Lesson 5: Box Plots and the Five-Number Summary

Property

A box plot (or box-and-whisker plot) is a visual representation of the five-number summary, and tells us much more about the spread of the data.
A box is drawn from the 25th percentile to the 75th percentile, and 'whiskers' are drawn from the lowest data value to the 25th percentile and from the 75th percentile to the highest data value.

Examples

• Given a five-number summary of {Min: 4, Q1: 7, Med: 10, Q3: 12, Max: 18}, a box plot would show a box extending from 7 to 12 with a line at 10, and whiskers reaching to 4 and 18.
• A data set of quiz scores is {5, 6, 6, 7, 8, 9, 10}. The five-number summary is {Min:5, Q1:6, Med:7, Q3:9, Max:10}. Its box plot has a box from 6 to 9 and whiskers from 5 to 10.
• If one box plot has a much longer box than another, it means the middle half of its data is more spread out. A longer right whisker means the top 25% of values are more spread out than the bottom 25%.

Explanation

A box plot turns the five-number summary into a picture. The 'box' contains the middle 50% of your data. The 'whiskers' stretch out to the highest and lowest values, showing the full range and how spread out everything is.

Property

The 5 number summary consists of five values: Minimum, Q1, Q2 (median), Q3, and Maximum. To find them:

1. Order the data from smallest to largest. The ends are the Minimum and Maximum.
2. Find the median of the entire data set (Q2).
3. Find the median of the lower half of the data (Q1).
4. Find the median of the upper half of the data (Q3).

The interquartile range (IQR) is the distance from Q1 to Q3, representing the middle 50% of the data. IQR = Q3 - Q1.

Examples

• For the data {2, 5, 6, 9, 11, 14, 17}, the 5 number summary is: Min=2, Q1=5, Q2=9, Q3=14, Max=17. The IQR is $14 - 5 = 9$.
• For the data {10, 20, 25, 35, 45, 50}, the median (Q2) is 30. The lower half is {10, 20, 25}, so Q1=20. The upper half is {35, 45, 50}, so Q3=45. The IQR is $45 - 20 = 25$.
• Given a 5 number summary of Min=5, Q1=12, Q2=18, Q3=22, Max=30, the range is $30-5=25$ and the interquartile range (IQR) is $22 - 12 = 10$.

Explanation

The 5 number summary provides a quick snapshot of your data's distribution. The IQR is a powerful measure of spread because it focuses on the middle half of the data, which means it isn't affected by unusually high or low outliers.