Influential Point vs Outlier

Influence is defined by a comparison, not by appearance: fit the line with the point, fit it again without, and see how much the answer moves. That makes influence a verdict you reach by refitting, unlike the two labels it gets confused with. An outlier in regression has a large residual, a high-leverage point has an $x$ far from $\bar{x}$ (x-bar), and neither one is automatically influential.

Start with six students who study 1, 2, 3, 4, 5 and 6 hours and score 60, 72, 65, 78, 71 and 86, so $\hat{y} = 58 + 4x$ and $r = 0.8107$. Add a seventh who studied 10 hours and scored 75. That point is 23 below the old line, and refitting drops the slope from 4 to 1.6144 and the correlation to 0.5681. Now instead add a seventh who studied 3 hours and scored 100, a residual of 30, larger than the first. Refitting moves the slope only to 3.2742. The bigger miss shifted the line about a third as far, because it sat next to $\bar{x}$ where the line can pivot around it.

So the sentence to drop is "that point has a huge residual, so it is influential." Residual size is not the measure. Distance from the mean of $x$ is what converts a miss into leverage over the whole fit.

Leverage on its own is not the verdict either. Add a seventh student at 10 hours who scored 98, which sits exactly on the original line: that point has more leverage than any other in the set, and the fit stays at $\hat{y} = 58 + 4x$, unchanged to the last digit. It does lift $r$ from 0.8107 to 0.9245, so it is influential for the correlation while having no influence at all on the line. Always say what it is influential for.

Finding an influential point is not permission to delete it. Investigate it, and report the fit both ways if you cannot resolve it.

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.