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?
Heights of 50 WNBA players
Time spend on last night’s homework of 55 seventh-grade students
Ages of 30 people at a concert
Activity 1: Mean or Median?
Problem 1
Your teacher will give you six cards. Each has either a dot plot or a histogram.
Sort the cards into two piles based on the distributions shown. Be prepared to explain your reasoning.
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.
Card A: What is a typical age of the dogs at the local dog park?
Card B: What is a typical number of people in the Croatian households?
Card C: What is a typical travel time for the Chinese students?
Card D: Would 18 years old be a good description of a typical age of the people who attended the birthday party?
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?
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?
Q3
Minimum
Maximum
Problem 3
What is the range of the data set?
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).
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.
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.