Read it again any time, including while you work through the exercises.
How can you compare at a glance the heights of two whole classes, or the temperatures of two cities over a year? Looking at every value one by one is impossible. The box plot (box-and-whisker diagram) summarizes an entire data set with only five numbers: the minimum, the first quartile Q1, the median, the third quartile Q3 and the maximum.
The picture speaks for itself: a box stretches from Q1 to Q3, with a vertical line inside it for the median. On each side, a 'whisker' connects the box to the minimum and the maximum. So the box contains roughly the central half of the values: 25% of the data lie below Q1, 25% above Q3.
The width of the box has a name: the interquartile range, equal to Q3 - Q1. It measures the spread of the central half of the data. Small interquartile range: the middle values are tightly packed. Large one: they are spread out.
Why not just use the range (maximum - minimum)? Because the range is fragile: one single extreme value, one exceptional heatwave day, and it blows up. The interquartile range ignores the lowest 25% and the highest 25%: it resists extreme values. It is a robust indicator.
Watch out for a frequent confusion: the line inside the box is the MEDIAN, not the mean. The mean cannot be read off a box plot.
Here are the ordered grades of 11 students: 2; 3; 5; 7; 8; 9; 11; 12; 14; 15; 20. Build the box plot summary and compute the interquartile range.
- 1
n = 11 ; médiane = 6e valeur = 9
With 11 values, the median is the 6th: there are 5 values before and 5 after. It is the line inside the box.
- 2
n/4 = 2.75 → Q1 = 3e valeur = 5
Q1 is the smallest value such that at least one quarter of the data is less than or equal to it: I take the 3rd value, which is 5.
- 3
3n/4 = 8.25 → Q3 = 9e valeur = 14
Q3 corresponds to three quarters: I take the 9th value, which is 14. So the box runs from 5 to 14.
- 4
Q3 - Q1 = 14 - 5 = 9
The interquartile range is the width of the box: 9. The whiskers reach the minimum 2 and the maximum 20.
Result: Min 2, Q1 = 5, median 9, Q3 = 14, max 20. Interquartile range: 9.