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

A value is an outlier if it 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.

Applying the Rule

This rule is part of SL 4.1: Introduction to Statistics in IB Mathematics: Applications and Interpretation. The interquartile range measures the spread of the middle half of the data, so it is less affected by extreme values than the range.

First calculate

$$\text{IQR}=Q_3-Q_1.$$

Then calculate the two outlier fences:

$$\text{Lower fence}=Q_1-1.5\times\text{IQR}$$ $$\text{Upper fence}=Q_3+1.5\times\text{IQR}.$$

Compare every observation with these fences.

Worked Example

Suppose $Q_1=12$ and $Q_3=20$. The interquartile range is

$$\text{IQR}=20-12=8.$$

Therefore, the fences are

$$12-1.5(8)=0$$

and

$$20+1.5(8)=32.$$

Values below $0$ or above $32$ are outliers. Thus, $-1$ and $35$ are outliers, while $0$ and $32$ are not because they lie exactly on the fences. Notice that the rule identifies possible unusual observations; it does not prove that a value is an error. In context, an outlier may be a valid observation or may result from incorrect measurement or recording.

A common misconception is measuring $1.5\times\text{IQR}$ from the median. This is incorrect: measure the lower distance from $Q_1$ and the upper distance from $Q_3$. Another error is using or to classify boundary values as outliers; the definition requires values to be strictly beyond the fences.

Exam Technique

Show $Q_1$, $Q_3$, the IQR, both fences, and the final comparisons. On a box-and-whisker diagram, plot outliers separately and end each whisker at the most extreme non-outlier. Use quartiles obtained consistently from the given data or your GDC, retain full precision during calculations, and round only the final results.

If asked to identify outliers from raw data, sort the data first and obtain $Q_1$ and $Q_3$ using one consistent quartile convention. Different valid conventions can produce slightly different quartiles, so use the values supplied by the question whenever available.