Lesson 15Quartiles and Interquartile Range

Learning Goal

Let’s look at other measures for describing distributions.

Learning Targets

  • I can use IQR to describe the spread of data.

  • I know what quartiles and interquartile range (IQR) measure and what they tell us about the data.

  • When given a list of data values or a dot plot, I can find the quartiles and interquartile range (IQR) for data.

Lesson Terms

  • median
  • quartile
  • range

Warm Up: Notice and Wonder: Two Parties

Problem 1

Here are two dot plots including the mean marked with a triangle. Each shows the ages of partygoers at a party.

Two dot plots for “age in years,” labeled “data set A” and “data set B”. On each dot plot, the numbers 5 through 45, in increments of 5, are indicated. There is a red triangle indicated at 15 on “data set A” and at 20 on “data set B”. The data for “data set A” are as follows: 8 years, 12 dots. 10 years, 3 dots. 12 years, 1 dot. 15 years, red triangle. 36 years, 1 dot. 42 years, 1 dot. 44 years, 2 dots. The data for “data set B” are as follows: 7 years, 1 dot. 8 years, 1 dot. 9 years, 1 dot. 10 years, 2 dots. 15 years, 1 dot. 16 years, 1 dot. 20 years, 2 dots and 1 red triangle. 22 years, 1 dot. 23 years, 1 dot. 24 years, 1 dot. 28 years, 1 dot. 30 years, 1 dot. 33 years, 1 dot. 35 years, 1 dot. 38 years, 1 dot. 42 years, 1 dot.

What do you notice and wonder about the distributions in the two dot plots?

Activity 1: The Five-Number Summary

Problem 1

Given are the ages of the people at one party, listed from least to greatest.

  1. Find the median of the data set and label it “50th percentile.” This splits the data into an upper half and a lower half.

    • 7 8 9 10 10 11 12 15 16 20 20 22 23 24 28 30 33 35 38 42

  2. Find the middle value of the lower half of the data, without including the median. Label this value “25th percentile.”

  3. Find the middle value of the upper half of the data, without including the median. Label this value “75th percentile.”

  4. You have split the data set into four pieces. Each of the three values that split the data is called a quartile.

    • We call the 25th percentile the first quartile. Write “Q1” next to that number.

    • The median can be called the second quartile. Write “Q2” next to that number.

    • We call the 75th percentile the third quartile. Write “Q3” next to that number.

  5. Label the lowest value in the set “minimum” and the greatest value “maximum.”

Problem 2

The values you have identified make up the five-number summary for the data set. Record them here.

Minimum: years  

Q1: years  

Q2: years  

Q3: years  

Maximum: years

Problem 3

The median of this data set is 20. This tells us that half of the people at the party were 20 years old or younger, and the other half were 20 or older. What do each of these other values tell us about the ages of the people at the party?

  1. the third quartile

  2. the minimum

  3. the maximum

Are you ready for more?

Problem 1

There was another party where 21 people attended. Here is the five-number summary of their ages.

minimum:   5       Q1:   6       Q2:   27       Q3:   32       maximum:   60  

  1. Do you think this party had more children or fewer children than the earlier one? Explain your reasoning.

  2. Were there more children or adults at this party? Explain your reasoning.

Activity 2: Range and Interquartile Range

Problem 1

Here is a dot plot you saw in an earlier task. It shows how long Elena’s bus rides to school took, in minutes, over 12 days.

A dot plot labeled “travel time in minutes.” The numbers 5 through 14 are indicated. The data is as follows. 5 minutes, 0 dots 6 minutes, 2 dots 7 minutes, 1 dot 8 minutes, 3 dots 9 minutes, 3 dots 10 minutes, 2 dots 11 minutes, 0 dots 12 minutes, 1 dot 13 minutes, 0 dots 14 minutes, 0 dots

Write the five-number summary for this data set by finding the minimum, Q1, Q2, Q3, and the maximum. Show your reasoning.

Problem 2

The range of a data set is one way to describe the spread of values in a data set. It is the difference between the greatest and least data values. What is the range of Elena’s data?

Problem 3

Another number that is commonly used to describe the spread of values in a data set is the interquartile range (IQR), which is the difference between Q1, the lower quartile, and Q3, the upper quartile.

  1. What is the interquartile range (IQR) of Elena’s data?

  2. What fraction of the data values are between the lower and upper quartiles?

Problem 4

Here are two dot plots that represent two data sets.

Two dot plots for two data sets, labeled “data set A” and “data set B”. On each dot plot, the numbers 14 through 28, in increments of 2, are indicated. The data for “data set A” are as follows: 14, 0 dots. 15, 1 dot. 16, 3 dots. 17, 3 dots. 18, 1 dot. 19, 0 dots. 20, 0 dots. 21, 3 dots. 22, 1 dot. 23, 2 dots. 24, 3 dots. 25, 3 dots. 26, 1 dot. 27, 3 dots. 28, 1 dot. The data for “data set B” are as follows: 14 through 22, 0 dots. 23, 3 dots. 24, 3 dots. 25, 8 dots. 26, 2 dots. 27, 6 dots. 28, 3 dots.

Without doing any calculations, predict:

  1. Which data set has the smaller range?

  2. Which data set has the smaller IQR?

Problem 5

Check your predictions by calculating the IQR and range for the data in each dot plot.

Lesson Summary

Earlier we learned that the mean is a measure of the center of a distribution and the MAD is a measure of the variability (or spread) that goes with the mean. There is also a measure of spread that goes with the median. It is called the interquartile range (IQR).

Finding the IQR involves splitting a data set into fourths. Each of the three values that splits the data into fourths is called a quartile. 

  • The median, or second quartile (Q2), splits the data into two halves.

  • The first quartile (Q1) is the middle value of the lower half of the data.

  • The third quartile (Q3) is the middle value of the upper half of the data. 

For example, here is a data set with 11 values.

Q1

Q2

Q3

  • The median is 33. 

  • The first quartile is 20. It is the median of the numbers less than 33. 

  • The third quartile 40. It is the median of the numbers greater than 33. 

The difference between the maximum and minimum values of a data set is the range. The difference between Q3 and Q1 is the interquartile range (IQR). Because the distance between Q1 and Q3 includes the middle two-fourths of the distribution, the values between those two quartiles are sometimes called the middle half of the data. 

The bigger the IQR, the more spread out the middle half of the data values are. The smaller the IQR, the closer together the middle half of the data values are. This is why we can use the IQR as a measure of spread.

A five-number summary can be used to summarize a distribution. It includes the minimum, first quartile, median, third quartile, and maximum of the data set. For the previous example, the five-number summary is 12, 20, 33, 40, and 49. These numbers are marked with diamonds on the dot plot.

A dot plot. The numbers 10 through 50, in increments of 5, are indicated. There are diamonds indicated at 12, 20, 33, 40 and 49. The data are as follows: 12, 1 dot; 19, 1 dot; 20, 1 dot; 21, 1 dot; 22, 1 dot; 33, 1 dot; 34, 1 dot; 35, 1 dot; 40, 1 dot; 49, 2 dots.

Different data sets can have the same five-number summary. For instance, here is another data set with the same minimum, maximum, and quartiles as the previous example.

A dot plot. The numbers 10 through 50, in increments of 5, are indicated. There are diamonds indicated at 12, 20, 33, 40, and 49. The data are as follows: 12, 1 dot; 14, 1 dot; 16, 1 dot; 18, 1 dot; 20, 1 dot; 24, 1 dot; 26, 1 dot; 28, 1 dot; 31, 1 dot; 33, 2 dots; 36, 1 dot; 38, 1 dot; 39, 1 dot; 40, 1 dot; 44, 1 dot; 46, 1 dot; 48, 1 dot; 49, 1 dot.