Lesson 15: Quartiles and Interquartile Range

Let's look at other measures for describing distributions.

15.1: Notice and Wonder: Two Parties

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?

15.2: The Five-Number Summary

Here are the ages of a group of the 20 partygoers you saw earlier, shown in order from least to greatest.

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

    1. Find and mark the median on the table, and label it “50th percentile.” The data is now partitioned into an upper half and a lower half.

    2. Find and mark the middle value of the lower half of the data, excluding the median. If there is an even number of values, find and write down the average of the middle two. Label this value “25th percentile.”

    3. Find and mark the middle value of the upper half of the data, excluding the median. If there is an even number of values, find and write down the average of the middle two. Label the value “75th percentile.”

    4. You have now partitioned the data set into four pieces. Each of the three values that “cut” the data is called a quartile.

      • The first (or lower) quartile is the 25th percentile mark. Write “Q1” next to “25th percentile.”
      • The second quartile is the median. Write “Q2” next to that label.
      • The third (or upper) quartile is the 75th percentile mark. Write “Q3” next to that label.

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

  2. Record the five values that you have just identified. They are the five-number summary of the data.

    Minimum: _____     Q1: _____     Q2: _____     Q3: _____     Maximum: _____

  3. The median (or Q2) value of this data set is 20. This tells us that half of the partygoers 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 partygoers?

    1. Q3
    2. Minimum
    3. Maximum

15.3: Range and Interquartile Range

  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.

  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?

  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? Use your answer to complete the following statement:

      The interquartile range (IQR) is the length that contains the middle ______ of the values in a data set.

  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:

    a.  Which data set has the smaller IQR? Explain your reasoning.

    b.  Which data set has the smaller range? Explain your reasoning.

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

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 called the interquartile range (IQR).

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

  • The median, which cuts the data into a lower half and an upper half, is the second quartile (Q2). 
  • 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. 

Here is a set of data with 11 values.

12 19 20 21 22 33 34 35 40 40 49
    Q1     Q2     Q3    
  • The median (Q2) is 33. 
  • The first quartile (Q1) is 20, the median of the numbers less than 33. 
  • The third quartile (Q3) is 40, the median of the numbers greater than 33. 

The difference between the minimum and maximum values of a data set is the range.

The difference between Q1 and Q3 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 are. The smaller the IQR, the closer the middle half of the data are. We consider the IQR a measure of spread for this reason.

A five-number summary, which includes the minimum, Q1, Q2, Q3, and the maximum, can be used to summarize a distribution.

The five numbers in this example are 12, 20, 33, 40, and 49. Their locations are marked with diamonds in the following 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 could have the same five-number summary. For instance, the following data has the same maximum, minimum, and quartiles as the one above.

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.

Practice Problems ▶

Glossary

range

range

The range of a data set is the difference between the maximum and the minimum. 

interquartile range (IQR)

interquartile range (IQR)

The interquartile range of a data set is a measure of spread of its distribution. It is the difference between the third quartile (Q3) and the first quartile (Q1).

quartile

quartile

The quartiles for a data set are three numbers that divide the data set into fourths. The median divides the set into two halves, and the first quartile (Q1) is the median of the lower half. The second quartile (Q2) is the median itself, and the third quartile (Q3) is the median of the upper half.