one more decimal place than the value with the most decimal places. For example, if the value with the most decimal places is [latex]6.1[/latex] and this is the smallest value, a convenient starting point is [latex]6.05[/latex] ([latex]6.1 – 0.05 = 6.05[/latex]). We say that [latex]6.05[/latex] has more precision. If the value with the most decimal places is [latex]2.23[/latex] and the lowest value is [latex]1.5[/latex], a convenient starting point is [latex]1.495[/latex] ([latex]1.5 – 0.005 = 1.495[/latex]). If the value with the most decimal places is [latex]3.234[/latex] and the lowest value is [latex]1.0[/latex], a convenient starting point is [latex]0.9995[/latex] ([latex]1.0 – 0.0005 = 0.9995[/latex]). If all the data happen to be integers and the smallest value is two, then a convenient starting point is [latex]1.5[/latex] ([latex]2 – 0.5 = 1.5[/latex]). Also, when the starting point and other boundaries are carried to one additional decimal place, no data value will fall on a boundary. The next two examples go into detail about how to construct a histogram using continuous data and how to create a histogram using discrete data.

Watch the following video for an example of how to draw a histogram.

The following data are the heights (in inches to the nearest half inch) of [latex]100[/latex] male semiprofessional soccer players. The heights are continuous data, since height is measured.

[latex]60[/latex]; [latex]60.5[/latex]; [latex]61[/latex]; [latex]61[/latex]; [latex]61.5[/latex]

[latex]63.5[/latex]; [latex]63.5[/latex]; [latex]63.5[/latex]

[latex]64[/latex]; [latex]64[/latex]; [latex]64[/latex]; [latex]64[/latex]; [latex]64[/latex]; [latex]64[/latex]; [latex]64[/latex]; [latex]64.5[/latex]; [latex]64.5[/latex]; [latex]64.5[/latex]; [latex]64.5[/latex]; [latex]64.5[/latex]; [latex]64.5[/latex]; [latex]64.5[/latex]; [latex]64.566[/latex]; [latex]66[/latex]; [latex]66[/latex]; [latex]66[/latex]; [latex]66[/latex]; [latex]66[/latex]; [latex]66[/latex]; [latex]66[/latex]; [latex]66[/latex]; [latex]66[/latex]; [latex]66.5[/latex]; [latex]66.5[/latex]; [latex]66.5[/latex]; [latex]66.5[/latex]; [latex]66.5[/latex]; [latex]66.5[/latex]; [latex]66.5[/latex]; [latex]66.5[/latex]; [latex]66.5[/latex]; [latex]66.5[/latex]; [latex]66.5[/latex]; [latex]67[/latex]; [latex]67[/latex]; [latex]67[/latex]; [latex]67[/latex]; [latex]67[/latex]; [latex]67[/latex]; [latex]67[/latex]; [latex]67[/latex]; [latex]67[/latex]; [latex]67[/latex]; [latex]67[/latex]; [latex]67[/latex]; [latex]67.5[/latex]; [latex]67.5[/latex]; [latex]67.5[/latex]; [latex]67.5[/latex]; [latex]67.5[/latex]; [latex]67.5[/latex]; [latex]67.5[/latex]

[latex]68[/latex]; [latex]68[/latex]; [latex]69[/latex]; [latex]69[/latex]; [latex]69[/latex]; [latex]69[/latex]; [latex]69[/latex]; [latex]69[/latex]; [latex]69[/latex]; [latex]69[/latex]; [latex]69[/latex]; [latex]69[/latex]; [latex]69.5[/latex]; [latex]69.5[/latex]; [latex]69.5[/latex]; [latex]69.5[/latex]; [latex]69.5[/latex]

[latex]70[/latex]; [latex]70[/latex]; [latex]70[/latex]; [latex]70[/latex]; [latex]70[/latex]; [latex]70[/latex]; [latex]70.5[/latex]; [latex]70.5[/latex]; [latex]70.5[/latex]; [latex]71[/latex]; [latex]71[/latex]; [latex]71[/latex]

[latex]72[/latex]; [latex]72[/latex]; [latex]72[/latex]; [latex]72.5[/latex]; [latex]72.5[/latex]; [latex]73[/latex]; [latex]73.5[/latex]; [latex]74[/latex]

The smallest data value is [latex]60[/latex]. Since the data with the most decimal places has one decimal (for instance, [latex]61.5[/latex]), we want our starting point to have two decimal places. Since the numbers [latex]0.5[/latex], [latex]0.05[/latex], [latex]0.005[/latex], etc. are convenient numbers, use [latex]0.05[/latex] and subtract it from [latex]60[/latex], the smallest value, for the convenient starting point.

[latex]60 – 0.05 = 59.95[/latex] which is more precise than, say, [latex]61.5[/latex] by one decimal place. The starting point is, then, [latex]59.95[/latex].

The largest value is [latex]74[/latex], so [latex]74 + 0.05 = 74.05[/latex] is the ending value.

Next, calculate the width of each bar or class interval. To calculate this width, subtract the starting point from the ending value and divide by the number of bars (you must choose the number of bars you desire). Suppose you choose eight bars.

[latex]\displaystyle\frac{{{74.05}-{59.95}}}{{8}}={1.76}[/latex]

We will round up to two and make each bar or class interval two units wide. Rounding up to two is one way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using [latex]1.76[/latex] as the width would also work. A guideline that is followed by some for the width of a bar or class interval is to take the square root of the number of data values and then round to the nearest whole number, if necessary. For example, if there are [latex]150[/latex] values of data, take the square root of [latex]150[/latex] and round to [latex]12[/latex] bars or intervals.

The boundaries are:

• [latex]59.95[/latex]
• [latex]59.95 + 2 = 61.95[/latex]
• [latex]61.95 + 2 = 63.95[/latex]
• [latex]63.95 + 2 = 65.95[/latex]
• [latex]65.95 + 2 = 67.95[/latex]
• [latex]67.95 + 2 = 69.95[/latex]
• [latex]69.95 + 2 = 71.95[/latex]
• [latex]71.95 + 2 = 73.95[/latex]
• [latex]73.95 + 2 = 75.95[/latex]

The heights [latex]60[/latex] through [latex]61.5[/latex] inches are in the interval [latex]59.95–61.95[/latex]. The heights that are [latex]63.5[/latex] are in the interval [latex]61.95–63.95[/latex]. The heights that are [latex]64[/latex] through [latex]64.5[/latex] are in the interval [latex]63.95–65.95[/latex]. The heights [latex]66[/latex] through [latex]67.5[/latex] are in the interval [latex]65.95–67.95[/latex]. The heights [latex]68[/latex] through [latex]69.5[/latex] are in the interval [latex]67.95–69.95[/latex]. The heights [latex]70[/latex] through [latex]71[/latex] are in the interval [latex]69.95–71.95[/latex]. The heights [latex]72[/latex] through [latex]73.5[/latex] are in the interval [latex]71.95–73.95[/latex]. The height [latex]74[/latex] is in the interval [latex]73.95–75.95[/latex].

The following histogram displays the heights on the [latex]x[/latex]-axis and relative frequency on the [latex]y[/latex]-axis.

The following data are the shoe sizes of [latex]50[/latex] male students. The sizes are continuous data since shoe size is measured. Construct a histogram and calculate the width of each bar or class interval. Suppose you choose six bars.

[latex]9[/latex]; [latex]9[/latex]; [latex]9.5[/latex]; [latex]9.5[/latex]; [latex]10[/latex]; [latex]10[/latex]; [latex]10[/latex]; [latex]10[/latex]; [latex]10[/latex]; [latex]10[/latex]; [latex]10.5[/latex]; [latex]10.5[/latex]; [latex]10.5[/latex]; [latex]10.5[/latex]; [latex]10.5[/latex]; [latex]10.5[/latex]; [latex]10.5[/latex]; [latex]10.5[/latex]

[latex]11[/latex]; [latex]11[/latex]; [latex]11[/latex]; [latex]11[/latex]; [latex]11[/latex]; [latex]11[/latex]; [latex]11[/latex]; [latex]11[/latex]; [latex]11[/latex]; [latex]11[/latex]; [latex]11[/latex]; [latex]11[/latex]; [latex]11[/latex]; [latex]11.5[/latex]; [latex]11.5[/latex]; [latex]11.5[/latex]; [latex]11.5[/latex]; [latex]11.5[/latex]; [latex]11.5[/latex]; [latex]11.5[/latex]

[latex]12[/latex]; [latex]12[/latex]; [latex]12[/latex]; [latex]12[/latex]; [latex]12[/latex]; [latex]12[/latex]; [latex]12[/latex]; [latex]12.5[/latex]; [latex]12.5[/latex]; [latex]12.5[/latex]; [latex]12.5[/latex]; [latex]14[/latex]

Smallest value: [latex]9[/latex]

Largest value: [latex]14[/latex]

Convenient starting value: [latex]9 – 0.05 = 8.95[/latex]

Convenient ending value: [latex]14 + 0.05 = 14.05[/latex]

[latex]\displaystyle\frac{{{14.05}-{8.95}}}{{6}}={0.85}[/latex]

The calculations suggest using [latex]0.85[/latex] as the width of each bar or class interval. You can also use an interval with a width equal to one.

The following data are the number of books bought by 50 part-time college students at ABC College. The number of books is discrete data , since books are counted.

[latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]

[latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]

[latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]

[latex]4[/latex]; [latex]4[/latex]; [latex]4[/latex]; [latex]4[/latex]; [latex]4[/latex]; [latex]4[/latex]

[latex]5[/latex]; [latex]5[/latex]; [latex]5[/latex]; [latex]5[/latex]; [latex]5[/latex]

[latex]6[/latex]; [latex]6[/latex]

Eleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books.

Because the data are integers, subtract [latex]0.5[/latex] from [latex]1[/latex], the smallest data value and add [latex]0.5[/latex] to [latex]6[/latex], the largest data value. Then the starting point is [latex]0.5[/latex] and the ending value is [latex]6.5[/latex].

Next, calculate the width of each bar or class interval. If the data are discrete and there are not too many different values, a width that places the data values in the middle of the bar or class interval is the most convenient. Since the data consist of the numbers [latex]1[/latex], [latex]2[/latex], [latex]3[/latex], [latex]4[/latex], [latex]5[/latex], [latex]6[/latex], and the starting point is [latex]0.5[/latex], a width of one places the [latex]1[/latex] in the middle of the interval from [latex]0.5[/latex] to [latex]1.5[/latex], the [latex]2[/latex] in the middle of the interval from [latex]1.5[/latex] to [latex]2.5[/latex], the [latex]3[/latex] in the middle of the interval from [latex]2.5[/latex] to [latex]3.5[/latex], the [latex]4[/latex] in the middle of the interval from _______ to _______, the [latex]5[/latex] in the middle of the interval from _______ to _______, and the _______ in the middle of the interval from _______ to _______ .

Calculate the number of bars as follows:

[latex]\displaystyle\frac{{{6.5}-{0.5}}}{{\text{number of bars}}}={1}[/latex]

where [latex]1[/latex] is the width of a bar. Therefore, bars = [latex]6[/latex].

The following histogram displays the number of books on the [latex]x[/latex]-axis and the frequency on the [latex]y[/latex]-axis.

USING THE TI-83, 83+, 84, 84+ CALCULATOR

Create the histogram for Example 2.

• Press Y=. Press CLEAR to delete any equations.
• Press STAT 1:EDIT. If L1 has data in it, arrow up into the name L1, press CLEAR and then arrow down. If necessary, do the same for L2.
• Into L1, enter [latex]1[/latex], [latex]2[/latex], [latex]3[/latex], [latex]4[/latex], [latex]5[/latex], [latex]6[/latex].
• Into L2, enter [latex]11[/latex], [latex]10[/latex], [latex]16[/latex], [latex]6[/latex], [latex]5[/latex], [latex]2[/latex].
• Press WINDOW. Set Xmin = [latex].5[/latex], Xscl = [latex](6.5 – .5)/6[/latex], Ymin = [latex]–1[/latex], Ymax = [latex]20[/latex], Yscl = [latex]1[/latex], Xres = [latex]1[/latex].
• Press 2nd Y=. Start by pressing 4:Plotsoff ENTER.
• Press 2nd Y=. Press 1:Plot1. Press ENTER. Arrow down to TYPE. Arrow to the 3rd picture (histogram). Press ENTER.
• Arrow down to Xlist: Enter L1 (2nd 1). Arrow down to Freq. Enter L2 (2nd 2).
• Press GRAPH.
• Use the TRACE key and the arrow keys to examine the histogram.

The following data are the number of sports played by 50 student athletes. The number of sports is discrete data since sports are counted.

[latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]; [latex]1[/latex]

[latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]; [latex]2[/latex]

[latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]

[latex]20[/latex] student athletes play one sport. [latex]22[/latex] student athletes play two sports. Eight student athletes play three sports.

Fill in the blanks for the following sentence. Since the data consist of the numbers [latex]1[/latex], [latex]2[/latex], [latex]3[/latex], and the starting point is [latex]0.5[/latex], a width of one places the [latex]1[/latex] in the middle of the interval [latex]0.5[/latex] to _____, the [latex]2[/latex] in the middle of the interval from _____ to _____, and the [latex]3[/latex] in the middle of the interval from _____ to _____.

[latex]1.5[/latex] to [latex]2.5[/latex]

[latex]2.5[/latex] to [latex]3.5[/latex]

Using this data set, construct a histogram.

Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary, but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram.

The following data represent the number of employees at various restaurants in New York City. Using this data, create a histogram. [latex]22[/latex]; [latex]35[/latex]; [latex]15[/latex]; [latex]26[/latex]; [latex]40[/latex]; [latex]28[/latex]; [latex]18[/latex]; [latex]20[/latex]; [latex]25[/latex]; [latex]34[/latex]; [latex]39[/latex]; [latex]42[/latex]; [latex]24[/latex]; [latex]22[/latex]; [latex]19[/latex]; [latex]27[/latex]; [latex]22[/latex]; [latex]34[/latex]; [latex]40[/latex]; [latex]20[/latex]; [latex]38[/latex]; and [latex]28[/latex] Use [latex]10–19[/latex] as the first interval.

COLLABORATIVE EXERCISE

Count the money (bills and change) in your pocket or purse. Your instructor will record the amounts. As a class, construct a histogram displaying the data. Discuss how many intervals you think is appropriate. You may want to experiment with the number of intervals.

Frequency Polygons

Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons.

To construct a frequency polygon, first examine the data and decide on the number of intervals, or class intervals, to use on the [latex]x[/latex]-axis and [latex]y[/latex]-axis. After choosing the appropriate ranges, begin plotting the data points. After all the points are plotted, draw line segments to connect them.

The first label on the [latex]x[/latex]-axis is [latex]44.5[/latex]. This represents an interval extending from [latex]39.5[/latex] to [latex]49.5[/latex]. Since the lowest test score is [latex]54.5[/latex], this interval is used only to allow the graph to touch the [latex]x[/latex]-axis. The point labeled [latex]54.5[/latex] represents the next interval, or the first “real” interval from the table, and contains five scores. This reasoning is followed for each of the remaining intervals with the point [latex]104.5[/latex] representing the interval from [latex]99.5[/latex] to [latex]109.5[/latex]. Again, this interval contains no data and is only used so that the graph will touch the [latex]x[/latex]-axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.

Frequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets.

Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with this data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.

One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don‘t have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph.

Constructing a Time Series Graph

To construct a time series graph, we must look at both pieces of our paired data set . We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.

Uses of a Time Series Graph

Time series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot.

Concept Review

A histogram is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets. A frequency polygon can also be used when graphing large data sets with data points that repeat. The data usually goes on [latex]y[/latex]-axis with the frequency being graphed on the [latex]x[/latex]-axis. Time series graphs can be helpful when looking at large amounts of data for one variable over a period of time.

Data on annual homicides in Detroit, 1961–73, from Gunst & Mason’s book ‘Regression Analysis and its Application’, Marcel Dekker

“Timeline: Guide to the U.S. Presidents: Information on every president’s birthplace, political party, term of office, and more.” Scholastic, 2013. Available online at http://www.scholastic.com/teachers/article/timeline-guide-us-presidents (accessed April 3, 2013).

“Presidents.” Fact Monster. Pearson Education, 2007. Available online at http://www.factmonster.com/ipka/A0194030.html (accessed April 3, 2013).

“Food Security Statistics.” Food and Agriculture Organization of the United Nations. Available online at http://www.fao.org/economic/ess/ess-fs/en/ (accessed April 3, 2013).

“Consumer Price Index.” United States Department of Labor: Bureau of Labor Statistics. Available online at http://data.bls.gov/pdq/SurveyOutputServlet (accessed April 3, 2013).

“CO2 emissions (kt).” The World Bank, 2013. Available online at http://databank.worldbank.org/data/home.aspx (accessed April 3, 2013).

“Births Time Series Data.” General Register Office For Scotland, 2013. Available online at http://www.gro-scotland.gov.uk/statistics/theme/vital-events/births/time-series.html (accessed April 3, 2013).

“Demographics: Children under the age of 5 years underweight.” Indexmundi. Available online at http://www.indexmundi.com/g/r.aspx?t=50&v=2224&aml=en (accessed April 3, 2013).

Gunst, Richard, Robert Mason. Regression Analysis and Its Application: A Data-Oriented Approach . CRC Press: 1980.

“Overweight and Obesity: Adult Obesity Facts.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/obesity/data/adult.html (accessed September 13, 2013).

• OpenStax, Statistics, Histograms, Frequency Polygons, and Time Series Graphs. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:11/Introductory_Statistics . License : CC BY: Attribution
• Introductory Statistics . Authored by : Barbara Illowski, Susan Dean. Provided by : Open Stax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]
• Histograms. Authored by : Khan Academy. Located at : https://youtu.be/4eLJGG2Ad30 . License : All Rights Reserved . License Terms : Standard YouTube License

2.2 Histograms, Frequency Polygons, and Time Series Graphs

For most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets. A rule of thumb is to use a histogram when the data set consists of 100 values or more.

A histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data.

The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:

• f = frequency
• n = total number of data values (or the sum of the individual frequencies), and
• RF = relative frequency,

For example, if three students in an English class of 40 students received from 90% to 100%, then, --> f = 3, n = 40, and RF = f n f n = 3 40 3 40 = 0.075. 7.5% of the students received 90–100%. 90–100% are quantitative measures.

To construct a histogram , first decide how many bars or intervals , also called classes, represent the data. Many histograms consist of five to 15 bars or classes for clarity. The number of bars needs to be chosen. Choose a starting point for the first interval to be less than the smallest data value. A convenient starting point is a lower value carried out to one more decimal place than the value with the most decimal places. For example, if the value with the most decimal places is 6.1 and this is the smallest value, a convenient starting point is 6.05 (6.1 – 0.05 = 6.05). We say that 6.05 has more precision. If the value with the most decimal places is 2.23 and the lowest value is 1.5, a convenient starting point is 1.495 (1.5 – 0.005 = 1.495). If the value with the most decimal places is 3.234 and the lowest value is 1.0, a convenient starting point is 0.9995 (1.0 – 0.0005 = 0.9995). If all the data happen to be integers and the smallest value is two, then a convenient starting point is 1.5 (2 – 0.5 = 1.5). Also, when the starting point and other boundaries are carried to one additional decimal place, no data value will fall on a boundary. The next two examples go into detail about how to construct a histogram using continuous data and how to create a histogram using discrete data.

Example 2.7

The following data are the heights (in inches to the nearest half inch) of 100 semiprofessional soccer players. The heights are continuous data, since height is measured. 60; 60.5; 61; 61; 61.5 63.5; 63.5; 63.5 64; 64; 64; 64; 64; 64; 64; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5 66; 66; 66; 66; 66; 66; 66; 66; 66; 66; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5 68; 68; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69.5; 69.5; 69.5; 69.5; 69.5 70; 70; 70; 70; 70; 70; 70.5; 70.5; 70.5; 71; 71; 71 72; 72; 72; 72.5; 72.5; 73; 73.5 74

The smallest data value is 60. Since the data with the most decimal places has one decimal (for instance, 61.5), we want our starting point to have two decimal places. Since the numbers 0.5, 0.05, 0.005, etc. are convenient numbers, use 0.05 and subtract it from 60, the smallest value, for the convenient starting point.

60 – 0.05 = 59.95 which is more precise than, say, 61.5 by one decimal place. The starting point is, then, 59.95.

The largest value is 74, so 74 + 0.05 = 74.05 is the ending value.

Next, calculate the width of each bar or class interval. To calculate this width, subtract the starting point from the ending value and divide by the number of bars (you must choose the number of bars you desire). Suppose you choose eight bars.

We will round up to two and make each bar or class interval two units wide. Rounding up to two is one way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using 1.76 as the width would also work. A guideline that is followed by some for the number of bars or class intervals is to take the square root of the number of data values and then round to the nearest whole number, if necessary. For example, if there are 150 values of data, take the square root of 150 and round to 12 bars or intervals.

The boundaries are:

• 59.95 + 2 = 61.95
• 61.95 + 2 = 63.95
• 63.95 + 2 = 65.95
• 65.95 + 2 = 67.95
• 67.95 + 2 = 69.95
• 69.95 + 2 = 71.95
• 71.95 + 2 = 73.95
• 73.95 + 2 = 75.95

The heights 60 through 61.5 inches are in the interval 59.95–61.95. The heights that are 63.5 are in the interval 61.95–63.95. The heights that are 64 through 64.5 are in the interval 63.95–65.95. The heights 66 through 67.5 are in the interval 65.95–67.95. The heights 68 through 69.5 are in the interval 67.95–69.95. The heights 70 through 71 are in the interval 69.95–71.95. The heights 72 through 73.5 are in the interval 71.95–73.95. The height 74 is in the interval 73.95–75.95.

The following histogram displays the heights on the x -axis and relative frequency on the y -axis.

The following data are the shoe sizes of 50 students. The sizes are discrete data since shoe size is measured in whole and half units only. Construct a histogram and calculate the width of each bar or class interval. Suppose you choose six bars. 9; 9; 9.5; 9.5; 10; 10; 10; 10; 10; 10; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11.5; 11.5; 11.5; 11.5; 11.5; 11.5; 11.5 12; 12; 12; 12; 12; 12; 12; 12.5; 12.5; 12.5; 12.5; 14

Example 2.8

Create a histogram for the following data: the number of books bought by 50 part-time college students at ABC College. The number of books is discrete data , since books are counted. 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1 2; 2; 2; 2; 2; 2; 2; 2; 2; 2 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3 4; 4; 4; 4; 4; 4 5; 5; 5; 5; 5 6; 6

Eleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books.

Because the data are integers, subtract 0.5 from 1, the smallest data value and add 0.5 to 6, the largest data value. Then the starting point is 0.5 and the ending value is 6.5.

Next, calculate the width of each bar or class interval. If the data are discrete and there are not too many different values, a width that places the data values in the middle of the bar or class interval is the most convenient. Since the data consist of the numbers 1, 2, 3, 4, 5, 6, and the starting point is 0.5, a width of one places the 1 in the middle of the interval from 0.5 to 1.5, the 2 in the middle of the interval from 1.5 to 2.5, the 3 in the middle of the interval from 2.5 to 3.5, the 4 in the middle of the interval from _______ to _______, the 5 in the middle of the interval from _______ to _______, and the _______ in the middle of the interval from _______ to _______ .

Calculate the number of bars as follows:

where 1 is the width of a bar. Therefore, bars = 6.

The following histogram displays the number of books on the x -axis and the frequency on the y -axis.

Using the TI-83, 83+, 84, 84+ Calculator

Go to Appendix G NOTEs for the TI-83, 83+, 84, 84+ Calculators . There are calculator instructions for entering data and for creating a customized histogram. Create the histogram for Example 2.8 .

• Press Y=. Press CLEAR to delete any equations.
• Press STAT 1:EDIT . If L1 has data in it, arrow up into the name L1 , press CLEAR and then arrow down. If necessary, do the same for L2 .
• Into L1 , enter 1, 2, 3, 4, 5, 6.
• Into L2 , enter 11, 10, 16, 6, 5, 2.
• Press WINDOW. Set Xmin = .5 , Xmax = 6.5 , Xscl = (6.5 – .5)/6 , Ymin = –1 , Ymax = 20 , Yscl = 1 , Xres = 1 .
• Press 2 nd Y=. Start by pressing 4:Plotsoff ENTER.
• Press 2 nd Y=. Press 1:Plot1 . Press ENTER. Arrow down to TYPE. Arrow to the 3 rd picture (histogram). Press ENTER.
• Arrow down to Xlist: Enter L1 (2 nd 1). Arrow down to Freq. Enter L2 (2 nd 2).
• Press GRAPH.
• Use the TRACE key and the arrow keys to examine the histogram.

The following data are the number of sports played by 50 student athletes. The number of sports is discrete data since sports are counted.

1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2 3; 3; 3; 3; 3; 3; 3; 3 20 student athletes play one sport. 22 student athletes play two sports. Eight student athletes play three sports.

Fill in the blanks for the following sentence. Since the data consist of the numbers 1, 2, 3, and the starting point is 0.5, a width of one places the 1 in the middle of the interval 0.5 to _____, the 2 in the middle of the interval from _____ to _____, and the 3 in the middle of the interval from _____ to _____.

Example 2.9

Using this data set, construct a histogram.

Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary, but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram.

The following data represent the number of employees at various restaurants in New York City. Using this data, create a histogram.

22 ; 35 ; 15 ; 26 ; 40 ; 28 ; 18 ; 20 ; 25 ; 34 ; 39 ; 42 ; 24 ; 22 ; 19 ; 27 ; 22 ; 34 ; 40 ; 20 ; 38 ; and 28 Use 10–19 as the first interval.

Collaborative Exercise

Count the money (bills and change) in your pocket or purse. Your instructor will record the amounts. As a class, construct a histogram displaying the data. Discuss how many intervals you think is appropriate. You may want to experiment with the number of intervals.

Frequency Polygons

Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons.

To construct a frequency polygon, first examine the data and decide on the number of intervals, or class intervals, to use on the x -axis and y -axis. After choosing the appropriate ranges, begin plotting the data points. After all the points are plotted, draw line segments to connect them.

Example 2.10

A frequency polygon was constructed from the frequency table below.

The first label on the x -axis is 44.5. This represents an interval extending from 39.5 to 49.5. Since the lowest test score is 54.5, this interval is used only to allow the graph to touch the x -axis. The point labeled 54.5 represents the next interval, or the first “real” interval from the table, and contains five scores. This reasoning is followed for each of the remaining intervals with the point 104.5 representing the interval from 99.5 to 109.5. Again, this interval contains no data and is only used so that the graph will touch the x -axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.

Try It 2.10

Construct a frequency polygon of U.S. Presidents’ ages at inauguration shown in Table 2.15 .

Frequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets.

Example 2.11

We will construct an overlay frequency polygon comparing the scores from Example 2.10 with the students’ final numeric grade.

Try It 2.11

We will construct an overlay frequency polygon comparing the scores from Example 2.11 with the students’ final test scores in algebra.

Constructing a Time Series Graph

Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with this data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.

One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don‘t have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph.

To construct a time series graph, we must look at both pieces of our paired data set . We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.

Example 2.12

The following data shows the Annual Consumer Price Index, each month, for ten years. Construct a time series graph for the Annual Consumer Price Index data only.

Try It 2.12

The following table is a portion of a data set from www.worldbank.org. Use the table to construct a time series graph for CO 2 emissions for the United States.

Uses of a Time Series Graph

Time series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot.

How NOT to Lie with Statistics

It is important to remember that the very reason we develop a variety of methods to present data is to develop insights into the subject of what the observations represent. We want to get a "sense" of the data. Are the observations all very much alike or are they spread across a wide range of values, are they bunched at one end of the spectrum or are they distributed evenly and so on. We are trying to get a visual picture of the numerical data. Shortly we will develop formal mathematical measures of the data, but our visual graphical presentation can say much. It can, unfortunately, also say much that is distracting, confusing and simply wrong in terms of the impression the visual leaves. Many years ago Darrell Huff wrote the book How to Lie with Statistics . It has been through 25 plus printings and sold more than one and one-half million copies. His perspective was a harsh one and used many actual examples that were designed to mislead. He wanted to make people aware of such deception, but perhaps more importantly to educate so that others do not make the same errors inadvertently.

Again, the goal is to enlighten with visuals that tell the story of the data. Pie charts have a number of common problems when used to convey the message of the data. Too many pieces of the pie overwhelm the reader. More than perhaps five or six categories ought to give an idea of the relative importance of each piece. This is after all the goal of a pie chart, what subset matters most relative to the others. If there are more components than this then perhaps an alternative approach would be better or perhaps some can be consolidated into an "other" category. Pie charts cannot show changes over time, although we see this attempted all too often. In federal, state, and city finance documents pie charts are often presented to show the components of revenue available to the governing body for appropriation: income tax, sales tax motor vehicle taxes and so on. In and of itself this is interesting information and can be nicely done with a pie chart. The error occurs when two years are set side-by-side. Because the total revenues change year to year, but the size of the pie is fixed, no real information is provided and the relative size of each piece of the pie cannot be meaningfully compared.

Histograms can be very helpful in understanding the data. Properly presented, they can be a quick visual way to present probabilities of different categories by the simple visual of comparing relative areas in each category. Here the error, purposeful or not, is to vary the width of the categories. This of course makes comparison to the other categories impossible. It does embellish the importance of the category with the expanded width because it has a greater area, inappropriately, and thus visually "says" that that category has a higher probability of occurrence.

Time series graphs perhaps are the most abused. A plot of some variable across time should never be presented on axes that change part way across the page either in the vertical or horizontal dimension. Perhaps the time frame is changed from years to months. Perhaps this is to save space or because monthly data was not available for early years. In either case this confounds the presentation and destroys any value of the graph. If this is not done to purposefully confuse the reader, then it certainly is either lazy or sloppy work.

Changing the units of measurement of the axis can smooth out a drop or accentuate one. If you want to show large changes, then measure the variable in small units, penny rather than thousands of dollars. And of course to continue the fraud, be sure that the axis does not begin at zero, zero. If it begins at zero, zero, then it becomes apparent that the axis has been manipulated.

Perhaps you have a client that is concerned with the volatility of the portfolio you manage. An easy way to present the data is to use long time periods on the time series graph. Use months or better, quarters rather than daily or weekly data. If that doesn't get the volatility down then spread the time axis relative to the rate of return or portfolio valuation axis. If you want to show "quick" dramatic growth, then shrink the time axis. Any positive growth will show visually "high" growth rates. Do note that if the growth is negative then this trick will show the portfolio is collapsing at a dramatic rate.

Again, the goal of descriptive statistics is to convey meaningful visuals that tell the story of the data. Purposeful manipulation is fraud and unethical at the worst, but even at its best, making these type of errors will lead to confusion on the part of the analysis.

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• Book title: Introductory Statistics 2e
• Publication date: Dec 13, 2023
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• Book URL: https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
• Section URL: https://openstax.org/books/introductory-statistics-2e/pages/2-2-histograms-frequency-polygons-and-time-series-graphs

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2.4.3: Frequency Polygons

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Frequency Polygons

Another type of graph that can be drawn to represent the same set of data as a histogram represents is a  frequency polygon . A  frequency polygon  is a graph constructed by using lines to join the midpoints of each interval, or bin. The heights of the points represent the frequencies. A frequency polygon can be created from the histogram or by calculating the midpoints of the bins from the frequency distribution table. The  midpoint  of a bin is calculated by adding the upper and lower boundary values of the bin and dividing the sum by 2.

Constructing Frequency Polygons 

1. The following histogram represents the marks made by 40 students on a math 10 test.

Use the histogram to construct a frequency polygon to represent the data.

USGS -  http://earthquake.usgs.gov/earthquakes/eqarchives/year/graphs.php ; http://pixabay.com/en/server-computer-case-controller-40240/  - CC BY-NC

USGS -  http://earthquake.usgs.gov/earthquakes/eqarchives/year/graphs.php ; http://pixabay.com/en/server-computer-case-controller-40240/  - CC BY-NC

There is no data value greater than 0 and less than 20. The jagged line that is inserted on the x-axis is used to represent this fact. The area under the frequency polygon is the same as the area under the histogram and is, therefore, equal to the frequency values that would be displayed in a distribution table. The frequency polygon also shows the shape of the distribution of the data, and in this case, it resembles a bell curve.

2. The following distribution table represents the number of miles run by 20 randomly selected runners during a recent road race:

Using this table, construct a frequency polygon.

Step 1:  Calculate the midpoint of each bin by adding the 2 numbers of the interval and dividing the sum by 2.

Step 2:  Plot the midpoints on a grid, making sure to number the x-axis with a scale that will include the bin sizes. Join the plotted midpoints with lines.

A frequency polygon usually extends 1 unit below the smallest bin value and 1 unit beyond the greatest bin value. This extension gives the frequency polygon an appearance of having a starting point and an ending point, which provides a view of the distribution of data. If the data set were very large so that the number of bins had to be increased and the bin size decreased, the frequency polygon would appear as a smooth curve.

3. The histogram shown below represents the minutes spent practicing per day by a professional violinist for each of the last 80 days:

USGS -  http://earthquake.usgs.gov/earthquakes/eqarchives/year/graphs.php  - CC BY-NC

Construct a frequency polygon for the data.

From the histogram, the following distribution table can be constructed:

Now you can calculate the midpoint of each bin by adding the 2 numbers of the interval and dividing the sum by 2.

Finally, you can plot the points on a grid and join the points with lines.

USGS -  http://earthquake.usgs.gov/earthquakes/eqarchives/year/graphs.php  - CC BY-NC

Examples 

The frequency polygon below represents the heights, in inches, of a group of professional basketball players. Use the frequency polygon to answer the following questions:

How many players' heights were measured?

To find the number of players whose heights were measured, just add up all of the frequencies. This can be done as follows:

6+10+22+24+16+20+2=100

This means that 100 players' heights were measured.

What was the bin size of the histogram on which the frequency polygon is based? 

The bin size of the histogram on which the frequency polygon is based is 3. This is apparent from the fact that the points that were connected to create the frequency polygon are 3 inches apart on the horizontal axis.

What range of heights was most common among the basketball players?

Remember that the  x -coordinate of each of the points that were connected to create the frequency polygon is the midpoint of one of the bins of the corresponding histogram. It's obvious from the frequency polygon that 78 inches has the greatest frequency, but this doesn't necessarily mean that height most common among the basketball players was 78 inches. All it means is that the range of heights that was most common among the basketball players was 76.5 inches to 79.5 inches.

What range of heights was least common among the basketball players?

For the same reason that the range of heights that was most common among the basketball players was 76.5 inches to 79.5 inches, the range of heights that was least common among the basketball players was 85.5 inches to 88.5 inches. Remember that the points at 66 inches and 90 inches along the horizontal axis were just added to give the frequency polygon the appearance of having a starting point and an ending point.

What percentage of the basketball players measured had a height of less than 76.5

The point at 75 inches along the horizontal axis represents the bin [73.5, 76.5). Therefore, to find the number of basketball players measured who had a height of less than 76.5 inches, add the frequencies of the first 3 bins as follows:

This means that 38 players had a height of less than 76.5 inches, so the percentage of the basketball players measured who had a height of less than 76.5 inches is 38/100=0.38=38%.

Review 

• stem-and-leaf
• frequency polygon
• The following frequency polygon represents the weights of players who all participated in the same sport. Use the polygon to answer the following questions:

a. How many players played the sport? b. What was the most common weight for the players? c. What sport do you think the players may have been playing? d. What do the weights of 55 kg and 105 kg represent? e. What 2 weights have no recorded players weighing those amounts?

Suppose the points (10, 0), (20, 4), (30, 12), (40, 18), (50, 9), (60, 7), (70, 0) were connected to form a frequency polygon. Use this information to answer the following questions:

• What was the bin size of the histogram on which the frequency polygon is based?
• Which bin had the highest frequency?
• Which bin had the lowest frequency?
• What percentage of the data had a value below 55?

Use the histogram shown below to answer the following questions:

• How many points would be connected to form the corresponding frequency polygon? (Include the points added to give the frequency polygon the appearance of having a starting point and an ending point.)
• List the points.
• Create the frequency polygon.

Review (Answers)

To view the Review answers, open this  PDF file  and look for section 7.10. 

Additional Resources

PLIX: Play, Learn, Interact, eXplore - Constructing a Frequency Polygon

Video: Frequency Polygons Principles

Activities: Frequency Polygons Discussion Questions

Study Aids: Presenting Univariate Data

Lesson Plans: Frequency Polygons Lesson Plan

Practice: Frequency Polygons

Real World: Frequency Polygons

• Frequency Polygon

The relevance of presentation of data in the pictorial or graphical form is immense. Frequency polygons give an idea about the shape of the data and the trends that a particular data set follows. Let us learn the step by step process of drawing a frequency polygon, with or without a histogram .

Suggested Videos

Presentation of data.

The key objective of statistics is to collect and organize data . One of the basics of data organization comes from presentation of data in a recognizable form so that it can be interpreted easily. You can organize data in the form of tables or you can present it pictorially.

Pictorial representation of data takes the form of bar charts , pie charts, histograms or frequency polygons. The benefit of this is that data in the visual form is easy to understand in one glance.

Download the Cheat Sheet of Statistics by clicking on the button below

A frequency polygon is a graphical form of representation of data. It is used to depict the shape of the data and to depict trends. It is usually drawn with the help of a histogram but can be drawn without it as well. A histogram is a series of rectangular bars with no space between them and is used to represent frequency distributions .

Steps to Draw a Frequency Polygon

• Mark the class intervals for each class on the horizontal axis. We will plot the frequency on the vertical axis.
• Calculate the classmark for each class interval. The formula for class mark is:

Classmark = (Upper limit + Lower limit) / 2

• Mark all the class marks on the horizontal axis. It is also known as the mid-value of every class.
• Corresponding to each class mark, plot the frequency as given to you. The height always depicts the frequency. Make sure that the frequency is plotted against the class mark and not the upper or lower limit of any class.
• Join all the plotted points using a line segment. The curve obtained will be kinked.
• This resulting curve is called the frequency polygon.

Note that the above method is used to draw a frequency polygon without drawing a histogram . You can also draw a histogram first by drawing rectangular bars against the given class intervals. After this, you must join the midpoints of the bars to obtain the frequency polygon. Remember that the bars will have no spaces between them in a histogram.

Solved Example for You

Question 1: Construct a frequency polygon using the data given below:

Answer: We first need to calculate the cumulate frequency from the frequency given.

We now start by plotting the class marks such as 54.5, 64.5, 74.5 and so on till 94.5. Note that we will also plot the previous and next class marks to start and end the polygon, i.e. we plot 44.5 and 104.5 as well.

Then, the frequencies corresponding to the class marks are plotted against each class mark. Like you can see below, this makes sense as the frequency for class marks 44.5 and 104.5 are zero and touching the x-axis. These plot points are used only to give a closed shape to the polygon. The polygon looks like this:

(Source: Stats LibreTexts)

Question 2: Explain how to construct a frequency polygon?

Answer: In order to create a frequency polygon, one must follow these steps:

• Creation of a histogram.
• Finding the midpoints for each bar that exists on the histogram.
• Placing a point on the origin of the histogram and its end.
• Connection of the points.

Question 3: Explain what is a frequency polygon?

Answer: The frequency histogram has the similarity to a column graph without the presence of spaces between columns. The frequency polygon happens to be a special line graph whose use takes place in statistics. One can draw these graphs either separately or combined. One can make use of the information that is available in a frequency distribution table for drawings of these graphs.

Question 4: Differentiate between a frequency polygon and frequency curve?

Answer: The major difference between a frequency polygon and frequency curve is that the drawing of a frequency polygon by joining points by a straight line while the drawing of a frequency curve takes place by a smooth hand.

Question 5: What can you learn from a frequency polygon?

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Frequency Polygons

A frequency polygon is almost identical to a histogram, which is used to compare sets of data or to display a cumulative frequency distribution. It uses a line graph to represent quantitative data.

Statistics deals with the collection of data and information for a particular purpose. The tabulation of each run for each ball in cricket gives the statistics of the game. Tables, graphs, pie-charts, bar graphs, histograms, polygons etc. are used to represent statistical data pictorially.

Frequency polygons are a visually substantial method of representing quantitative data and its frequencies. Let us discuss how to represent a frequency polygon.

Steps to Draw Frequency Polygon

To draw frequency polygons, first we need to draw histogram and then follow the below steps:

• Step 1- Choose the class interval and mark the values on the horizontal axes
• Step 2- Mark the mid value of each interval on the horizontal axes.
• Step 3- Mark the frequency of the class on the vertical axes.
• Step 4- Corresponding to the frequency of each class interval, mark a point at the height in the middle of the class interval
• Step 5- Connect these points using the line segment.
• Step 6- The obtained representation is a frequency polygon.

Let us consider an example to understand this in a better way.

Example 1: In a batch of 400 students, the height of students is given in the following table. Represent it through a frequency polygon.

Solution:  Following steps are to be followed to construct a histogram from the given data:

• The heights are represented on the horizontal axes on a suitable scale as shown.
• The number of students is represented on the vertical axes on a suitable scale as shown.
• Now rectangular bars of widths equal to the class- size and the length of the bars corresponding to a frequency of the class interval is drawn.

ABCDEF represents the given data graphically in form of frequency polygon as:

Frequency polygons can also be drawn independently without drawing histograms. For this, the midpoints of the class intervals known as class marks are used to plot the points.

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4.2: Frequency Distributions and Statistical Graphs

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Once we have collected data, then we need to start analyzing the data. One way to display and summarize data is to use statistical graphing techniques. The type of graph we use depends on the type of data collected. Qualitative data use graphs like bar graphs and pie graphs. Quantitative data use graphs such as histograms and frequency polygons.

In order to create graphs, we must first organize and create a summary of the individual data values in the form of a frequency distribution . A frequency distribution is a listing all of the data values (or groups of data values) and how often those data values occur.

Frequency and Frequency Distributions

Frequency is the number of times a data value or groups of data values (called classes ) occur in a data set.

A frequency distribution is a listing of each data value or class of data values along with their frequencies.

Relative frequency is the frequency divided by \(n\), the size of the sample. This gives the proportion of the entire data set represented by each value or class. Relative frequencies are expressed as fractions, decimals, or percentages.

A relative frequency distribution is a listing of each data value or class of data values along with their relative frequencies.

The method of creating a frequency distribution depends on whether we are working with qualitative data or quantitative data . We will now look at how to create each type of frequency distribution according to the type of data and the graphs that go with them.

Organizing Qualitative Data

Qualitative data are pieces of information that allow us to classify the items under investigation into various categories. We usually begin working with qualitative data by giving the frequency distribution as a frequency table.

Frequency Table

A frequency table is a table with two columns. One column lists the categories, and another column gives the frequencies with which the items in the categories occur (how many data fit into each category).

An insurance company determines vehicle insurance premiums based on known risk factors. If a person is considered a higher risk, their premiums will be higher. One potential factor is the color of your car. The insurance company believes that people with some colors of cars are more likely to ve involved in accidents. To research this, the insurance company examines police reports for recent total-loss collisions. The data is summarized in the frequency table below.

\(\begin{array}{|c|c|} \hline \textbf { Color } & \textbf { Frequency } \\ \hline \text { Blue } & 25 \\ \hline \text { Green } & 52 \\ \hline \text { Red } & 41 \\ \hline \text { White } & 36 \\ \hline \text { Black } & 39 \\ \hline \text { Grey } & 23 \\ \hline \end{array}\)

Graphing Qualitative Data in Bar Graphs and Pie Charts

Once we have organized and summarized qualitative data into a frequency table, we are ready to graph the data. There are various ways to visualize qualitative data. In this section we will consider two common graphs: bar graphs and pie graphs .

A bar graph is displays a bar for each category. The length of each bar indicates the frequency of that category.

To construct a bar graph, we need to draw a vertical axis and a horizontal axis. The vertical direction has a scale and measures the frequency of each category. The horizontal axis has no scale in this instance but lists the categories. The construction of a bar chart is most easily described by use of an example.

Using the car color data from Example 1, note the highest frequency was 52, so the vertical axis needs to go from 0 to 52. We might as well use 0 to 55 so that we can put a hash mark every 5 units:

You should notice a few things about the correct construction of this bar graph.

• The height of each bar is determined by the frequency of the corresponding color.
• Both axes are labeled clearly.
• The bars do not touch and they are the same width.

The horizontal grid lines are a nice touch, but not necessary. In practice, you will find it useful to draw bar graphs on graph paper so the grid lines will already be in place or use technology to create the graph. Instead of grid lines, we might also list the frequencies at the top of each bar, like this:

In a survey, adults were asked whether they personally worried about a variety of environmental concerns. The numbers (out of 1012 surveyed) who indicated that they worried “a great deal” about some selected concerns are summarized below.

\(\begin{array}{|c|c|} \hline \textbf { Environmental Issue } & \textbf { Frequency } \\ \hline \text { Pollution of drinking water } & 597 \\ \hline \text { Contamination of soil and water by toxic waste } & 526 \\ \hline \text { Air pollution } & 455 \\ \hline \text { Global warming } & 354 \\ \hline \end{array}\)

Display the data using a bar graph.

Try it Now 1

A questionnaire on the makes of people's vehicles showed the following responses from 30 participants. Construct a frequency table and a bar graph to represent the data. ( F = Ford, H = Honda, V = Volkswagen, M = Mazda)

F M M M V M F M F V H H F V F H H F M M V H M V V F V H M F

A class was asked for their favorite soft drink with the following results:

• Create a frequency distribution table for the data.
• Create a relative frequency distribution table for the data.
• Draw a bar graph of the frequency distribution.
• Draw a bar graph of the relative frequency distribution.
• To make a frequency distribution table, list each drink type and and then count how often each drink occurs in the data. Notice that Coke happens 9 times in the data set, Pepsi happens 10 times, and so on.
• To make a relative frequency distribution table, use the previous results and divide each frequency by 33, which is the total number of data responses.
• Along the horizontal axis you place the drinks. Space these apart equally, and allow space to draw bars above the axis. The vertical axis shows the frequencies. Make sure you create a scale along that axis in which all of the frequencies will fit. Notice that the highest frequency is 10, so you want to make sure the vertical axis goes to at least 10, and you may want to count by two for every tick mark. Here is what the graph looks like using Excel.

• A bar graph for the relative frequency distribution is similar to the bar graph for the frequency distribution except that the relative frequencies are used along the vertical axis instead. Notice that the graph does not actually change except the numbers on the vertical scale.

Let's use the last example to introduce another way of visualizing data – a pie chart also known as circle graph .

A pie chart is a graph where the "pie" represents the entire sample and the "slices" represent the categories or classes. The size of the slice of the pie corresponds to the relative frequency for that category.

To find the angle that each “slice” takes up, multiply the relative frequency of that slice by 360°.

Note: Theoretically, the percentages of all slices of a pie chart must add to 100%. In practice, the percentages may add to be slightly more or less than 100% if percentages are rounded.

To draw a pie chart, multiply the relative frequencies of each drink by 360°. Then, use a protractor to mark off the corresponding angle in a circle. Usually it is easier to use Excel or some other spreadsheet program to draw the graph.

The pie graph from Excel is shown below.

Try it Now 2

The Red Cross Blood Donor Clinic had a very successful morning collecting blood donations. Within 3 hours, many people had made donations. The table shows the frequency distribution of the blood types of the donations. Construct a pie chart to display the relative frequency distribution.

Organizing Quantitative Data

Quantitative is data that is the result of counting or measuring some aspect of items under investigation. For this reason, this type of data is also known as numerical data. Quantitative data can also be summarized in a table to show its frequency distribution.

A teacher records scores on a 20-point quiz for the 30 students in his class. The scores are

19 20 18 18 17 18 19 17 20 18 20 16 20 15 17 12 18 19 18 19 17 20 18 16 15 18 20 5 0 0

These scores could be summarized into a frequency table by counting how many times each particular data value occurs.

\(\begin{array}{|c|c|} \hline \textbf { Score } & \textbf { Frequency } \\ \hline 0 & 2 \\ \hline 5 & 1 \\ \hline 12 & 1 \\ \hline 15 & 2 \\ \hline 16 & 2 \\ \hline 17 & 4 \\ \hline 18 & 8 \\ \hline 19 & 4 \\ \hline 20 & 6 \\ \hline \end{array}\)

In the previous example, the table listed every different data value that occurred and how often each value occurred. We call this type of frequency distribution presentation ungrouped . Sometimes it is helpful to group the data into classes to observe information about the distribution of data that otherwise wouldn't be noticeable. This is particularly true if there are many different values or each value only occurs once. You can think about classes as "bins" that we create to sort the data. When we group the data into classes, we call this type of frequency distribution presentation grouped .

When data are grouped, the following guidelines about the classes should be followed

• Classes should have the same width.
• Classes should not overlap.
• Each piece of data should belong to only one class.

Let's use the data from the previous example to create a grouped frequency distribution.

Create a grouped frequency table in two ways:

• with classes of width 5 beginning at a score of 0, and
• with classes of width 6 beginning at a score of 0.
• The first class contains the scores 0, 1, 2, 3, and 4 -- if any occur. Likewise, the second class will contain scores 5, 6, 7, 8, and 9 -- if any occur. This pattern continues until classes are no longer needed.

The first two columns of the table shows the classes and the frequency of the data in each class.

In the first column, the numbers 0, 4, 10, 15, and 20 are called the lower class limits and the numbers 4, 9, 14, 19, and 24 are the upper class limits. You can see these limits increase by 5. The class width can be determined as the difference between any two consecutive lower or upper class limits. The class mark is the midpoint of the class and is determined by averaging the lower and upper limits of the class. The class marks are shown in the third column of the table.

The modal class of a frequency distribution is the class with the highest frequency. Here the modal class is 15-19 with a frequency of 20 students. This grouping of the data allows us to more clearly see the grade distribution. Always be sure that the sum of the frequencies is the number of data values.

When the data are grouped using this structure, the modal class is 18–23.

Try it Now 3

The data below indicates number of children in a sample of 16 families:

2 1 2 1 2 5 5 3 2 3 5 2 5 2 2 1

• Create a non-grouped frequency table for the data.
• Create a grouped frequency table with first class 0-2. Identify the class width, the class mark for each class, and the modal class.

There is a "sort feature" on the TI calculator that sorts data in ascending or descending order for you. This makes organizing data and counting frequencies much easier. The steps for entering data and sorting it is shown here for the data presented in Try it 3 .

Let's consider the reverse situation when we have a frequency table with grouped data and determine information about the original data. This scenario is important because you will often see grouped data due to data storage capacities.

Answer the questions using the frequency table.

• What is the total number of data values in this data distribution?

Adding the frequencies of each class, we have \(4 + 7 + 1 + 0 +3 +5 = 20\).

• What class width is used to group the data?

Subtract any two consecutive lower class limits or any two consecutive upper class limits. For example, \(16 – 9 = 7\).

• What is the class mark of the second class ?

The class mark is the midpoint of the class. Average the lower and upper class limit: \(\frac{16+22}{2} = 19\).

• What is the modal class?

The class with the highest frequency is 16-22.

• If an additional class were added to the end of the table, what would be the upper and lower class limits?

Add the class width 7 to the last lower and upper class limits to get 51-57.

Graphing Quantitative Data in Histograms and Frequency Polygons

A histogram is a statistical graph commonly used to visualize frequency distributions of quantitative data. A histogram is like a bar graph, but where the horizontal axis is a number line.

A histogram is a graph with observed values or classes of values along the horizontal axis and frequencies along the vertical axis. A bar with a height equal to the frequency (or relative frequency) is built above each observed value or class.

In a histogram, classes may be identified by their class marks (midpoints of the classes) or by their class limits. The horizontal scale may or may not begin at 0, and but the vertical scale should always start at zero. The bars generally touch in a histogram - unless the frequency is 0 for a particular data value or class of values.

Let's illustrate how a histogram is constructed with the following example.

Each member of a class is asked how many plastic beverage bottles they use and discard in a week. Suppose the following (hypothetical) data are collected.

First, we organize the data by grouping it and presenting it in a frequency table. The classes have width 2 and begin at 1.

Next, we draw a bar for each class so that its height represents the frequency of students using those numbers of bottles. We label the midpoints of each bar with the class marks along the horizontal axis.

Graphing data can get tedious and complicated, especially if there are lots of data to organize. Excel and other software can easily make graphs. So can a TI graphing calculator. The steps to creating a histogram for these data is given below.

Suppose that we have collected weights from 100 male subjects as part of a nutrition study. For our weight data, we have values ranging from a low of 121 pounds to a high of 263 pounds, giving a total span of 263-121 = 142. We could create 7 intervals with a width of around 20, 14 intervals with a width of around 10, or somewhere in between. Often time we have to experiment with a few possibilities to find something that represents the data well. Let us try using an interval width of 15. We could start at 121, or at 120 since it is a nice round number.

\(\begin{array}{|c|c|} \hline \textbf { Interval } & \textbf { Frequency } \\ \hline 120-134 & 4 \\ \hline 135-149 & 14 \\ \hline 150-164 & 16 \\ \hline 165-179 & 28 \\ \hline 180-194 & 12 \\ \hline 195-209 & 8 \\ \hline 210-224 & 7 \\ \hline 225-239 & 6 \\ \hline 240-254 & 2 \\ \hline 255-269 & 3 \\ \hline \end{array}\)

A histogram of this data would look like

You can see the modal class is 165-179. You can also conclude there is a higher frequency of males in the lower part of the distribution of weights because the bars are taller there.

Try it Now 4

Create a histogram for the data given in Example 5 using the frequency table of ungrouped data.

Another way to visualize frequency distribution data is to construct a frequency polygon .

Frequency Polygon

An alternative representation of a histogram is a frequency polygon . A frequency polygon starts like a histogram, but instead of drawing a bar, a point is placed at the midpoint of each interval at a height equal to the frequency. Typically, the points are connected with straight lines to emphasize the shape of the data distribution.

The following example illustrates the relationship between a histogram and a frequency polygon for the same data.

Ms. Winter made a histogram and frequency polygon of the science test scores from 5 th period.

From either the histogram or the frequency polygon, we can see the class width is 10 points. We can also see that the modal class is 80-89. Finally, you can conclude that there is a larger frequency of students who scored high on the test than low on the test because the bars of the histogram and peak on the frequency polygon are taller on the right side of the horizontal axis.

The data below came from a task in which the goal is to move a computer mouse to a target on the screen as fast as possible. On 20 of the trials, the target was a small rectangle; on the other 20, the target was a large rectangle. Time to reach the target was recorded on each trial.

\(\begin{array}{|c|c|c|} \hline \begin{array}{c} \textbf { Interval } \\ \textbf { (milliseconds) } \end{array} & \begin{array}{c} \textbf { Frequency } \\ \textbf { small target } \end{array} & \begin{array}{c} \textbf { Frequency } \\ \textbf { large target } \end{array} \\ \hline 300-399 & 0 & 0 \\ \hline 400-499 & 1 & 5 \\ \hline 500-599 & 3 & 10 \\ \hline 600-699 & 6 & 5 \\ \hline 700-799 & 5 & 0 \\ \hline 800-899 & 4 & 0 \\ \hline 900-999 & 0 & 0 \\ \hline 1000-1099 & 1 & 0 \\ \hline 1100-1199 & 0 & 0 \\ \hline \end{array}\)

One option to represent this data would be a comparative histogram or bar chart, in which bars for the small target group and large target group are placed next to each other.

A pair of frequency polygons in the same graph for the same two sets of data makes it easier to see that reaction times were generally shorter for the larger target, and that the reaction times for the smaller target were more spread out.

In the next section, we will begin to analyze and describe data distributions numerically rather than graphically.

Data Presentation: Graphs, Frequency Tables and Histograms

• First Online: 17 August 2016

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Illustrations are generally not used enough as a tool for analysing data and it is often the case that, on their own, large batches of raw data can appear overwhelming to the observer. If a body of information is presented using a simple diagram or graph then it is, as a rule, more readily understandable. In general diagrams are much easier on the eye, being more visually attractive. They are also easier on the brain, in that they are less difficult to comprehend at a glance. Another advantage is that they may also reveal aspects of data that might otherwise be overlooked. Their disadvantage lies in the fact that they are not always precise and often don’t allow for further statistical analysis.

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Özdemir, D. (2016). Data Presentation: Graphs, Frequency Tables and Histograms. In: Applied Statistics for Economics and Business. Springer, Cham. https://doi.org/10.1007/978-3-319-26497-4_2

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Chapter 2.3: Histograms, Frequency Polygons, and Time Series Graphs

For most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets. A rule of thumb is to use a histogram when the data set consists of 100 values or more.

A histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data.

The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:

• f = frequency
• n = total number of data values (or the sum of the individual frequencies), and
• RF = relative frequency,

For example, if three students in Mr. Ahab’s English class of 40 students received from 90% to 100%, then,

To construct a histogram , first decide how many bars or intervals , also called classes, represent the data. Many histograms consist of five to 15 bars or classes for clarity. The number of bars needs to be chosen. Choose a starting point for the first interval to be less than the smallest data value. A convenient starting point is a lower value carried out to one more decimal place than the value with the most decimal places. For example, if the value with the most decimal places is 6.1 and this is the smallest value, a convenient starting point is 6.05 (6.1 – 0.05 = 6.05). We say that 6.05 has more precision. If the value with the most decimal places is 2.23 and the lowest value is 1.5, a convenient starting point is 1.495 (1.5 – 0.005 = 1.495). If the value with the most decimal places is 3.234 and the lowest value is 1.0, a convenient starting point is 0.9995 (1.0 – 0.0005 = 0.9995). If all the data happen to be integers and the smallest value is two, then a convenient starting point is 1.5 (2 – 0.5 = 1.5). Also, when the starting point and other boundaries are carried to one additional decimal place, no data value will fall on a boundary. The next two examples go into detail about how to construct a histogram using continuous data and how to create a histogram using discrete data.

The following data are the heights (in inches to the nearest half inch) of 100 male semiprofessional soccer players. The heights are continuous data, since height is measured. 60;  60.5;  61;  61;  61.5 63.5;  63.5;  63.5 64;  64;  64;  64;  64;  64;  64;  64.5;  64.5;  64.5;  64.5;  64.5;  64.5;  64.5;  64.5 66;  66;  66;  66;  66;  66;  66;  66;  66;  66;  66.5;  66.5;  66.5;  66.5;  66.5;  66.5;  66.5;  66.5;  66.5;  66.5;  66.5;  67;  67;  67;  67;  67;  67;  67;  67;  67;  67;  67;  67;  67.5;  67.5;  67.5;  67.5;  67.5;  67.5;  67.5 68;  68;  69;  69;  69;  69;  69;  69;  69;  69;  69;  69;  69.5;  69.5;  69.5;  69.5;  69.5 70;  70;  70;  70;  70;  70;  70.5;  70.5;  70.5;  71;  71;  71 72;  72;  72;  72.5;  72.5;  73;  73.5 74

The smallest data value is 60. Since the data with the most decimal places has one decimal (for instance, 61.5), we want our starting point to have two decimal places. Since the numbers 0.5, 0.05, 0.005, etc. are convenient numbers, use 0.05 and subtract it from 60, the smallest value, for the convenient starting point.

60 – 0.05 = 59.95 which is more precise than, say, 61.5 by one decimal place. The starting point is, then, 59.95.

The largest value is 74, so 74 + 0.05 = 74.05 is the ending value.

Next, calculate the width of each bar or class interval. To calculate this width, subtract the starting point from the ending value and divide by the number of bars (you must choose the number of bars you desire). Suppose you choose eight bars.

We will round up to two and make each bar or class interval two units wide. Rounding up to two is one way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using 1.76 as the width would also work. A guideline that is followed by some for the number of bars or class intervals is to take the square root of the number of data values and then round to the nearest whole number, if necessary. For example, if there are 150 values of data, take the square root of 150 and round to 12 bars or intervals.

The boundaries are:

• 59.95 + 2 = 61.95
• 61.95 + 2 = 63.95
• 63.95 + 2 = 65.95
• 65.95 + 2 = 67.95
• 67.95 + 2 = 69.95
• 69.95 + 2 = 71.95
• 71.95 + 2 = 73.95
• 73.95 + 2 = 75.95

The heights 60 through 61.5 inches are in the interval 59.95–61.95. The heights that are 63.5 are in the interval 61.95–63.95. The heights that are 64 through 64.5 are in the interval 63.95–65.95. The heights 66 through 67.5 are in the interval 65.95–67.95. The heights 68 through 69.5 are in the interval 67.95–69.95. The heights 70 through 71 are in the interval 69.95–71.95. The heights 72 through 73.5 are in the interval 71.95–73.95. The height 74 is in the interval 73.95–75.95.

The following histogram displays the heights on the x -axis and relative frequency on the y -axis.

The following data are the shoe sizes of 50 male students. The sizes are discrete data since shoe size is measured in whole and half units only. Construct a histogram and calculate the width of each bar or class interval. Suppose you choose six bars. 9;  9;  9.5;  9.5;  10;  10;  10;  10;  10;  10;  10.5;  10.5;  10.5;  10.5;  10.5;  10.5;  10.5;  10.5 11;  11;  11;  11;  11;  11;  11;  11;  11;  11;  11;  11;  11;  11.5;  11.5;  11.5;  11.5;  11.5;  11.5;  11.5 12;  12;  12;  12;  12;  12;  12;  12.5;  12.5;  12.5;  12.5;  14

Create a histogram for the following data: the number of books bought by 50 part-time college students at ABC College.the number of books bought by 50 part-time college students at ABC College. The number of books is discrete data , since books are counted. 1;  1;  1;  1;  1;  1;  1;  1;  1;  1;  1 2;  2;  2;  2;  2;  2;  2;  2;  2;  2 3;  3;  3;  3;  3;  3;  3;  3;  3;  3;  3;  3;  3;  3;  3;  3 4;  4;  4;  4;  4;  4 5;  5;  5;  5;  5 6;  6

Eleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books.

Because the data are integers, subtract 0.5 from 1, the smallest data value and add 0.5 to 6, the largest data value. Then the starting point is 0.5 and the ending value is 6.5.

Next, calculate the width of each bar or class interval. If the data are discrete and there are not too many different values, a width that places the data values in the middle of the bar or class interval is the most convenient. Since the data consist of the numbers 1, 2, 3, 4, 5, 6, and the starting point is 0.5, a width of one places the 1 in the middle of the interval from 0.5 to 1.5, the 2 in the middle of the interval from 1.5 to 2.5, the 3 in the middle of the interval from 2.5 to 3.5, the 4 in the middle of the interval from _______ to _______, the 5 in the middle of the interval from _______ to _______, and the _______ in the middle of the interval from _______ to _______ .

Calculate the number of bars as follows:

where 1 is the width of a bar. Therefore, bars = 6.

The following histogram displays the number of books on the x -axis and the frequency on the y -axis.

Go to (Figure) . There are calculator instructions for entering data and for creating a customized histogram. Create the histogram for (Figure) .

• Press Y=. Press CLEAR to delete any equations.
• Press STAT 1:EDIT. If L1 has data in it, arrow up into the name L1, press CLEAR and then arrow down. If necessary, do the same for L2.
• Into L1, enter 1, 2, 3, 4, 5, 6.
• Into L2, enter 11, 10, 16, 6, 5, 2.
• Press WINDOW. Set Xmin = .5, Xmax = 6.5, Xscl = (6.5 – .5)/6, Ymin = –1, Ymax = 20, Yscl = 1, Xres = 1.
• Press 2 nd Y=. Start by pressing 4:Plotsoff ENTER.
• Press 2 nd Y=. Press 1:Plot1. Press ENTER. Arrow down to TYPE. Arrow to the 3 rd picture (histogram). Press ENTER.
• Arrow down to Xlist: Enter L1 (2 nd 1). Arrow down to Freq. Enter L2 (2 nd 2).
• Press GRAPH.
• Use the TRACE key and the arrow keys to examine the histogram.

The following data are the number of sports played by 50 student athletes. The number of sports is discrete data since sports are counted.

1;  1;  1;  1;  1;  1;  1;  1;  1;  1;  1;  1;  1;  1;  1;  1;  1;  1;  1;  1 2;  2;  2;  2;  2;  2;  2;  2;  2;  2;  2;  2;  2;  2;  2;  2;  2;  2;  2;  2;  2;  2 3;  3;  3;  3;  3;  3;  3;  3 20 student athletes play one sport. 22 student athletes play two sports. Eight student athletes play three sports.

Fill in the blanks for the following sentence. Since the data consist of the numbers 1, 2, 3, and the starting point is 0.5, a width of one places the 1 in the middle of the interval 0.5 to _____, the 2 in the middle of the interval from _____ to _____, and the 3 in the middle of the interval from _____ to _____.

Using this data set, construct a histogram.

Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary, but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram.

The following data represent the number of employees at various restaurants in New York City. Using this data, create a histogram.

22  35  15  26  40  28  18  20  25  34  39  42  24  22  19  27  22  34  40  20  38  and  28 Use 10–19 as the first interval.

Count the money (bills and change) in your pocket or purse. Your instructor will record the amounts. As a class, construct a histogram displaying the data. Discuss how many intervals you think is appropriate. You may want to experiment with the number of intervals.

Frequency Polygons

Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons.

To construct a frequency polygon, first examine the data and decide on the number of intervals, or class intervals, to use on the x -axis and y -axis. After choosing the appropriate ranges, begin plotting the data points. After all the points are plotted, draw line segments to connect them.

A frequency polygon was constructed from the frequency table below.

The first label on the x -axis is 44.5. This represents an interval extending from 39.5 to 49.5. Since the lowest test score is 54.5, this interval is used only to allow the graph to touch the x -axis. The point labeled 54.5 represents the next interval, or the first “real” interval from the table, and contains five scores. This reasoning is followed for each of the remaining intervals with the point 104.5 representing the interval from 99.5 to 109.5. Again, this interval contains no data and is only used so that the graph will touch the x -axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.

Construct a frequency polygon of U.S. Presidents’ ages at inauguration shown in (Figure) .

Frequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets.

We will construct an overlay frequency polygon comparing the scores from (Figure) with the students’ final numeric grade.

Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with this data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.

One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don‘t have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph.

Constructing a Time Series Graph

To construct a time series graph, we must look at both pieces of our paired data set . We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.

The following data shows the Annual Consumer Price Index, each month, for ten years. Construct a time series graph for the Annual Consumer Price Index data only.

The following table is a portion of a data set from www.worldbank.org. Use the table to construct a time series graph for CO 2 emissions for the United States.

Uses of a Time Series Graph

Time series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot.

Data on annual homicides in Detroit, 1961–73, from Gunst & Mason’s book ‘Regression Analysis and its Application’, Marcel Dekker

“Timeline: Guide to the U.S. Presidents: Information on every president’s birthplace, political party, term of office, and more.” Scholastic, 2013. Available online at http://www.scholastic.com/teachers/article/timeline-guide-us-presidents (accessed April 3, 2013).

“Presidents.” Fact Monster. Pearson Education, 2007. Available online at http://www.factmonster.com/ipka/A0194030.html (accessed April 3, 2013).

“Food Security Statistics.” Food and Agriculture Organization of the United Nations. Available online at http://www.fao.org/economic/ess/ess-fs/en/ (accessed April 3, 2013).

“Consumer Price Index.” United States Department of Labor: Bureau of Labor Statistics. Available online at http://data.bls.gov/pdq/SurveyOutputServlet (accessed April 3, 2013).

“CO2 emissions (kt).” The World Bank, 2013. Available online at http://databank.worldbank.org/data/home.aspx (accessed April 3, 2013).

“Births Time Series Data.” General Register Office For Scotland, 2013. Available online at http://www.gro-scotland.gov.uk/statistics/theme/vital-events/births/time-series.html (accessed April 3, 2013).

“Demographics: Children under the age of 5 years underweight.” Indexmundi. Available online at http://www.indexmundi.com/g/r.aspx?t=50&v=2224&aml=en (accessed April 3, 2013).

Gunst, Richard, Robert Mason. Regression Analysis and Its Application: A Data-Oriented Approach . CRC Press: 1980.

“Overweight and Obesity: Adult Obesity Facts.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/obesity/data/adult.html (accessed September 13, 2013).

Chapter Review

A histogram is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets. A frequency polygon can also be used when graphing large data sets with data points that repeat. The data usually goes on y -axis with the frequency being graphed on the x -axis. Time series graphs can be helpful when looking at large amounts of data for one variable over a period of time.

Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars. Complete the table.

What does the frequency column in (Figure) sum to? Why?

What does the relative frequency column in (Figure) sum to? Why?

What is the difference between relative frequency and frequency for each data value in (Figure) ?

The relative frequency shows the proportion of data points that have each value. The frequency tells the number of data points that have each value.

What is the difference between cumulative relative frequency and relative frequency for each data value?

To construct the histogram for the data in (Figure) , determine the appropriate minimum and maximum x and y values and the scaling. Sketch the histogram. Label the horizontal and vertical axes with words. Include numerical scaling.

Answers will vary. One possible histogram is shown:

Construct a frequency polygon for the following:

Construct a frequency polygon from the frequency distribution for the 50 highest ranked countries for depth of hunger.

Find the midpoint for each class. These will be graphed on the x -axis. The frequency values will be graphed on the y -axis values.

Use the two frequency tables to compare the life expectancy of men and women from 20 randomly selected countries. Include an overlayed frequency polygon and discuss the shapes of the distributions, the center, the spread, and any outliers. What can we conclude about the life expectancy of women compared to men?

Construct a times series graph for (a) the number of male births, (b) the number of female births, and (c) the total number of births.

The following data sets list full time police per 100,000 citizens along with homicides per 100,000 citizens for the city of Detroit, Michigan during the period from 1961 to 1973.

• Construct a double time series graph using a common x -axis for both sets of data.
• Which variable increased the fastest? Explain.
• Did Detroit’s increase in police officers have an impact on the murder rate? Explain.

1) Often, cruise ships conduct all on-board transactions, with the exception of gambling, on a cashless basis. At the end of the cruise, guests pay one bill that covers all onboard transactions. Suppose that 60 single travelers and 70 couples were surveyed as to their on-board bills for a seven-day cruise from Los Angeles to the Mexican Riviera. Following is a summary of the bills for each group.

• Fill in the relative frequency for each group.
• Construct a histogram for the singles group. Scale the x -axis by $50 widths. Use relative frequency on the y -axis.
• Construct a histogram for the couples group. Scale the x -axis by $50 widths. Use relative frequency on the y -axis.
• List two similarities between the graphs.
• List two differences between the graphs.
• Overall, are the graphs more similar or different?
• Construct a new graph for the couples by hand. Since each couple is paying for two individuals, instead of scaling the x -axis by $50, scale it by $100. Use relative frequency on the y -axis.
• How did scaling the couples graph differently change the way you compared it to the singles graph?
• Based on the graphs, do you think that individuals spend the same amount, more or less, as singles as they do person by person as a couple? Explain why in one or two complete sentences.

2) Suppose that three book publishers were interested in the number of fiction paperbacks adult consumers purchase per month. Each publisher conducted a survey. In the survey, adult consumers were asked the number of fiction paperbacks they had purchased the previous month. The results are as follows:

• Find the relative frequencies for each survey. Write them in the charts.
• Using either a graphing calculator, computer, or by hand, use the frequency column to construct a histogram for each publisher’s survey. For Publishers A and B, make bar widths of one. For Publisher C, make bar widths of two.
• In complete sentences, give two reasons why the graphs for Publishers A and B are not identical.
• Would you have expected the graph for Publisher C to look like the other two graphs? Why or why not?
• Make new histograms for Publisher A and Publisher B. This time, make bar widths of two.
• Now, compare the graph for Publisher C to the new graphs for Publishers A and B. Are the graphs more similar or more different? Explain your answer.

3) Use the following information to answer the next two exercises: Suppose one hundred eleven people who shopped in a special t-shirt store were asked the number of t-shirts they own costing more than $19 each.

The percentage of people who own at most three t-shirts costing more than $19 each is approximately:

• Cannot be determined

4)  If the data were collected by asking the first 111 people who entered the store, then the type of sampling is:

• simple random
• convenience

5) Following are the 2010 obesity rates by U.S. states and Washington, DC.

Construct a bar graph of obesity rates of your state and the four states closest to your state. Hint: Label the x -axis with the states.

Answers to odd questions

• See (Figure) and (Figure) .

• Both graphs have a single peak.
• Both graphs use class intervals with width equal to ?50.
• The couples graph has a class interval with no values.
• It takes almost twice as many class intervals to display the data for couples.
• Answers may vary. Possible answers include: The graphs are more similar than different because the overall patterns for the graphs are the same.
• Check student’s solution.
• Both graphs display 6 class intervals.
• Both graphs show the same general pattern.
• Answers may vary. Possible answers include: Although the width of the class intervals for couples is double that of the class intervals for singles, the graphs are more similar than they are different.
• Answers may vary. Possible answers include: You are able to compare the graphs interval by interval. It is easier to compare the overall patterns with the new scale on the Couples graph. Because a couple represents two individuals, the new scale leads to a more accurate comparison.
• Answers may vary. Possible answers include: Based on the histograms, it seems that spending does not vary much from singles to individuals who are part of a couple. The overall patterns are the same. The range of spending for couples is approximately double the range for individuals.

5) Answers will vary.

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Data Presentation as a Histogram or a Frequency Polygon

A histogram shows the distribution of numerical data in the form of a graph. However, it is very similar to a bar chart, a histogram groups data into intervals. To construct a histogram, you need to establish all the intervals of data, commonly known as bins. The intervals should capture all the data points and also be non-overlapping.

The intervals appear on the horizontal axis, while the absolute frequencies appear on the vertical axis. For a histogram with equal intervals in size, a rectangle should be erected over the interval, with its height being proportional to the absolute frequency. If intervals are unequal in size, the erected rectangle has an area proportional to the absolute frequency of that particular interval. We would have the vertical axis labeled as ‘density’ instead of frequency in such a case. There should be no space between bars to indicate that the intervals are continuous .

Example 1: Histogram

Consider the previous example of the returns offered by a stock. To bring you up to speed, these were the intervals and the corresponding frequencies:

$$ \begin{array}{c|c} \textbf{Interval} & \textbf{Tally} & \textbf{Frequency} \\ \hline -30\% \leq R_t \leq -20\% & \text{II} & \text{2} \\ -20\% \leq R_t \leq -10\% & \text{I} & \text{1} \\ -10\% \leq R_t \leq 0\% & \text{III} & \text{3} \\ 0\% \leq R_t \leq 10\% & \text{IIIIII} & \text{6} \\ 10\% \leq R_t \leq 20\% & \text{IIIIIII} & \text{7} \\ 20\% \leq R_t \leq 30\% & \text{IIIII} & \text{5} \\ 30\% \leq R_t \leq 40\% & \text{I} & \text{1} \\ \textbf{Total} & \text{} & \textbf{25} \\ \end{array} $$

Frequency Polygon

It is also used to represent the distribution of data graphically. However, it has a major difference when compared to the histogram. Instead of having the class intervals on the horizontal axis clearly showing their upper and lower limits, a frequency polygon uses the midpoints of the class intervals.

$$ \text{Midpoint of a class interval} =\text {Lower limit} + \cfrac { (\text{Upper limit} – \text{Lower limit}) }{ 2 } $$

The vertical axis features the absolute frequencies, which are then joined using straight lines and markers.

Example 2: Frequency Polygon

Going back to the stock return data, we could come up with a frequency polygon.

To come up with the midpoints, we use the formula above. As an example, the midpoint of the interval -30% ≤ R t ≤ -20% is:

$$ \text{Midpoint} = -30 + \cfrac {(-20 – – 30)}{2} = -25 $$

We can calculate the midpoints for the other intervals in a similar manner. The final frequency polygon should look like this:

The frequency polygon is important because it shows the shape of a distribution of data. It can also be very useful when comparing two sets of data side-by-side.

Note: The endpoints touch the X-axis. The vertical scale can also be positioned at the left margin.

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Frequency Polygon

Here we will learn about frequency polygons, including what they are and how to construct them.

There are also frequency polygon worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is a frequency polygon?

A frequency polygon is a graph that shows the frequencies of grouped data. It is a type of frequency diagram that plots the midpoints of the class intervals against the frequencies and then joins up the points with straight lines.

Frequency polygons are used to display data and show the frequency distribution over a continuous scale.

To construct a frequency polygon we need to know the value for the frequency of each group. 

For example,

Below is an example of a frequency polygon, with the associated data table.

The grouped data is presented as a continuous scale on the horizontal axis.

Step-by-step guide: Frequency diagrams

How to draw a frequency polygon

In order to draw a frequency polygon:

Calculate the midpoint of each class interval.

Plot the class frequency at the midpoint for the class.

Connect the plotted values using straight line segments.

Explain how to draw a frequency polygon

Frequency polygons worksheet

Get your free frequency polygons worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Related lessons on representing data

Frequency polygon  is part of our series of lessons to support revision on  representing data . You may find it helpful to start with the main representing data lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Representing data
• Scatter graphs
• Two way tables
• Frequency diagram
• Frequency graph
• x and y axis
• Time series graph
• Tally chart
• Stem and leaf diagram

Frequency polygon examples

Example 1: standard problem.

The length L\;cm of 16 carrots were measured and recorded into a grouped frequency table.

Draw a frequency polygon to represent the grouped data. Use the axes below.

The midpoints can be found by adding the lower limit of the class interval to the upper limit of the class interval and dividing by 2.

2 Plot the class frequency at the midpoint for the class.

The midpoints are the x -coordinate and the frequencies are the y -coordinate.

3 Connect the plotted values using straight line segments.

Join the first point to the second point with a straight line. Keep joining the points in order with straight lines.

Example 2: large class intervals

18 students were asked to hop on one leg as far as they can in 3 minutes. The results were grouped into the table below.

Draw a frequency polygon on the axes provided to represent this data.

The midpoints can be found by adding the lower limit of the class interval to the upper limit of the class interval and dividing by 2 .

Example 3: standard problem

Draw a frequency polygon for the following grouped data on the axes provided showing the average speed of vehicles travelling down a road.

Example 4: worded problem

An electrician is measuring the output voltage of plug sockets. He collects the data into groups shown in the table below.

Draw a frequency polygon to represent the data on the set of axes below.

Example 5: standard problem – 5 rows

Data for the test score as a percentage is given in the table below.

Use the set of axes below to draw a frequency polygon for the data. Label each axis.

Don’t forget to label the axes titles as well.

Example 6: different class intervals

The time T seconds for a raft of penguins to dive into water from a ledge was recorded in a grouped frequency table below. Draw a frequency polygon on the axes below for the following grouped data.

Common misconceptions

• Joining the points

Do not join the last point to the first point.

• Using the endpoints, not the midpoints

The coordinate must be plotted at the centre of each class interval, at the given frequency.

• Frequencies are whole numbers

Since frequencies are a count of how many times an item occurs, they will always be integers. They are not decimals. The midpoint can be a decimal.

• Straight line segments only

The coordinates of a frequency polygon are joined using a straight line segment. Do not connect them using freehand or draw them curved like a cumulative frequency curve; they should be drawn using a ruler!

• The graph is not a frequency polygon

A frequency polygon is a specific type of frequency diagram. Do not draw a bar chart or a vertical line diagram – they are not distinctly frequency polygons.

Practice frequency polygon questions

1. Which is the correct frequency polygon for the grouped frequency table?

Plotting the coordinates and joining them with straight line segments, we get

2. Which is the correct frequency polygon for the grouped frequency table?

Plotting the coordinates and connecting them using straight line segments, we get

3. Which is the correct frequency polygon for the grouped frequency table?

These coordinates need joining up with straight line segments to achieve the solution.

4. Which is the correct frequency polygon for the grouped frequency table?

These coordinates need joining up with straight line segments to get the solution.

5. Which is the correct frequency polygon for the grouped frequency table?

Each coordinate is to be plotted and connected with a straight line segment to the next coordinate.

6. Which is the correct frequency polygon for the grouped frequency table?

These need joining up with straight line segments to get the solution.

Frequency polygon GCSE questions

1. The frequency table shows some information about 54 test scores.

On the grid, draw a frequency polygon for the information in the table.

Midpoints: 10, 30, 50, 70 and 90.

All coordinates plotted accurately.

All coordinates connected with a straight line segment.

2. A farmer weighs 50 potatoes.

The frequency table shows the information.

Midpoints: 20, 40, 60, 80 and 100.

3. The frequency table shows some information about the heights of 80 plants.

Midpoints: 5, 15, 25, 35, 45 and 55.

Learning checklist

You have now learned how to:

• Interpret and present discrete and continuous data using appropriate graphical methods (frequency polygons)

The next lessons are

• Mean, median, mode
• Frequency table
• Cumulative frequency

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2.2: Histograms, Frequency Polygons, and Time Series Graphs

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For most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets. A rule of thumb is to use a histogram when the data set consists of 100 values or more.

A  histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data.

To construct a histogram , first decide how many bars or intervals , also called classes, represent the data. Many histograms consist of five to 15 bars or classes for clarity. The number of bars needs to be chosen. Choose a starting point for the first interval to be less than the smallest data value. A convenient starting point is a lower value carried out to one more decimal place than the value with the most decimal places.

For example:

• If the value with the most decimal places is 6.1 and this is the smallest value, a convenient starting point is 6.05 (6.1 – 0.05 = 6.05). We say that 6.05 has more precision.
• If the value with the most decimal places is 2.23 and the lowest value is 1.5, a convenient starting point is 1.495 (1.5 – 0.005 = 1.495).
• If the value with the most decimal places is 3.234 and the lowest value is 1.0, a convenient starting point is 0.9995 (1.0 – 0.0005 = 0.9995).
• If all the data happen to be integers and the smallest value is two, then a convenient starting point is 1.5 (2 – 0.5 = 1.5).

Also, when the starting point and other boundaries are carried to one additional decimal place, no data value will fall on a boundary. The next two examples go into detail about how to construct a histogram using continuous data and how to create a histogram using discrete data.

Watch the following video for an example of how to draw a histogram.

A YouTube element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/esm/?p=44

The following histogram displays the heights on the x -axis and relative frequency on the y -axis.

[/hidden-answer]

Create the histogram for Example 2 by using TI-Calculator:

• Press Y=. Press CLEAR to delete any equations.
• Press STAT 1:EDIT. If L1 has data in it, arrow up into the name L1, press CLEAR and then arrow down. If necessary, do the same for L2.
• Into L1, enter 1, 2, 3, 4, 5, 6.
• Into L2, enter 11, 10, 16, 6, 5, 2.
• Press WINDOW. Set Xmin = .5, Xscl = (6.5 – .5)/6, Ymin = –1, Ymax = 20, Yscl = 1, Xres = 1.
• Press 2nd Y=. Start by pressing 4:Plotsoff ENTER.
• Press 2nd Y=. Press 1:Plot1. Press ENTER. Arrow down to TYPE. Arrow to the 3rd picture (histogram). Press ENTER.
• Arrow down to Xlist: Enter L1 (2nd 1). Arrow down to Freq. Enter L2 (2nd 2).
• Press GRAPH.
• Use the TRACE key and the arrow keys to examine the histogram.

[reveal-answer q=”288404″]Show Answer[/reveal-answer] [hidden-answer a=”288404″]

Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary, but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram.[/hidden-answer]

Smallest value: 9

Largest value: 14

Convenient starting value: 9 – 0.05 = 8.95

Convenient ending value: 14 0.05 = 14.05

You can also use an interval with a width equal to one.

Practice Problem 2:

The following data are the number of sports played by 50 student athletes. The number of sports is discrete data since sports are counted.

1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1

2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2

3; 3; 3; 3; 3; 3; 3; 3

20 student athletes play one sport. 22 student athletes play two sports. Eight student athletes play three sports.

Fill in the blanks for the following sentence. Since the data consist of the numbers 1, 2, 3, and the starting point is 0.5, a width of one places the 1 in the middle of the interval 0.5 to _____, the 2 in the middle of the interval from _____ to _____, and the 3 in the middle of the interval from _____ to _____. [reveal-answer q=”917196″]Show Answer[/reveal-answer] [hidden-answer a=”917196″]

2.5 to 3.5[/hidden-answer]

Frequency Polygons

Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons.

To construct a frequency polygon, first examine the data and decide on the number of intervals, or class intervals, to use on the x -axis and y -axis. After choosing the appropriate ranges, begin plotting the data points. After all the points are plotted, draw line segments to connect them.

A frequency polygon was constructed from the frequency table below.

The first label on the x -axis is 44.5. This represents an interval extending from 39.5 to 49.5. Since the lowest test score is 54.5, this interval is used only to allow the graph to touch the x -axis. The point labeled 54.5 represents the next interval, or the first “real” interval from the table, and contains five scores. This reasoning is followed for each of the remaining intervals with the point 104.5 representing the interval from 99.5 to 109.5. Again, this interval contains no data and is only used so that the graph will touch the x -axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.

We will construct an overlay frequency polygon comparing the scores with the students’ final numeric grade.

Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with this data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.

Frequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets.

Constructing a Time Series Graph

The following data shows the Annual Consumer Price Index, each month, for ten years. Construct a time series graph for the Annual Consumer Price Index data only.

The following table is a portion of a data set from www.worldbank.org. Use the table to construct a time series graph for CO 2 emissions for the United States.

Uses of a Time Series Graph

Time series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot.

Data on annual homicides in Detroit, 1961–73, from Gunst & Mason’s book ‘Regression Analysis and its Application’, Marcel Dekker

“Timeline: Guide to the U.S. Presidents: Information on every president’s birthplace, political party, term of office, and more.” Scholastic, 2013. Available online at http://www.scholastic.com/teachers/a...-us-presidents (accessed April 3, 2013).

“Presidents.” Fact Monster. Pearson Education, 2007. Available online at http://www.factmonster.com/ipka/A0194030.html (accessed April 3, 2013).

“Food Security Statistics.” Food and Agriculture Organization of the United Nations. Available online at http://www.fao.org/economic/ess/ess-fs/en/ (accessed April 3, 2013).

“Consumer Price Index.” United States Department of Labor: Bureau of Labor Statistics. Available online at http://data.bls.gov/pdq/SurveyOutputServlet (accessed April 3, 2013).

“CO2 emissions (kt).” The World Bank, 2013. Available online at http://databank.worldbank.org/data/home.aspx (accessed April 3, 2013).

“Births Time Series Data.” General Register Office For Scotland, 2013. Available online at http://www.gro-scotland.gov.uk/stati...me-series.html (accessed April 3, 2013).

“Demographics: Children under the age of 5 years underweight.” Indexmundi. Available online at http://www.indexmundi.com/g/r.aspx?t=50&v=2224&aml=en (accessed April 3, 2013).

Gunst, Richard, Robert Mason. Regression Analysis and Its Application: A Data-Oriented Approach . CRC Press: 1980.

“Overweight and Obesity: Adult Obesity Facts.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/obesity/data/adult.html (accessed September 13, 2013).

Concept Review

A histogram is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets. A frequency polygon can also be used when graphing large data sets with data points that repeat. The data usually goes on y -axis with the frequency being graphed on the x -axis. Time series graphs can be helpful when looking at large amounts of data for one variable over a period of time.

• Histogram (Show Sizes of 50 Male Students). Authored by : Eng Hong SIn. Located at : https://courses.lumenlearning.com/atd-odessa-statistics/wp-content/uploads/sites/2290/2017/02/Shoe-Sizes-of-50-male-students.png . License : CC BY: Attribution
• OpenStax, Statistics, Histograms, Frequency Polygons, and Time Series Graphs. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:11/Introductory_Statisticshttps://cnx.org/contents/[email protected]:38NI55rV@8/Histograms-Frequency-Polygons- . License : CC BY: Attribution
• Introductory Statistics . Authored by : Barbara Illowski, Susan Dean. Provided by : Open Stax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected] .
• Histograms. Authored by : Khan Academy. Located at : https://youtu.be/4eLJGG2Ad30 . License : All Rights Reserved . License Terms : Standard YouTube License

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Frequency polygons (Drawing and interpreting)

Subject: Mathematics

Age range: 16+

Resource type: Lesson (complete)

Last updated

21 May 2019

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One FULL LESSON on drawing and interpreting frequency polygons. Dependent on ability, this lesson could be split into two lessons. The second half of the lesson could present a good opportunity to teach students how to calculate the mean value in a calculator using the statistics mode.

Contents of download:

• Clicker version : Normal PowerPoint lesson with which you can use a clicker / mouse / keyboard to continue animations and show solutions.
• Worksheets (including example and extension).

We are learning about: Frequency Polygons We are learning to: Draw and interpret frequency polygons.

Differentiated objectives:

• Developing learners will be able to complete a grouped frequency table for a set of data.
• Secure learners will be draw a frequency polygon from a grouped frequency table.
• Excelling learners will be able to solve unfamiliar problems involving frequency polygons.

Main: Walkthrough examples followed by practice questions on worksheets. Starts with basic calculating the area moving on to finding a missing side and angle. All solutions given on PPT.

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MrKWilliams

Superb resource.

Another fantastic set of resources. Thank you so much for sharing.

Excellent, well explained resource - thank you for sharing!

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