Lesson 8Summarizing Distributions

Learning Goal

Let’s explore the mean and median.

Let’s summarize distributions.

Learning Targets

  • I can determine when the mean, median, range, or interquartile range is more appropriate to describe the data.

  • I can explain how the distribution of data affects the mean and the median.

  • I can use a dot plot, histogram, and box plot to answer questions about a data set.

  • I can use the medians and the IQR to compare groups.

Lesson Terms

  • interquartile range (IQR)
  • quartile

Warm Up: Mean vs Median

Problem 1

For each dot plot or histogram:

  • Predict if the mean is greater than, less than, or approximately equal to the median. Explain your reasoning.

  • Which measure of center–the mean or the median–better describes a typical value for the following distributions?

  1. Heights of 50 WNBA players

    A dot plot of height of WNBA players ranging from 75 to 82.

  2. Time spend on last night’s homework of 55 seventh-grade students

    A histogram of time spent on homework (in minutes) ranging from 0 to 60.

  3. Ages of 30 people at a concert

    A histogram for “Ages of 30 people at a family dinner party.” The horizontal axis is labeled “ages in years,” with the numbers 0 through 55, in increments of 5, indicated. The vertical axis has the numbers 0 through 12, in increments of 2, indicated. The approximate height of the bars are as follows: From 5 up to 10 years, 2; From 10 up to 15 years, 3; From 15 up to 20 years, 1; From 20 up to 25 years, 1; From 25 up to 30 years, 2; From 30 up to 35 years, 3; From 35 up to 40 years, 2. From 40 up to 45 years, 5; From 45 up to 50 years, 11.

Activity 1: Mean or Median?

Problem 1

Your teacher will give you six cards. Each has either a dot plot or a histogram.

  1. Sort the cards into two piles based on the distributions shown. Be prepared to explain your reasoning.

  2. Discuss your sorting decisions with another group. Did you have the same cards in each pile? If so, did you use the same sorting categories? If not, how are your categories different?

    Pause here for a class discussion.

Problem 2

Use the information on the cards to answer the following questions.

  1. Card A: What is a typical age of the dogs at the local dog park?

  2. Card B: What is a typical number of people in the Croatian households?

  3. Card C: What is a typical travel time for the Chinese students?

  4. Card D: Would 18 years old be a good description of a typical age of the people who attended the birthday party?

  5. Card E: Is 12.5 minutes or 24.5 minutes a better description of a typical time it takes the students in Argentina to get to school?

  6. Card F: Would 18.6 years old be a good description of a typical age of the people who went on a field trip to Chicago?

Problem 3

How did you decide which measure of center to use for the dot plots on Cards A–C? What about for those on Cards D–F?

Activity 2: The Five-Number Summary

Here are the ages of a group of the 20 people playing an online game, shown in order from least to greatest.

5

7

9

10

10

11

13

16

16

20

20

21

23

24

27

31

33

35

38

45

Problem 1

Provide the following information for the data.

Minimum: Q1: Q2: Q3: Maximum:

Problem 2

The median (or Q2) value of this data set is 20. This tells us that half of the players are 20 or younger, and that the other half are 20 or older. What does each of the following values tell us about the ages of the players?

  1. Q3

  2. Minimum

  3. Maximum

Problem 3

  1. What is the range of the data set?

  2. What is the IQR of the data set?

Lesson Summary

Both the mean and the median are ways of measuring the center of a distribution. They tell us slightly different things, however.

The dot plot shows the weights of 30 cookies. The mean weight is 21 grams (marked with a triangle). The median weight is 20.5 grams (marked with a diamond).

A dot plot for cookie weights in grams. The numbers 8 through 34, in increments of 2, are indicated. A diamond is indicated at 20.5 grams and a triangle is indicated at 21 grams. Data are as follows: 9 grams, 1 dot; 10 grams, 1 dot; 11 grams, 2 dots; 12 grams, 1 dot; 14 grams, 1 dot; 16 grams, 2 dots; 17 grams, 1 dot; 18 grams, 2 dots; 19 grams, 1 dot; 20 grams, 3 dots; 21 grams, 1 dot; 22 grams, 3 dots; 23 grams, 1 dot; 24 grams, 2 dots; 26 grams, 2 dots; 28 grams, 1 dot; 30 grams, 1 dot; 32 grams, 2 dots; 33 grams, 1 dot; 34 grams, 1 dot.

The mean tells us that if the weights of all cookies were distributed so that each one weighed the same, that weight would be 21 grams. We could also think of 21 grams as a balance point for the weights of all of the cookies in the set. 

The median tells us that half of the cookies weigh more than 20.5 grams and half weigh less than 20.5 grams. In this case, both the mean and the median could describe a typical cookie weight because they are fairly close to each other and to most of the data points.

Here is a different set of 30 cookies. It has the same mean weight as the first set, but the median weight is 23 grams.

A dot plot for “cookie weights in grams.” The numbers 8 through 34, in increments of 2, are indicated. A triangle is indicated at 21 grams, and a diamond is indicated at 23 grams. The data are as follows: 9 grams, 1 dot; 10 grams, 1 dot; 13 grams, 1 dot; 14 grams, 1 dot; 16 grams, 1 dot; 17 grams, 1 dot; 19 grams, 1 dot; 20 grams, 2 dots; 21 grams, 2 dots; 22 grams, 3 dots; 23 grams, 6 dots; 24 grams, 5 dots; 25 grams, 4 dots; 26 grams, 1 dot.

In this case, the median is closer to where most of the data points are clustered and is therefore a better measure of center for this distribution. That is, it is a better description of a typical cookie weight. The mean weight is influenced (in this case, pulled down) by a handful of much smaller cookies, so it is farther away from most data points.

In general, when a distribution is symmetrical or approximately symmetrical, the mean and median values are close. But when a distribution is not roughly symmetrical, the two values tend to be farther apart.