Mean vs median: which to use, and what skew does

Mean vs median: the short answer

The mean is the arithmetic average: add up every value and divide by how many values there are. The median is the middle value after you sort the data from low to high. For a sample of size $n$, the mean is written $\bar{x}$ and read "x-bar". The median has no standard single symbol, so it is usually written as the word itself or as $M$.

Here is the difference that decides almost everything about mean vs median. The median is resistant, which means outliers and skew barely move it. The mean is not resistant, so a single extreme value can pull it a long way from the center of the data. Every other rule in this guide follows from that one fact.

How to find the mean and the median

The sample mean adds the values and divides by the count:

The symbol $\sum x_i$ means "add up all the data values," and $n$ is the number of values.

To find the median, sort the data from smallest to largest first. If $n$ is odd, the median is the single middle value. If $n$ is even, the median is the average of the two middle values. For example, with the six sorted values 3, 5, 8, 9, 10, 14 the two middle values are 8 and 9, so the median is $\frac{8+9}{2} = 8.5$.

Notice what the median does not use: the actual size of the largest or smallest values. It only counts positions in the sorted list. The mean, by contrast, uses every value's exact size, which is why a far-out value has so much more effect on the mean than on the median.

How the mean and median behave in a skewed distribution

Skew describes a distribution with a longer tail on one side. In a right-skewed distribution the tail stretches toward the high values, and the mean is usually above the median. In a left-skewed distribution the tail stretches toward the low values, and the mean is usually below the median. When a distribution is roughly symmetric the mean and median land close together, and for a perfectly symmetric shape such as the normal curve they are equal.

The tail is the reason. A few unusually large values in a long right tail pull the mean toward them, while the median stays down near the bulk of the data. The stronger the skew, the wider the gap between the two, so the size of that gap is itself a rough signal of how lopsided the distribution is. This is why you see median income and median home price in news reports rather than the mean: those distributions are right-skewed, and the median describes a typical case better than the mean does.

One honest caution. "Right skew means the mean is greater than the median" is a strong tendency, not a law. It holds for almost every distribution you will meet in this course, but statisticians have built right-skewed distributions where the mean actually falls below the median. Treat it as a reliable guide for reasoning, not as a theorem you can lean on in every single case.

Why the mean moves but the median holds

Each value contributes an equal share, $1/n$, to the mean. So if you change one value by an amount $d$ and leave the rest alone, the mean shifts by exactly $d/n$. Push one value far out to the right and the mean follows it, scaled down by the sample size but never ignoring it.

The median behaves differently. It is fixed by whichever value sits in the middle position, so it responds to how many values fall above and below it, not to how large those values are. You can replace the largest value with something ten times bigger and the median will not move, because the middle position has not changed. That is what resistance means in practice, and it is the mechanical reason the two measures can disagree.

When to report the mean and when to report the median

Use the median when the data are skewed or contain outliers. Good examples include incomes and home prices, where a handful of very large values would drag the mean away from a typical case.

Use the mean when the distribution is roughly symmetric with no extreme values. The mean also feeds directly into the standard deviation and into most of the inference you do later in the course, so it is the natural default whenever the shape allows it. On the AP exam, questions often ask you to justify a choice, so name the reason: point to skew or an outlier when you pick the median, and to a symmetric shape when you pick the mean.

When you are unsure, report both. A mean well above the median points to right skew or a high outlier. A mean well below the median points to left skew or a low outlier.

Comparing the two is a fast shape check, and a boxplot or the 1.5 times IQR rule for outliers can confirm what the gap is telling you. You can build intuition in the descriptive statistics sandbox by dragging points around, or run the numbers on a data set in the five-number summary calculator.

Mistakes to avoid

A few errors show up often.

• Do not assume the median is always the better choice: for symmetric data the mean uses more information and is what the rest of the AP course is built on.
• Do not report the mean alone for money data without checking the shape, since one large value can make it misleading.
• Do not forget to sort before finding the median, because reading the middle of an unsorted list gives the wrong answer.
• Remember that the mean and median are both measures of center; they say nothing about spread, so pair them with a standard deviation or an interquartile range when you summarize a distribution.