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Statistics and probability

Course: Statistics and probability > Unit 3

Lesson 7: Box and whisker plots
Math
Statistics and probability
Summarizing quantitative data
Box and whisker plots
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Box plot review

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What is a box and whisker plot?

A box and whisker plot—also called a box plot—displays the five-number summary of a set of data. The five-number summary is the minimum, first quartile, median, third quartile, and maximum.
In a box plot, we draw a box from the first quartile to the third quartile. A vertical line goes through the box at the median. The whiskers go from each quartile to the minimum or maximum.
A box and whisker plot with the left end of the whisker labeled min, the right end of the whisker is labeled max. The beginning of the box is labeled Q 1. The end of the box is labeled Q 3. The line that divides the box is labeled median.

Example: Finding the five-number summary

A sample of 10 boxes of raisins has these weights (in grams):
25, 28, 29, 29, 30, 34, 35, 35, 37, 38
Make a box plot of the data.
Step 1: Order the data from smallest to largest.
Our data is already in order.
25, 28, 29, 29, 30, 34, 35, 35, 37, 38
Step 2: Find the median.
The median is the mean of the middle two numbers:
25, 28, 29, 29, 30, 34, 35, 35, 37, 38
start fraction, 30, plus, 34, divided by, 2, end fraction, equals, 32
The median is 32.
Step 3: Find the quartiles.
The first quartile is the median of the data points to the left of the median.
25, 28, 29, 29, 30
Q, start subscript, 1, end subscript, equals, 29
The third quartile is the median of the data points to the right of the median.
34, 35, 35, 37, 38
Q, start subscript, 3, end subscript, equals, 35
Step 4: Complete the five-number summary by finding the min and the max.
The min is the smallest data point, which is 25.
The max is the largest data point, which is 38.
The five-number summary is 25, 29, 32, 35, 38.

Example (continued): Making a box plot

Let's make a box plot for the same dataset from above.
Step 1: Scale and label an axis that fits the five-number summary.
A number line labeled weight in grams. It is numbered from 25 to 40.
Step 2: Draw a box from Q, start subscript, 1, end subscript to Q, start subscript, 3, end subscript with a vertical line through the median.
Recall that Q, start subscript, 1, end subscript, equals, 29, the median is 32, and Q, start subscript, 3, end subscript, equals, 35, point
The box of a box and whisker plot without the whiskers. The beginning of the box is labeled Q 1 at 29. The end of the box is labeled Q 3 at 35. The vertical line that divides the box is labeled median at 32.
Step 3: Draw a whisker from Q, start subscript, 1, end subscript to the min and from Q, start subscript, 3, end subscript to the max.
Recall that the min is 25 and the max is 38.
A box and whisker plot. The beginning of the box is labeled Q 1 at 29. The end of the box is labeled Q 3 at 35. The vertical line that divides the box is labeled median at 32. The left part of the whisker is labeled min at 25. The right part of the whisker is labeled max 38.
We don't need the labels on the final product:
A box and whisker plot. The beginning of the box is at 29. The end of the box is at 35. The vertical line that divides the box is at 32. The left part of the whisker is at 25. The right part of the whisker is at 38.
Want to learn more about making box and whisker plots? Check out this video.
Want to practice making box plots? Check out this exercise.

Interpreting quartiles

The five-number summary divides the data into sections that each contain approximately 25, percent of the data in that set.
A box and whisker plot with the left end of the whisker labeled min, the right end of the whisker is labeled max. The beginning of the box is labeled Q 1. The end of the box is labeled Q 3. The line that divides the box is labeled median. The distance from the min to the Q 1 is twenty five percent. The distance from the Q 1 to the Q 2 is twenty five percent. The distance from the Q 2 to the Q 3 is twenty five percent. The distance from the Q 3 is Max is twenty five percent.

Example: Interpreting quartiles

About what percent of the boxes of raisins weighed more than 29 grams?
A box and whisker plot. The beginning of the box is at 29. The end of the box is at 35. The vertical line that divides the box is at 32. The left part of the whisker is at 25. The right part of the whisker is at 38.
Since Q, start subscript, 1, end subscript, equals, 29, about 25, percent of data is lower than 29 and about 75, percent is above is 29.
A box and whisker plot. The beginning of the box is labeled Q 1 at 29. The end of the box is at 35. The vertical line that divides the box is at 32. The left part of the whisker is at 25. The right part of the whisker is at 38. The distance from the Q 1 to the dividing vertical line is twenty five percent. The distance from the vertical line to the end of the box is twenty five percent. The distance from the Q 3 is Max is twenty five percent.
About 75, percent of the boxes of raisins weighed more than 29 grams.
Want to learn more about interpreting quartiles? Check out this video.
Want to practice more problems like this? Check out this exercise.

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