Lesson 16Box Plots

Learning Goal

Let’s explore how box plots can help us summarize distributions.

Learning Targets

  • I can use the five-number summary to draw a box plot.

  • I know what information a box plot shows and how it is constructed.

Lesson Terms

  • box plot
  • interquartile range (IQR)
  • median
  • quartile
  • range

Warm Up: Notice and Wonder: Puppy Weights

Problem 1

Here are the birth weights, in ounces, of all the puppies born at a kennel in the past month.

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  • 14

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  • 20

What do you notice and wonder about the distribution of the puppy weights?

Activity 1: Human Box Plot

Problem 1

Your teacher will give you the data on the lengths of names of students in your class. Write the five-number summary by finding the data set’s minimum, Q1, Q2, Q3, and the maximum.

Pause for additional instructions from your teacher.

Activity 2: Studying Blinks

Twenty people participated in a study about blinking. The number of times each person blinked while watching a video for one minute was recorded. The data values are shown here, in order from smallest to largest.

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  • 6

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  • 11

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  • 32

  • 36

  • 51

Problem 1

  1. Use the grid and axis to make a dot plot of this data set.

    A blank grid for “number of blinks” with the numbers 0 through 55, in increments of 5 indicated along the horizontal axis. Between each number indicated there are 4 evenly spaced vertical gridlines. There are 13 horizontal gridlines.

  2. Find the median (Q2) and mark its location on the dot plot.

  3. Find the first quartile (Q1) and the third quartile (Q3). Mark their locations on the dot plot.

  4. What are the minimum and maximum values?

Problem 2

A box plot can be used to represent the five-number summary graphically. Let’s draw a box plot for the number-of-blinks data. On the grid, above the dot plot:

  1. Draw a box that extends from the first quartile (Q1) to the third quartile (Q3). Label the quartiles.

  2. At the median (Q2), draw a vertical line from the top of the box to the bottom of the box. Label the median.

  3. From the left side of the box (Q1), draw a horizontal line (a whisker) that extends to the minimum of the data set. On the right side of the box (Q3), draw a similar line that extends to the maximum of the data set.

Problem 3

You have now created a box plot to represent the number of blinks data. What fraction of the data values are represented by each of these elements of the box plot?

  1. The left whisker

  2. The box

  3. The right whisker

Are you ready for more?

Problem 1

Suppose there were some errors in the data set: the smallest value should have been 6 instead of 3, and the largest value should have been 41 instead of 51. Determine if any part of the five-number summary would change. If you think so, describe how it would change. If not, explain how you know.

Lesson Summary

A box plot represents the five-number summary of a data set.

It shows the first quartile (Q1) and the third quartile (Q3) as the left and right sides of a rectangle or a box. The median (Q2) is shown as a vertical segment inside the box. On the left side, a horizontal line segment—a “whisker”—extends from Q1 to the minimum value. On the right, a whisker extends from Q3 to the maximum value.

The rectangle in the middle represents the middle half of the data. Its width is the IQR. The whiskers represent the bottom quarter and top quarter of the data set.

Earlier we saw dot plots representing the weights of pugs and beagles. The box plots for these data sets are shown above the corresponding dot plots.

Box plots and dot plots for two sets of data: "pug weights in kilograms” and "beagle weights in kilograms". The numbers 6 through 11 are indicated and there are tick marks midway between each indicated number. Each box plot is above it's corresponding dot plot. The approximate data for the box plot for "pug weights in kilograms" are as follows: Minimum value, 6. Maximum value, 8. Q1, 6.5. Q2, 7. Q3, 7.3. The approximate data for the dot plot for "pug weights in kilograms" are as follows: 6 kilograms, 1 dot. 6.2 kilograms, 2 dots. 6.4 kilograms, 2 dots. 6.6 kilograms, 2 dots. 6.8 kilograms, 2 dots. 7 kilograms, 3 dots. 7.2 kilograms, 3 dots. 7.4 kilograms, 1 dot. 7.6 kilograms, 2 dots. 7.8 kilograms, 1 dot. 8 kilograms, 1 dot. The approximate data for the box plot for "beagle weights in kilograms" are as follows: Minimum value, 9. Maximum value, 11. Q1, 9.6. Q2, 10. Q3, 10.5. The approximate data for the dot plot for "beagle weights in kilograms" are as follows: 9 kilograms, 1 x. 9.2 kilograms, 2 x's. 9.4 kilograms, 1 x. 9.6 kilograms, 3 x's. 9.8 kilograms, 1 x. 10 kilograms, 3 x's. 10.2 kilograms, 3 x's. 10.4 kilograms, 1 x. 10.6 kilograms, 2 x's. 10.8 kilograms, 2 x's. 11 kilograms, 1 x.

We can tell from the box plots that, in general, the pugs in the group are lighter than the beagles: the median weight of pugs is 7 kilograms and the median weight of beagles is 10 kilograms. Because the two box plots are on the same scale and the rectangles have similar widths, we can also tell that the IQRs for the two breeds are very similar. This suggests that the variability in the beagle weights is very similar to the variability in the pug weights.