What Counts as an Outlier (\(1.5 \times \mathrm{IQR}\) Rule)?

An outlier is a data value that lies more than $1.5$ times the interquartile range below the lower quartile or above the upper quartile. A value exactly on either boundary is not an outlier.

This is part of IB Mathematics: Analysis and Approaches, SL 4.1: Concepts, reliability and sampling techniques, and may be assessed on Paper 1 or Paper 2.

The interquartile range measures the spread of the middle half of the data:

$$\mathrm{IQR}=Q_3-Q_1$$

Quartiles divide an ordered data set into four parts. $Q_1$ marks the lower quarter and $Q_3$ marks the upper quarter, so the IQR is resistant to extreme observations. This makes it more appropriate than the range for detecting values separated from the central spread.

Use it to calculate two outlier fences:

$$\text{lower fence}=Q_1-1.5\left(\mathrm{IQR}\right)$$ $$\text{upper fence}=Q_3+1.5\left(\mathrm{IQR}\right)$$

The phrase “from the nearest quartile” explains why the lower fence starts at $Q_1$ and the upper fence starts at $Q_3$. Do not measure every observation from the median, and do not use $1.5$ times the full range.

For example, suppose $Q_1=5$ and $Q_3=13$. Then $\mathrm{I}\mathrm{Q}\mathrm{R}$, giving fences of and . Therefore, and are outliers, but and are not.

In a box-and-whisker diagram, identified outliers are plotted separately with crosses. However, an outlier is not automatically an error: it may be a valid extreme observation, so any interpretation must use the context and reliability of the data source.

A common misconception is that a value equal to a fence is an outlier. The definition says more than, so strict inequalities are required.

In an exam, show the IQR and both fences before listing outliers. For calculate, working is required; do not provide only the final values.