IQR fence (1.5 IQR rule)

The fences are the two cutoffs the 1.5 IQR rule measures data against. The lower fence is $Q_1 - 1.5 \times \text{IQR}$ and the upper fence is $Q_3 + 1.5 \times \text{IQR}$, with $\text{IQR} = Q_3 - Q_1$ and both quartiles found by the median-excluded rule the TI-84 uses: when $n$ is odd, drop the overall median before taking the median of each half. A value is an outlier when it lies strictly beyond a fence.

Nine reaction times: 8, 15, 19, 21, 22, 22, 23, 27, 37. The median is 22, the fifth value. Leave it out and the lower half 8, 15, 19, 21 gives $Q_1 = 17$, while the upper half 22, 23, 27, 37 gives $Q_3 = 25$. So the IQR is 8, $1.5 \times 8 = 12$, and the fences sit at $17 - 12 = 5$ and $25 + 12 = 37$.

The largest value is 37, landing exactly on the upper fence. "It reached the fence, so it is an outlier." The rule is more than 1.5 IQRs beyond the quartile, so a value sitting on a fence is inside it and this set has no outliers. The other frequent slip is multiplying the wrong quantity: the 1.5 multiplies the IQR, never the quartile. Here $1.5 \times 25 = 37.5$ sits close enough to 37 to hide the error, and on other data it is nowhere near.

Fences do not have to fall inside the data. The lower fence of 5 sits below the minimum of 8, so no low value here can be flagged at all. A fence marks points that deserve a second look, not points that are wrong.

The fences are built from quartiles, which are resistant, so an extreme value has little pull on the criterion judging it. Add a tenth reaction time of 200 and the upper fence moves only from 37 to 39, while the cutoff for the other rule the Fall 2026 course lists, more than 2 standard deviations from the mean, jumps from 37.43 to 153.25.