IQR Fence vs Outlier

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.

An outlier is a value that stands apart from the rest of the data, and the AP course gives two rules for deciding when apart counts. The first flags any value more than $1.5 \times \text{IQR}$ above the third quartile $Q_3$ or below the first quartile $Q_1$. The second flags any value more than 2 standard deviations from the mean. Neither rule is the definition of an outlier. They are criteria, and they do not always agree.

Take 12, 13, 19, 19, 20, 23, 25, 27, 35, 40. The quartiles are $Q_1 = 19$ and $Q_3 = 27$, so the IQR is 8 and the upper fence sits at $27 + 1.5(8) = 39$. The value 40 clears it, so the IQR rule calls 40 an outlier. Now the other rule: the mean is 23.3 and $s = 8.91$, so two standard deviations reach $23.3 + 2(8.91) = 41.12$, and 40 does not get there. Same data, two rules, opposite answers. Say which rule you used.

"There is an outlier, so it must be a mistake and I should drop it." An outlier is a statement about position inside this one data set and nothing more. A 40 from a broken sensor is an error; a 40 from an unusually productive week is the most informative point you have. Removing a flagged value is a decision about the study that needs a reason you can write down, not an arithmetic step.

The 2 standard deviation rule quietly fails on small samples. No observation among $n$ values can sit more than $\frac{n-1}{\sqrt{n}}$ sample standard deviations from the mean, because the value doing the stretching also inflates $s$. With $n = 5$ that ceiling is 1.79, so the rule flags nothing at all however extreme one of the five values is. The 1.5 IQR rule has no such blind spot.

Outliers sit in topic 1.7 of the Fall 2026 course, which is where both rules are listed, and topic 1.6 names outliers, gaps, and clusters as the unusual features every description of a distribution has to mention in context.