define measurement homework

Measurement (Introduction)

Measurement is finding a number that shows the size or amount of something.

We can measure:

Length is how far from end to end. Also called Distance.

The length of this guitar is about 1 meter (slightly more than 1 yard ).

Area is the size of a surface: how much is inside the boundary of a flat (2-dimensional) object such as a triangle or circle.

Here Ariel the Dog is waiting patiently inside 1 square meter .

Volume is the amount of 3-dimensional space an object occupies. Also called Capacity.

This jug has exactly 1 liter of water in it (slightly more than 2 pints )

Mass (Weight)

Mass is how much matter something contains.

This bar of Gold has a Mass of 1 kilogram (slightly more than 2 pounds )

(Weight is "Heaviness" ... the downward force caused by gravity on an object.)

Temperature

How hot or cold a thing is.

Temperature is measured using a thermometer, usually in the Celsius or Fahrenheit scale.

This block of ice would measure 0° Celsius , or 32° Fahrenheit

Time is the ongoing sequence of events taking place.

We measure time using seconds, minutes, hours, days, weeks, months and years.

There are lots of other things we can measure, but those are the most common.

Systems of Measurement

There are two main "Systems of Measurement":

Metric System (meters, kilograms, liters, etc.)

US Standard Units (feet, pounds, pints, etc.)

History of Measurement

Years ago people came up with standard length measures, but they didn't all agree on one system.

The Imperial System (which uses yards, feet, inches, etc to measure length) was developed over hundreds of years in the UK

Then the French came up with the Metric System (origins in 1670, but developed in the 1790s), which soon spread through Europe, and then most of the world, even to England itself in 1965.

The USA developed their own version of the Imperial system ( US Standard Units ), but the Metric System is also used in the USA, particularly in Science.

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Unit 14: Units of measurement

About this unit.

Did you know units of measurement are all around us? From how much milk is in your cereal bowl to how far you run in gym class, these measurements help us make sense of the world. In this unit, you'll learn how to estimate and convert measurements for mass, volume, and length.

Estimating mass

• Units of measurement: FAQ (Opens a modal)
• Metric system: units of weight (Opens a modal)
• Choose pounds or ounces to measure weight (Opens a modal)
• U.S. customary units: weight (Opens a modal)
• Estimate mass (grams and kilograms) Get 3 of 4 questions to level up!
• Estimating mass (ounces and pounds) Get 3 of 4 questions to level up!

Estimating volume

• Metric system: units of volume (Opens a modal)
• U.S. customary units: fluid volume (Opens a modal)
• Estimate volume (milliliters and liters) Get 3 of 4 questions to level up!
• Estimating volume (cups, pints, quarts, and gallons) Get 5 of 7 questions to level up!

Estimating length

• Metric system: units of distance (Opens a modal)
• U.S. customary units: distance (Opens a modal)
• Estimating length (mm, cm, m, km) Get 5 of 7 questions to level up!
• Estimating length (in, ft, yd, and mi) Get 5 of 7 questions to level up!
• Finding reasonable unit of measurement example (Opens a modal)
• Time differences example (Opens a modal)
• Time word problem: puzzle (Opens a modal)
• Time word problem: travel time (Opens a modal)
• Estimating time (seconds, minutes, and hours) Get 3 of 4 questions to level up!
• Time differences Get 3 of 4 questions to level up!
• Telling time word problems Get 3 of 4 questions to level up!

Converting units of mass

• How to convert kg to mg and T to oz (Opens a modal)
• Converting pounds to ounces (Opens a modal)
• Metric units of mass review (g and kg) (Opens a modal)
• US Customary units of weight review (oz & lb) (Opens a modal)
• Convert to smaller units (g and kg) Get 3 of 4 questions to level up!
• Convert to smaller units (oz and lb) Get 3 of 4 questions to level up!

Converting units of volume

• Convert liters to milliliters (Opens a modal)
• Converting US customary units of volume (Opens a modal)
• Metric units of volume review (L and mL) (Opens a modal)
• US Customary units of volume review (c, pt, qt, & gal) (Opens a modal)
• Convert to smaller units (mL and L) Get 3 of 4 questions to level up!
• Convert to smaller units (c, pt, qt, & gal) Get 5 of 7 questions to level up!

Converting units of length

• Converting metric units of length (Opens a modal)
• Converting feet to inches (Opens a modal)
• Converting yards into inches (Opens a modal)
• Metric units of length review (mm, cm, m, & km) (Opens a modal)
• US Customary units of length review (in, ft, yd, & mi) (Opens a modal)
• Convert to smaller units (mm, cm, m, & km) Get 5 of 7 questions to level up!
• Convert to smaller units (in, ft, yd, & mi) Get 5 of 7 questions to level up!

Converting units of time

• Converting units of time (Opens a modal)
• Time word problem: Susan's break (Opens a modal)
• Converting units of time review (seconds, minutes, & hours) (Opens a modal)
• Convert to smaller units (sec, min, & hr) Get 5 of 7 questions to level up!
• Time conversion word problems Get 3 of 4 questions to level up!

Money word problems

• Word problem: making change (Opens a modal)
• Currency conversion word problem (Opens a modal)
• Convert money word problems Get 3 of 4 questions to level up!

Conversion word problems

• Multi-step unit conversion examples (metric) (Opens a modal)
• Multi-step unit conversion examples (U.S. Customary) (Opens a modal)
• Metric conversions word problems Get 3 of 4 questions to level up!
• US customary conversion word problems Get 3 of 4 questions to level up!

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5: Measurement

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• Page ID 59950

• Darlene Diaz
• Santiago Canyon College via ASCCC Open Educational Resources Initiative
• 5.1: U.S. Customary Measurement System Measurement is a number that describes the size or amount of something. You can measure many things like length, area, capacity, weight, temperature and time. In the United States, two main systems of measurement are used: the metric system and the U.S. customary measurement system. This topic addresses the measurement of length using the U.S. customary measurement system.
• 5.2: Metric Units of Measurement In the United States, both the U.S. customary measurement system and the metric system are used, especially in medical, scientific, and technical fields. In most other countries, the metric system is the primary system of measurement. People in many countries use words like “kilometer,” “liter,” and “milligram” to measure the length, volume, and weight of different objects. These measurement units are part of the metric system.
• 5.3: Temperature Scales A TV weatherman in San Diego may forecast a high of 89°, but a similar forecaster in Tijuana, Mexico—which is only 20 miles south—may look at the same weather pattern and say that the day’s high temperature is going to be 32°. What’s going on here? The difference is that the two countries use different temperature scales. In the U.S, temperatures are usually measured using the Fahrenheit scale, while most countries that use the metric system use the Celsius scale to record temperatures.
• 5.4: Exercises This page contains 31 exercise problems related to the material from Chapter 5.

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Measurement Worksheets

Length, weight, capacity & temperature.

Our measurement worksheets begin with the concepts of size comparison (e.g. " longer vs shorter ") and measuring in non-standard units (" The pencil is 3 erasers long ") and progresses to measuring length, weight, capacity and temperature in customary and metric units.

Choose your grade / topic:

Kindergarten measurement worksheets, kindergarten size comparison worksheets, grade 1 measurement worksheets, grade 2 measurement worksheets, grade 3 measurement worksheets, grade 4 measurement worksheets, grade 5 measurement worksheets, grade 6 measurement worksheets.

Topics include:

• Measuring lengths of an object pictured beside a scale (no units)
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• Measure lengths with a ruler (inches or centimeters)
• Bigger vs smaller
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• Taller vs shorter
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• Longer vs shorter
• Longest and shortest
• Same size vs different size
• Which object is shorter / longer?
• Ordering 3 objects by length
• Measuring length in non-standard units
• Measuring length in inches, centimeters
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• Differences in length
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• Comparing weights using a balance scale
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• Capacity - more or less than 1 gallon
• Capacity - more or less than 1 liter
• Temperature - reading a thermometer
• Units of length (customary - inches, feet, yards, miles)
• Units of length (metric - cm, m, km)
• Measuring length to the nearest quarter inch (or millimeter)
• Converting lengths between inches, feet and yards
• Converting lengths between meters, centimeters and millimeters
• Weights - reading and using a scale
• Units of weight (ounces and pounds)
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• Convert weights (between ounces and pounds or gm and kg)
• Capacity - reading a measuring cup
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• Convert volumes between cups, pints, quarts and gallons
• Convert volumes between liters and milliliters
• Weather and temperature
• Length word problems
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• Customary units of length (inches, feet, yards, miles)
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• Customary units of weight (ounces, pounds, tons)
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• Reading and balancing scales
• Comparing weights
• Convert between customary units of weight
• Convert between metric units of weight
• Customary units of capacity (teaspoons to gallons)
• Metric units of capacity (ml, L)
• Capacity charts
• Reading measuring cups
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• Convert between customary units of capacity
• Convert between metric units of capacity
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• Mixed units of measurement
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• Measurement projects
• Convert lengths (customary units)
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• Convert lengths between customary and metric units
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1: Introduction - Matter and Measurement

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Chemistry is the study of matter and the changes that material substances undergo. Of all the scientific disciplines, it is perhaps the most extensively connected to other fields of study. As you begin your study of college chemistry, those of you who do not intend to become professional chemists may well wonder why you need to study chemistry. You will soon discover that a basic understanding of chemistry is useful in a wide range of disciplines and career paths. You will also discover that an understanding of chemistry helps you make informed decisions about many issues that affect you, your community, and your world. A major goal of this text is to demonstrate the importance of chemistry in your daily life and in our collective understanding of both the physical world we occupy and the biological realm of which we are a part.

• 1.1: Studying Chemistry An understanding of chemistry is essential for understanding much of the natural world and is central to many other disciplines. Chemistry is the study of matter and the changes material substances undergo. It is essential for understanding much of the natural world and central to many other scientific disciplines, including astronomy, geology, paleontology, biology, and medicine.
• 1.2: Classification of Matter Matter can be classified according to physical and chemical properties. Matter is anything that occupies space and has mass. The three states of matter are solid, liquid, and gas. A physical change involves the conversion of a substance from one state of matter to another, without changing its chemical composition. Most matter consists of mixtures of pure substances, which can be homogeneous (uniform in composition) or heterogeneous (different regions possess different compositions & properties.
• 1.3: Properties of Matter All matter has physical and chemical properties. Physical properties are characteristics that scientists can measure without changing the composition of the sample under study, such as mass, color, and volume (the amount of space occupied by a sample). Chemical properties describe the characteristic ability of a substance to react to form new substances; they include its flammability and susceptibility to corrosion.
• 1.4: Units of Measurement The natural sciences begin with observation, and this usually involves numerical measurements of quantities such as length, volume, density, and temperature. Most of these quantities have units of some kind associated with them, and these units must be retained when you use them in calculations. Measuring units can be defined in terms of a very small number of fundamental ones that, through "dimensional analysis", provide insight into their derivation and meaning.
• 1.5: Uncertainty in Measurement Measurements may be accurate, meaning that the measured value is the same as the true value; they may be precise, meaning that multiple measurements give nearly identical values (i.e., reproducible results); they may be both accurate and precise; or they may be neither accurate nor precise. The goal of scientists is to obtain measured values that are both accurate and precise.
• 1.6: Dimensional Analysis Dimensional analysis is used in numerical calculations, and in converting units. It can help us identify whether an equation is set up correctly (i.e. the resulting units should be as expected). Units are treated similarly to the associated numerical values, i.e., if a variable in an equation is supposed to be squared, then the associated dimensions are squared, etc.
• 1.E: Matter and Measurement (Exercises) These are homework exercises to accompany the Textmap created for "Chemistry: The Central Science" by Brown et al.
• 1.S: Matter and Measurement (Summary) This is the summary Module for the chapter "Matter and Measurement" in the Brown et al. General Chemistry Textmap.

Thumbnail: Chemicals in flasks (including Ammonium hydroxide and Nitric acid) lit in different colours (CC BY 2.0 Generic; Joe Sullivan via Flickr )

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• Units Of Measurement
• Accuracy Precision Measurement

Accuracy and Precision - The Art of Measurement

Measurement is essential for us to understand the external world, and through millions of years of life, we have developed a sense of measurement. Measurements require tools that provide scientists with a quantity. The problem here is that the result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is referred to as an error. Accuracy and precision are two important factors to consider while taking measurements. Both these terms reflect how close a measurement is to a known or accepted value. In this article, let us learn in detail about precision and accuracy.

The ability of an instrument to measure the accurate value is known as accuracy. In other words, it is the  the closeness of the measured value to a standard or true value . Accuracy is obtained by taking small readings. The small reading reduces the error of the calculation. The accuracy of the system is classified into three types as follows:

• Point Accuracy

The accuracy of the instrument only at a particular point on its scale is known as point accuracy. It is important to note that this accuracy does not give any information about the general accuracy of the instrument.

• Accuracy as Percentage of Scale Range

The uniform scale range determines the accuracy of a measurement. This can be better understood with the help of the following example: Consider a thermometer having the scale range up to 500 ºC . The thermometer has an accuracy of ±0.5 percent of scale range i.e. 0.005 x 500 = ± 2.5 ºC. Therefore, the reading will have a maximum error of ± 2.5 ºC.

• Accuracy as Percentage of True Value

Such type of accuracy of the instruments is determined by identifying the measured value regarding their true value. The accuracy of the instruments is neglected up to ±0.5 percent from the true value.

The closeness of two or more measurements to each other is known as the precision of a substance. If you weigh a given substance five times and get 3.2 kg each time, then your measurement is very precise but not necessarily accurate. Precision is independent of accuracy. The below examples will tell you about how you can be precise but not accurate and vice versa. Precision is sometimes separated into:

• Repeatability

The variation arising when the conditions are kept identical and repeated measurements are taken during a short time period.

• Reproducibility

The variation arises using the same measurement process among different instruments and operators, and over longer time periods.

Accuracy is the degree of closeness between a measurement and its true value. Precision is the degree to which repeated measurements under the same conditions show the same results.

Since you are here, you might want to check out the following articles:

• Errors in Measurement
• Measurement of length

Accuracy and Precision Examples

The top left image shows the target hit at high precision and accuracy. The top right image shows the target hit at a high accuracy but low precision. The bottom left image shows the target hit at a high precision but low accuracy. The bottom right image shows the target hit at low accuracy and low precision.

More Examples

• If the weather temperature reads 28 °C outside and it is 28 °C outside, then the measurement is said to be accurate. If the thermometer continuously registers the same temperature for several days, the measurement is also precise.
• If you take the measurement of the mass of a body of 20 kg and you get 17.4,17,17.3 and 17.1, your weighing scale is precise but not very accurate. If your scale gives you values of 19.8, 20.5, 21.0, and 19.6, it is more accurate than the first balance but not very precise.

Difference between Accuracy and Precision

In the previous few sections having discussed what each term means, let us now look at their differences.

Practice Questions

Q1) The volume of a liquid is 26 mL. A student measures the volume and finds it to be 26.2 mL, 26.1 mL, 25.9 mL, and 26.3 mL in the first, second, third, and fourth trial, respectively. Which of the following statements is true for his measurements?

a. They are neither precise nor accurate. b. They have poor accuracy. c. They have good precision. d. They have poor precision.

Answer: They have good precision.

Q2) The volume of a liquid is 20.5 mL. Which of the following sets of measurement represents the value with good accuracy? 18.6 mL, 17.8 mL, 19.6 mL, 17.2 mL 19.2 mL, 19.3 mL, 18.8 mL, 18.6 mL 18.9 mL, 19.0 mL, 19.2 mL, 18.8 mL 20.2 mL, 20.5 mL, 20.3 mL, 20.1 mL Answer: The set 20.2 mL, 20.5 mL, 20.3 mL, 20.1 mL represents the value with good accuracy.

Frequently Asked Questions – FAQs

What is meant by accuracy.

Accuracy refers to the closeness of the measured value to a standard or true value.

What is the classification of accuracy of the system?

Accuracy of the system are classified into:

State true or false: Multiple measurements or factors are needed.

What is meant by precision.

Precision is the degree to which repeated measurements under the same conditions show the same results.

What is meant by error

The difference between the actual value and the measured value is known as error.

Write the two factors to be considered while taking measurements?

What is point accuracy, can the results be precise and accurate, define reproducibility., if the player shoots ball into the goal in one shot, he is said to be accurate or precise, watch the video to find out what are base measurements.

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3.1: Measures of Center

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• Kathryn Kozak
• Coconino Community College

This section focuses on measures of central tendency. Many times you are asking what to expect on average. Such as when you pick a major, you would probably ask how much you expect to earn in that field. If you are thinking of relocating to a new town, you might ask how much you can expect to pay for housing. If you are planting vegetables in the spring, you might want to know how long it will be until you can harvest. These questions, and many more, can be answered by knowing the center of the data set. There are three measures of the “center” of the data. They are the mode, median, and mean. Any of the values can be referred to as the “average.”

• The mode is the data value that occurs the most frequently in the data. To find it, you count how often each data value occurs, and then determine which data value occurs most often.
• The median is the data value in the middle of a sorted list of data. To find it, you put the data in order, and then determine which data value is in the middle of the data set.
• The mean is the arithmetic average of the numbers. This is the center that most people call the average, though all three – mean, median, and mode – really are averages.

There are no symbols for the mode and the median, but the mean is used a great deal, and statisticians gave it a symbol. There are actually two symbols, one for the population parameter and one for the sample statistic. In most cases you cannot find the population parameter, so you use the sample statistic to estimate the population parameter.

Definition \(\PageIndex{1}\): Population Mean

The population mean is given by

\(\mu=\dfrac{\sum x}{N}\), pronounced mu

• \(N\) is the size of the population.
• \(x\) represents a data value.
• \(\sum x\) means to add up all of the data values.

Definition \(\PageIndex{2}\): Sample Mean

Sample Mean:

\(\overline{x}=\dfrac{\sum x}{n}\), pronounced x bar, where

• \(n\) is the size of the sample.

The value for \(\overline{x}\) is used to estimate \(\mu\) since \(\mu\) can't be calculated in most situations.

Example \(\PageIndex{1}\) finding the mean, median, and mode

Suppose a vet wants to find the average weight of cats. The weights (in pounds) of five cats are in Example \(\PageIndex{1}\).

Find the mean, median, and mode of the weight of a cat.

Before starting any mathematics problem, it is always a good idea to define the unknown in the problem. In this case, you want to define the variable. The symbol for the variable is \(x\).

The variable is \(x =\) weight of a cat

\(\overline{x}=\dfrac{6.8+8.2+7.5+9.4+8.2}{5}=\dfrac{40.1}{5}=8.02\) pounds

You need to sort the list for both the median and mode. The sorted list is in Example \(\PageIndex{2}\).

There are 5 data points so the middle of the list would be the 3rd number. (Just put a finger at each end of the list and move them toward the center one number at a time. Where your fingers meet is the median.)

The median is therefore 8.2 pounds.

This is easiest to do from the sorted list that is in Example \(\PageIndex{2}\). Which value appears the most number of times? The number 8.2 appears twice, while all other numbers appear once.

Mode = 8.2 pounds.

A data set can have more than one mode. If there is a tie between two values for the most number of times then both values are the mode and the data is called bimodal (two modes). If every data point occurs the same number of times, there is no mode. If there are more than two numbers that appear the most times, then usually there is no mode.

In Example \(\PageIndex{1}\), there were an odd number of data points. In that case, the median was just the middle number. What happens if there is an even number of data points? What would you do?

Example \(\PageIndex{2}\) finding the median with an even number of data points

Suppose a vet wants to find the median weight of cats. The weights (in pounds) of six cats are in Example \(\PageIndex{4}\). Find the median.

Variable: \(x =\) weight of a cat

First sort the list if it is not already sorted.

There are 6 numbers in the list so the number in the middle is between the 3rd and 4th number. Use your fingers starting at each end of the list in Example \(\PageIndex{5}\) and move toward the center until they meet. There are two numbers there.

To find the median, just average the two numbers.

median \(=\dfrac{7.5+8.2}{2}=7.85\) pounds

The median is 7.85 pounds.

Example \(\PageIndex{3}\) finding mean and median using technology

Suppose a vet wants to find the median weight of cats. The weights (in pounds) of six cats are in Example \(\PageIndex{4}\). Find the median

Variable: \(x=\) weight of a cat

You can do the calculations for the mean and median using the technology.

The procedure for calculating the sample mean ( \(\overline{x}\) ) and the sample median (Med) on the TI-83/84 is in Figures 3.1.1 through 3.1.4 . First you need to go into the STAT menu, and then Edit. This will allow you to type in your data (see Figure \(\PageIndex{1}\) ).

Once you have the data into the calculator, you then go back to the STAT menu, move over to CALC, and then choose 1-Var Stats (see Figure \(\PageIndex{2}\) ). The calculator will now put 1-Var Stats on the main screen. Now type in L1 (2nd button and 1) and then press ENTER. (Note if you have the newer operating system on the TI-84, then the procedure is slightly different.) If you press the down arrow, you will see the rest of the output from the calculator. The results from the calculator are in Figure \(\PageIndex{3}\) .

The commands for finding the mean and median using R are as follows:

variable<-c(type in your data with commas in between) To find the mean, use mean(variable) To find the median, use median(variable)

So for this example, the commands would be

weights<-c(6.8, 8.2, 7.5, 9.4, 8.2, 6.3) mean(weights) [1] 7.733333 median(weights) [1] 7.85

Example \(\PageIndex{4}\) affect of extreme values on mean and median

Suppose you have the same set of cats from Example \(\PageIndex{1}\) but one additional cat was added to the data set. Example \(\PageIndex{6}\) contains the six cats’ weights, in pounds.

Find the mean and the median.

mean \(=\overline{x}=\dfrac{6.8+7.5+8.2+8.2+9.4+22.1}{6}=10.37\) pounds

The data is already in order, thus the median is between 8.2 and 8.2.

median \(=\dfrac{8.2+8.2}{2}=8.2\) pounds

The mean is much higher than the median. Why is this? Notice that when the value of 22.1 was added, the mean went from 8.02 to 10.37, but the median did not change at all. This is because the mean is affected by extreme values, while the median is not. The very heavy cat brought the mean weight up. In this case, the median is a much better measure of the center.

An outlier is a data value that is very different from the rest of the data. It can be really high or really low. Extreme values may be an outlier if the extreme value is far enough from the center. In Example \(\PageIndex{4}\), the data value 22.1 pounds is an extreme value and it may be an outlier.

If there are extreme values in the data, the median is a better measure of the center than the mean. If there are no extreme values, the mean and the median will be similar so most people use the mean.

The mean is not a resistant measure because it is affected by extreme values. The median and the mode are resistant measures because they are not affected by extreme values.

As a consumer you need to be aware that people choose the measure of center that best supports their claim. When you read an article in the newspaper and it talks about the “average” it usually means the mean but sometimes it refers to the median. Some articles will use the word “median” instead of “average” to be more specific. If you need to make an important decision and the information says “average”, it would be wise to ask if the “average” is the mean or the median before you decide.

As an example, suppose that a company wants to use the mean salary as the average salary for the company. This is because the high salaries of the administration will pull the mean higher. The company can say that the employees are paid well because the average is high. However, the employees want to use the median since it discounts the extreme values of the administration and will give a lower value of the average. This will make the salaries seem lower and that a raise is in order.

Why use the mean instead of the median? The reason is because when multiple samples are taken from the same population, the sample means tend to be more consistent than other measures of the center. The sample mean is the more reliable measure of center.

To understand how the different measures of center related to skewed or symmetric distributions, see Figure \(\PageIndex{5}\) . As you can see sometimes the mean is smaller than the median and mode, sometimes the mean is larger than the median and mode, and sometimes they are the same values.

One last type of average is a weighted average. Weighted averages are used quite often in real life. Some teachers use them in calculating your grade in the course, or your grade on a project. Some employers use them in employee evaluations. The idea is that some activities are more important than others. As an example, a fulltime teacher at a community college may be evaluated on their service to the college, their service to the community, whether their paperwork is turned in on time, and their teaching. However, teaching is much more important than whether their paperwork is turned in on time. When the evaluation is completed, more weight needs to be given to the teaching and less to the paperwork. This is a weighted average.

Definition \(\PageIndex{3}\)

Weighted Average

\(\dfrac{\sum x w}{\sum w}\) where \(w\) is the weight of the data value, \(x\).

Example \(\PageIndex{5}\) weighted average

In your biology class, your final grade is based on several things: a lab score, scores on two major tests, and your score on the final exam. There are 100 points available for each score. The lab score is worth 15% of the course, the two exams are worth 25% of the course each, and the final exam is worth 35% of the course. Suppose you earned scores of 95 on the labs, 83 and 76 on the two exams, and 84 on the final exam. Compute your weighted average for the course.

Variable: \(x=\) score

The weighted average is \(\dfrac{\sum x w}{\sum w}=\dfrac{\text { sum of the scores times their weights }}{\text { sum of all the weights }}\)

weighted average \(=\dfrac{95(0.15)+83(0.25)+76(0.25)+84(0.35)}{0.15+0.25+0.25+0.35}=\dfrac{83.4}{1.00}=83.4 \%\)

A weighted average can be found using technology.

The procedure for calculating the weighted average on the TI-83/84 is in Figures 3.1.6 through 3.1.9 . First you need to go into the STAT menu, and then Edit. This will allow you to type in the scores into L1 and the weights into L2 (see Figure \(\PageIndex{6}\) ).

Once you have the data into the calculator, you then go back to the STAT menu, move over to CALC, and then choose 1-Var Stats (see Figure \(\PageIndex{7}\) ). The calculator will now put 1-Var Stats on the main screen. Now type in L1 (2nd button and 1), then a comma (button above the 7 button), and then L2 (2nd button and 2) and then press ENTER. (Note if you have the newer operating system on the TI-84, then the procedure is slightly different.) The results from the calculator are in Figure \(\PageIndex{9}\) . The \(\overline{x}\) is the weighted average.

x<-c(type in your data with commas in between) w<-c(type in your weights with commas in between weighted.mean(x,w)

x<-c(95, 83, 76, 84) w<-c(.15, .25, .25, .35) weighted.mean(x,w) [1] 83.4

Example \(\PageIndex{6}\) weighted average

The faculty evaluation process at John Jingle University rates a faculty member on the following activities: teaching, publishing, committee service, community service, and submitting paperwork in a timely manner. The process involves reviewing student evaluations, peer evaluations, and supervisor evaluation for each teacher and awarding him/her a score on a scale from 1 to 10 (with 10 being the best). The weights for each activity are 20 for teaching, 18 for publishing, 6 for committee service, 4 for community service, and 2 for paperwork.

• One faculty member had the following ratings: 8 for teaching, 9 for publishing, 2 for committee work, 1 for community service, and 8 for paperwork. Compute the weighted average of the evaluation.
• Another faculty member had ratings of 6 for teaching, 8 for publishing, 9 for committee work, 10 for community service, and 10 for paperwork. Compute the weighted average of the evaluation.
• Which faculty member had the higher average evaluation?

a. Variable: \(x=\) rating

evaluation \(=\dfrac{8(20)+9(18)+2(6)+1(4)+8(2)}{20+18+6+4+2}=\dfrac{354}{50}=7.08\)

b. evaluation \(=\dfrac{6(20)+8(18)+9(6)+10(4)+10(2)}{20+18+6+4+2}=\dfrac{378}{50}=7.56\)

c. The second faculty member has a higher average evaluation.

You can find a weighted average using technology. The last thing to mention is which average is used on which type of data.

Mode can be found on nominal, ordinal, interval, and ratio data, since the mode is just the data value that occurs most often. You are just counting the data values. Median can be found on ordinal, interval, and ratio data, since you need to put the data in order. As long as there is order to the data you can find the median. Mean can be found on interval and ratio data, since you must have numbers to add together.

Exercise \(\PageIndex{1}\)

Table \(\PageIndex{11}\) : Salaries of Print-O-Matic Printing Company Employees a. Find the mean and median. b. Find the mean and median with the CEO's salary removed. c. What happened to the mean and median when the CEO’s salary was removed? Why? d. If you were the CEO, who is answering concerns from the union that employees are underpaid, which average of the complete data set would you prefer? Why? e. If you were a platen worker, who believes that the employees need a raise, which average would you prefer? Why?

Table \(\PageIndex{12}\) : Fixed Costs for Print-O-Matic Printing Company a. Find the mean and median. b. Find the mean and median with the bank charger removed. c. What happened to the mean and median when the bank charger was removed? Why? d. If it is your job to oversee the fixed costs, which average using te complete data set would you prefer to use when submitting a report to administration to show that costs are low? Why? e. If it is your job to find places in the budget to reduce costs, which average using the complete data set would you prefer to use when submitting a report to administration to show that fixed costs need to be reduced? Why?

• You collect data on people’s likelihood (very likely, likely, neutral, unlikely, very unlikely) to vote for a candidate.
• You collect data on the diameter at breast height of trees in the Coconino National Forest.
• You collect data on the year wineries were started.
• You collect the drink types that people in Sydney, Australia drink.
• You collect data on the height of plants using a new fertilizer.
• You collect data on the cars that people drive in Campbelltown, Australia.
• You collect data on the temperature at different locations in Antarctica.
• You collect data on the first, second, and third winner in a beer competition.

• An employee at Coconino Community College (CCC) is evaluated based on goal setting and accomplishments toward the goals, job effectiveness, competencies, and CCC core values. Suppose for a specific employee, goal 1 has a weight of 30%, goal 2 has a weight of 20%, job effectiveness has a weight of 25%, competency 1 has a goal of 4%, competency 2 has a goal has a weight of 3%, competency 3 has a weight of 3%, competency 4 has a weight of 3%, competency 5 has a weight of 2%, and core values has a weight of 10%. Suppose the employee has scores of 3.0 for goal 1, 3.0 for goal 2, 2.0 for job effectiveness, 3.0 for competency 1, 2.0 for competency 2, 2.0 for competency 3, 3.0 for competency 4, 4.0 for competency 5, and 3.0 for core values. Find the weighted average score for this employee. If an employee has a score less than 2.5, they must have a Performance Enhancement Plan written. Does this employee need a plan?
• An employee at Coconino Community College (CCC) is evaluated based on goal setting and accomplishments toward goals, job effectiveness, competencies, CCC core values. Suppose for a specific employee, goal 1 has a weight of 20%, goal 2 has a weight of 20%, goal 3 has a weight of 10%, job effectiveness has a weight of 25%, competency 1 has a goal of 4%, competency 2 has a goal has a weight of 3%, competency 3 has a weight of 3%, competency 4 has a weight of 5%, and core values has a weight of 10%. Suppose the employee has scores of 2.0 for goal 1, 2.0 for goal 2, 4.0 for goal 3, 3.0 for job effectiveness, 2.0 for competency 1, 3.0 for competency 2, 2.0 for competency 3, 3.0 for competency 4, and 4.0 for core values. Find the weighted average score for this employee. If an employee that has a score less than 2.5, they must have a Performance Enhancement Plan written. Does this employee need a plan?
• A statistics class has the following activities and weights for determining a grade in the course: test 1 worth 15% of the grade, test 2 worth 15% of the grade, test 3 worth 15% of the grade, homework worth 10% of the grade, semester project worth 20% of the grade, and the final exam worth 25% of the grade. If a student receives an 85 on test 1, a 76 on test 2, an 83 on test 3, a 74 on the homework, a 65 on the project, and a 79 on the final, what grade did the student earn in the course?
• A statistics class has the following activities and weights for determining a grade in the course: test 1 worth 15% of the grade, test 2 worth 15% of the grade, test 3 worth 15% of the grade, homework worth 10% of the grade, semester project worth 20% of the grade, and the final exam worth 25% of the grade. If a student receives a 92 on test 1, an 85 on test 2, a 95 on test 3, a 92 on the homework, a 55 on the project, and an 83 on the final, what grade did the student earn in the course?

1. mean = 253.93, median = 268, mode = none

3. mean = 67.68 km, median = 64 km, mode = 56 and 64 km

5. a. mean = $89,370.42, median = $75,311, b. mean = $79,196.56, median = $74,773, c. See solutions, d. See solutions, e. See solutions

7. a. ordinal- median and mode, b. ratio – all three, c. interval – all three, d. nominal – mode

9. Skewed right, mean higher