8.1 Measures of Central Tendency

Learning Objectives

You will design simple experiments and collect data.
You will determine mean, median and range for quantitative data and from data represented in a display.
You will use quantities to draw conclusions about the data, compare different data sets, and make predictions.
You will describe the impact that inserting or deleting a data point has on the mean or median of a data set.
You will know how to create data displays using a spreadsheet.

Introduction

The Iditarod

The students in Mr. Hawkins class are studying about the Iditarod. Some of the students had never heard of the Iditarod before, so Mr. Hawkins started off his class by showing them this video about the race.

http://www.iditarod.com/

The students sat through the video in awe. When it was over, the room was so quiet that you could have heard a pin drop. Marcus was the first one to raise his hand.

“How far is it?” he asked.

“That is a great question,” Mr. Hawkins said. “The race is 1,150 miles from Anchorage Alaska to Nome Alaska. Men and women have finished it and won it too. This year, there were 10 men who finished on top. One finished in 8 days and the rest in 9 days.”

“How fast did they go?” Karen asked from the back of the room. “I mean, you can’t go very fast on a dog sled, right?”

“Well, for you and me it might not seem fast, but for those dogs I am sure that it is. This leads us to a great math problem. Here are the speeds of the top 10 finishers. What is the average speed here?”

Mr. Hawkins wrote these speeds on the board.

4.81 mph, 4.79 mph, 4.76 mph, 4.67 mph, 4.66 mph, 4.64 mph, 4.62 mph, 4.6 mph, 4.58 mph, 4.55 mph

“Take out a piece of paper and figure this out.”

Marcus took out a piece of paper, but he couldn’t remember how to figure out the average score.

If you remember how to do it, figure it out now. Then go through this lesson to begin learning all about data and statistics. When you are finished this lesson, you can check your work with Marcus’ work and see if you have the correct average speed. Look for this problem again at the end of the lesson.

Guided Learning

Find the Mean of a Set of Data

Data is one of those words that we hear all the time, especially in math class. We hear about collecting data, organizing data, analyzing data, etc. But....

What is data?

Data is numerical information collected in a set. When we look at data in math and science, we look at information that has been gathered over time or that has been gathered to evaluate a topic. Learning to look at data is part of the work of people in math and science. Analyzing data can help scientists predict future events too.

We can analyze numerical data several different ways. We can look for the mean, the median, the mode and the range of a set of data.

Let’s start with the mean.

The Mean is sometimes referred to as the average of a set of data. The mean is the sum of the data values divided by the number of data values. You often hear about averages with grades or speeds. What is your average grade in math class? That number determines your final grade or the grade that you received on a test or quiz. We use averages all the time. Let’s look at the steps to figuring out the mean or average through the following example.

Example A

The chart below depicts the daily temperature in San Diego for the first seven days in August. Calculate the mean temperature for the first seven days in August.

Date:	Temperature:
Sunday 8/1	$88^\circ F$
Monday 8/2	$83^\circ F$
Tuesday 8/3	$87^\circ F$
Wednesday 8/4	$89^\circ F$
Thursday 8/5	$82^\circ F$
Friday 8/6	$79^\circ F$
Saturday 8/7	$87^\circ F$

Step 1: Add to determine the sum of the data values.

88 + 83 + 87 + 89 + 82 + 79 + 87 = 595

Step 2: Divide the sum, 595, by 7 since there are seven numbers in the given data set.

$595 \div 7 = 85$

The mean temperature for the first week in August was $85^\circ F$ .

This was a real life example of how averages help us figure out weather. If you think about the weather forecast, you will often hear "average" being mentioned. The meteorologist will talk about average snowfall or average temperature or average rainfall.

Sometimes, an average will not be a whole number. When this happens, you may need to round to the nearest whole number.

Find the mean for each data set below.

11, 13, 14, 15, 16, 22, 24, 25, 30, 32
34, 36, 38, 41, 43, 44, 50, 53, 50, 50, 62, 66
8, 16, 24, 32, 40

Take a minute to check your answers with a peer.

Find the Median of a Set of Data

Now that you have learned about the mean of a set of data, let’s move on to the median. If you think about the word “median” you can think about the median in a road or street. The median of a street is in the middle of the street. Just like the median of a road, the median of a set of data is the middle value of the set of numbers.

The Median is the middle number when the values are arranged in order from the least to the greatest.

Notice that a key to finding the median is that the values must be arranged in order from least to greatest. If they are not arranged in this way, you will not be able to determine an accurate median score!

Example B

The chart below depicts the daily temperature in San Diego for the first seven days in August. Determine the median temperature.

Date:	Temperature:
Sunday 8/1	$88^\circ F$
Monday 8/2	$83^\circ F$
Tuesday 8/3	$87^\circ F$
Wednesday 8/4	$89^\circ F$
Thursday 8/5	$82^\circ F$
Friday 8/6	$79^\circ F$
Saturday 8/7	$87^\circ F$

Step 1: Arrange the temperatures in order from least to greatest.

79, 82, 83, 87, 87, 88, 89

Step 2: Determine the data value in the middle of the data set. In this case, 87 is the median temperature.

The answer is $87^\circ F$ .

Example C

Katie’s first four test scores are 75%, 81%, 80%, and 84%. Determine the median of Katie’s test scores.

Step 1: Arrange the test scores in order from least to greatest.

75, 80, 81, 84

Step 2: In this case, there are two data values in the middle of the data set. To find the median, find the average of the two data values. Recall that to find the mean, determine the sum of the numbers and then divide by two.

$80 + 81 &= 161\\161 \div 2 &= 80.5$

The median of Katie’s test scores is 80.5%.

Sometimes, you will have median scores that are not whole numbers. When this happens, you will likely need to include the decimal in your answer. This means that the median score is between two whole numbers.

Find the median score for each data set.

12, 14, 15, 16, 18, 20
14. 18, 19, 34, 32, 30, 41, 50
5, 10, 23, 20, 7, 9, 11, 18, 35, 16, 22

Take a minute to check your answers with a peer.

Identify the Range of a Set of Data

The range of a set of data is the difference between the greatest and least values.

To find the range of a set of data, subtract the smallest data value from the largest data value.

Example D

Determine the range for the set of data: 47, 56, 51, 45, and 41.

Subtract the smallest data value, in this case 541 from the largest data value of 56.

56 - 41 = 15

The difference between the largest and smallest number is 15, therefore the range for this set of data is 15.

Example E

The chart below depicts the daily temperature in San Diego for the first seven days in August. Identify the range in temperatures.

Date:	Temperature:
Sunday 8/1	$88^\circ F$
Monday 8/2	$83^\circ F$
Tuesday 8/3	$87^\circ F$
Wednesday 8/4	$89^\circ F$
Thursday 8/5	$82^\circ F$
Friday 8/6	$79^\circ F$
Saturday 8/7	$87^\circ F$

Subtract the smallest value 79 from the largest value 89.

89 - 79 = 10

The range in temperatures is $10^\circ F$ .

Find the range of each data set.

12, 14, 15, 16, 18, 20
14, 18, 19, 34, 32, 30, 41, 50
5, 10, 23, 20, 7, 9, 11, 18, 35, 16, 22

Check your answers with a peer.

Select among Mean, Median, Mode, and Range to Describe a Set of Data

Different situations require different analysis. Sometimes it makes more sense to describe data using one term versus of another term.

For example, if you are looking for the difference between speeds or times, using the range would make the most sense. The average wouldn’t help you to understand the difference.

When analyzing the daily temperatures for the month of August, the best way to examine the data would be to calculate the mean because there shouldn’t be much variance in the daily temperature.

If you are looking for the middle value in, say, sales for a store, then the median would be the best way to analyze the data.

Let’s look at some examples.

Example F

Should mean, median, mode, or range be determined to best analyze the following set of data?

Student:	Number of minutes:
1	29
2	32
3	40
4	33
5	38

Since there is not a number that occurs most often, the mode is not the best way to analyze the data.

Calculating the range of this data just allows one to see the difference between the students who finished the exam first and last.

In this case, determining the mean is the best way to analyze this data. The mean gives the average amount of time it took for the five students to complete the exam. In this case, the mean is 34.4 minutes. It can also be helpful to look at the median of this data, 33. In this case, the mean and median are close in value.

Remember to think about what the data describes and what your objective is in analyzing the data and this will help you to choose the best method for analyzing the data.

Use a Stem-and-Leaf Plot to Find the Mean, Median, Mode and Range of a Set of Data

Now that you know how to create a stem-and-leaf plot, let’s look at how we can use it to analyze data and draw conclusions. First, let’s review some of the vocabulary words that we used in the first lesson of this chapter.

The mean is sometimes also called the average of a set of data. To find the mean, add the data values and then divide the sum by the number of data values.

The median is the data value in the middle when the data is ordered from least to greatest. Since the data is ordered from least to greatest on a stem-and-leaf plot, find the data value in the middle of the stem-and-leaf plot.

The mode is the data value that occurs most often. On a stem-and-leaf plot, the mode is the most repeated leaf.

The range is the difference between the highest and the lowest data value.

Data from a stem-and-leaf plot can be used to determine the mean, median, mode, and range for a set of data. Let’s look at how we can do this.

Example G

The stem-and-leaf plot below depicts the weight (in pounds) of the ten trout caught in a fishing competition. Determine the mean, median, mode, and range of the data on the stem-and-leaf plot.

Stem	Leaf
2	9
3	1
4	0 5
5	2
6	2
7	6
8	3
9	2 2
Key: $2 \big \| 9 = 2.9$

Step 1: Using the key, combine the stem with each of its leaves. The values are in order from least to greatest on the stem-and-leaf plot. Therefore, keep them in order as you list the data values.

2.9, 3.1, 4.0, 4.5, 5.2, 6.2, 7.6, 8.3, 9.2, 9.2

Step 2: Recall that to determine the mean you add the data values and then divide the sum by the number of data values.

$2.9 + 3.1 + 4.0 + 4.5 + 5.2 + 6.2 + 7.6 + 8.3 + 9.2 + 9.2 &= 60.2\\60.2 \div 10 &= 6.2\\\text{Mean} &= 6.2 \ pounds$

Step 3: The data is already arranged in order from least to greatest. Therefore, to determine the median, identify the number in the middle of the data set. In this case, two data values share the middle position. To find the median, find the mean of these two data values.

$& 2.9, \ 3.1, \ 4.0, \ 4.5, \ 5.2, \ 6.2, \ 7.6, \ 8.3, \ 9.2, \ 9.2\\& \qquad \qquad \quad 5.2 + 6.2 = 11.4\\& \qquad \qquad \quad 11.4 \div 2 = 5.7\\& \qquad \qquad \quad \text{Median} = 5.7 \ pounds$

Step 4: Recall that the mode is the data value that occurs most. Looking at the stem-and-leaf plot, you can see that the data value 9.2 appears twice. Therefore, the mode is 9.2.

Mode = 9.2 pounds

Step 5: Recall that the range is the difference of the greatest and least values. On the stem-and-leaf plot, the greatest value is the last value; the smallest value is the first value.

$9.2 - 2.9 &= 6.3\\\text{Range} &= 6.3$

Answer

Mean = 6.2 pounds

Median = 5.7 pounds

Mode = 9.2 pounds

Range = 6.3 pounds

Try this out on your own. Here is a set of data to use.

Determine the mean, median, mode, and range for the data on the stem-and-leaf plot.

Stem	Leaf
1	4 6 7 8 8
2	0 2 4 9
3	1 3
Key: $2 \big \| 0 = 20$

Use a Box-and-Whisker Plot to Find the Median, Quartiles, and Extremes of a Set of Data

Now that you understand how to create a box-and-whisker plot, we can also use created ones to examine data. Using box-and-whisker plots we can draw inferences and make conclusions.

Let’s look at an example.

Example H

The weight of bears varies between species. Weight also varies within species as a result of habitat and diet. The box-and-whisker plot was created after recording the weight (in pounds) of several black bears across the country. Use the box-and-whisker plot to answer the questions below.

The number line is labeled by tens. Notice that each section on the number line has been divided into fifths. Therefore, each mark on the number line represents two. This is important to note prior to answering the questions below.

What are the highest and lowest weights represented on the box-and-whisker plot? The lowest value or weight is 127 pounds. The highest value or weight is 201 pounds.

What is the median weight for a black bear? The median weight is 163 pounds.

What is the median weight for the lower quartile? The median weight of the lower quartile is 129 pounds.

What is the median weight for the upper quartile? The median weight of the upper quartile is 196 pounds.

Example I

The box-and-whisker plot below was created after recording amount of time it took for several runners to finish a 5K race. Use the box-and-whisker plot to answer the questions below.

The number line on the box-and-whisker plot is labeled by twos. Notice that there is only one section in between each labeled value. Therefore, each mark on the number line represents one. This is important to note when answering the questions below.

Identify the first and last finish times for the race. The first finish time or the smallest value identified on the box-and-whisker plot is 12 minutes. The last finish time or largest value on the box-and-whisker plot is 26 minutes.

Identify the median finish time for the race. The median finish time is 17 minutes.

What was the median finishing time in the lower quartile? The median of the lower quartile is 14 minutes.

What was the median finishing time in the upper quartile? The median of the upper quartile is 21 minutes.

Example J

The box-and-whisker plot below depicts the number of books students read during summer vacation. Write a paragraph to describe the data on the box-and-whisker plot.

Here is a paragraph that could be written to describe the data.

The least amount of books read is one. The greatest amount of books read is ten. Five is the median number of books read. Because the number line is labeled by ones, each section in between each number represents one-half. Therefore, the median number of books read in the lower quartile is two and one-half. The median number of books read in the upper quartile is eight. You can see that the data is split evenly. Fifty percent of the students read more than five books. Fifty percent of the students read less than five books.

Real-Life Example Completed

The Iditarod

Here is the original problem once again. Reread it and then compare your answer for the mean with the given solution.

http://www.iditarod.com/

The students sat through the video in awe. When it was over, the room was so quiet that you could have heard a pin drop. Marcus was the first one to raise his hand.

“How far is it?” he asked.

“How fast did they go?” Karen asked from the back of the room. “I mean, you can’t go very fast on a dog sled, right?”

4.81 mph, 4.79 mph, 4.76 mph, 4.67 mph, 4.66 mph, 4.64 mph, 4.62 mph, 4.6 mph, 4.58 mph, 4.55 mph.

“Take out a piece of paper and figure this out.”

Marcus took out a piece of paper, but he couldn’t remember how to figure out the average score.

By now you understand that the average is the same thing as the mean. The students have been asked to find the mean speed of the dog sleds on the 2010 Iditarod. They have been given the speeds of the top ten finishers. This is the data that we will use to figure out the mean.

First, add up all of the speeds.

4.81 + 4.79 + 4.76 + 4.67 + 4.66 + 4.64 + 4.62 + 4.6 + 4.58 + 4.55 = 46.68

Next, we divide this sum by 10 because there were ten dog sleds, so there were 10 different speeds.

$46.68 \div 10 = 4.668$ rounds to 4.67

The average speed is 4.67 mph.

Now that you understand mean, median, mode, and range, you can figure out the median, mode, and range of the data as well. Take a few minutes to do that.

The median is the middle speed. This is between 4.66 and 4.64. We can say that the median speed is 4.65 mph.

There isn’t a mode for this data.

The range is the difference between the fastest speed and the slowest.

$4.81 - 4.55 = .26 \ mph$

You can see that there wasn’t a huge difference between the fastest time and the slowest time. But that .26 was enough to make the difference between first and tenth place!

Review

Another term for mean is average.
A frequent mistake for mean is forgetting to divide the sum that is calculated for mean, so make sure you complete all steps until you find the average.

Data

Data is numerical information collected in a set.

Lower Quartile

The lower quartile is the median of the lower half of the data.

Mean

The mean is the sum of the data values divided by the number of data values.

Median

The median is the middle number when the values are arranged in order from the least to the greatest.

Mode

The mode is the data value that occurs most often.

Range

The range of a set of data is the difference between the greatest and least values.

Stem & Leaf Plot

Stem-and-Leaf plot is a method of organizing numerical data in order of place value. The

“ones digit” and the”'tens digit and greater” of each data item is separated as leaves and

stems respectively.

Upper Quartile

The upper quartile is the median of the upper half of the data.

Video Resources

James Sousa, Example of Finding the Mean of a Data Set

James Sousa, Example of Finding the Median of a Data Set

http://www.mathplayground.com/howto_mode.html

Khan Academy, Reading Box and Whisker Plots

Khan Academy, Stem and Leaf Plots

http://www.mathplayground.com/howto_stemleaf.html