Skewness vs Outlier

Skewness is a statement about the tail. A distribution is skewed right when the values thin out gradually toward the high end, and skewed left when they thin out toward the low end. The name follows the tail, never the peak, so a right-skewed graph has its tall bars on the left and its thin straggle on the right.

Take 1, 2, 3, 4, 5, 6, 21. The bulk sits between 1 and 6 with one value far above, so the shape is skewed right. The median is 4, the fourth of seven ordered values. The mean is $42/7 = 6$, and only the 21 sits above it. The long tail pulled the mean toward it and left the median alone, because the median counts only how many values lie on each side of it.

The sentence to unlearn is "it is skewed right because most of the data are on the right." That describes a left skew. The pile shows where values are common; the skew is named for where they run out. Trace the tail with a finger and read off the direction it points.

The mean landing above the median is a consequence of right skew, not its definition, and it is a strong tendency rather than a theorem: statisticians have built right-skewed distributions whose mean falls below their median. Use the comparison to confirm a shape you have already read off a graph, not to decide the shape without one.

Two boundaries. Skew is a word for single-peaked quantitative distributions, so a graph with two clear peaks gets called bimodal instead, and one distant point in an otherwise balanced set is usually reported as roughly symmetric with an outlier, since skew describes a tail that thins out gradually rather than a lone gap. And skew means nothing for categorical data: the bars of a bar graph can be reordered at will, so any lopsidedness you see there is an artifact of the order you chose. Shape vocabulary is Unit 1 topic 1.6.

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.