Outliers & Boxplots

You’ll be able to

Apply the 1.5 × IQR rule to identify outliers
Construct and read a boxplot using the five-number summary
Explain how outliers affect resistant versus non-resistant statistics

The five-number summary and the boxplot

The five-number summary — minimum, Q1, median, Q3, maximum — divides the data into four quarters. A boxplot draws a box from Q1 to Q3 (spanning the middle 50%, the IQR) with a line at the median, and "whiskers" extending to the most extreme non-outlier values. Boxplots are ideal for comparing several groups side by side, but they hide the number of peaks — a boxplot cannot reveal whether a distribution is bimodal.

The 1.5 × IQR rule

The standard rule flags a value as an outlier if it falls more than 1.5 × IQR below Q1 or above Q3. You compute two fences: a lower fence at Q1 − 1.5·IQR and an upper fence at Q3 + 1.5·IQR. Any value beyond a fence is an outlier. On a modified boxplot, outliers are plotted as individual dots and the whiskers stop at the last value inside the fences.

Outlier fences (1.5 × IQR rule)

lower fence = Q1 − 1.5·IQR · upper fence = Q3 + 1.5·IQR · IQR = Q3 − Q1

A data value is an outlier if it is below the lower fence or above the upper fence. The fences themselves are not necessarily data values.

Worked example

A data set has Q1 = 20, Q3 = 32. Determine the outlier fences, and decide whether the values 2 and 48 are outliers.

1.Compute the IQR: IQR = Q3 − Q1 = 32 − 20 = 12.
2.Lower fence = Q1 − 1.5·IQR = 20 − 1.5(12) = 20 − 18 = 2.
3.Upper fence = Q3 + 1.5·IQR = 32 + 1.5(12) = 32 + 18 = 50.
4.Compare: 2 is not below the lower fence of 2 (it equals it, so not beyond it), and 48 is below the upper fence of 50, so it is not an outlier.

Answer: IQR = 12, fences at 2 and 50. The value 48 is inside the upper fence (50), so it is not an outlier, and 2 is not below the lower fence (2), so it is not an outlier either. A value would need to be below 2 or above 50 to be flagged.

Checkpoint

A distribution has Q1 = 40 and Q3 = 60. Using the 1.5 × IQR rule, a value is an outlier if it is:

On the exam

Show the fence arithmetic explicitly on free response: state IQR = Q3 − Q1, multiply by 1.5, then add to Q3 and subtract from Q1. Graders want to see the boundary values, not just the word "outlier."

Checkpoint

Which statistic is most affected when a single extreme high outlier is added to a data set?

Answer the 2 checkpoints as you read.