How to read a box plot

• The median (the line inside the box) splits the data in half.
• The box runs from the lower quartile Q1 to the upper quartile Q3. It contains the middle 50% of the data.
• The whiskers reach to the smallest and largest values (or, in some conventions, to the furthest values that are not outliers).
• The interquartile range, IQR = Q3 − Q1, is the width of the box: a measure of spread that ignores extreme values.

Using the data above, already in order:

2.1, 2.5, 2.8, 3.0, 3.1, 3.3, 3.4, 3.6, 3.9, 4.4, 4.6, 5.2

1. Minimum 2.1, maximum 5.2.
2. There are 12 values, so the median is the mean of the 6th and 7th: (3.3 + 3.4)/2 = 3.35.
3. The lower half is the first six values; its median is (2.8 + 3.0)/2 = 2.9. That is Q1.
4. The upper half is the last six; its median is (3.9 + 4.4)/2 = 4.15. That is Q3.
5. IQR = 4.15 − 2.9 = 1.25.

With an odd number of values, the median is the middle value and is left out of both halves. This is the convention TI calculators and most schools use; spreadsheets sometimes interpolate instead and get slightly different quartiles. In the calculator, quartile(D, 1) and median(D) give these values directly.

• Symmetric: the median near the middle of the box and whiskers of similar length.
• Skewed right: a long right whisker and the median closer to the left of the box. Incomes and house prices often look like this.
• Skewed left: the mirror image, a long left whisker.

A common rule flags a value as an outlier if it is more than 1.5 × IQR below Q1 or above Q3. For the data above, the fences are 2.9 − 1.875 = 1.025 and 4.15 + 1.875 = 6.025, so there are no outliers. Some box plots draw outliers as separate dots and stop the whiskers at the fences; the calculator draws whiskers to the minimum and maximum, so compute the fences yourself if you need them.

Box plots drawn on the same scale make comparisons immediate. Compare the medians for a typical value, the boxes for the spread of the middle half, and the overlap of the boxes for how different the groups really are. If one group’s median lies outside the other group’s box, the difference is usually worth taking seriously.