Skewed left vs skewed right: what happens to the mean

By Jude Wallis · Published July 28, 2026

The tail names the skew: a right-skewed (positively skewed) distribution has a longer right tail toward larger values, and a left-skewed (negatively skewed) one has a longer left tail. The long tail usually pulls the mean toward it, so right skew tends to put the mean above the median.

AP Statistics: Unit 1 (topics 1.6 Descriptions for One Quantitative Variable Distributions, 1.7 Summary Statistics for One Quantitative Variable, 1.8 Graphical Representations of Summary Statistics for One Quantitative Variable). Reading skew and its effect on the mean sits in Unit 1 of the Fall 2026 AP Statistics course: describing a quantitative distribution's shape (topic 1.6), choosing resistant summaries such as the median and IQR (1.7), and the mean-median relationship under skew (1.8).

The tail names the skew

The direction of a skew is named for its longer tail, not for the side where the peak sits. A distribution is skewed right, also called positively skewed, when the right tail toward the larger values is longer than the left. It is skewed left, or negatively skewed, when the left tail toward the smaller values is longer than the right. When the left half is roughly a mirror image of the right half, the distribution is approximately symmetric.

The names feel backward to a lot of students, because a right-skewed pile of data leans to the left: most values bunch low and a thin tail reaches high. Household income is the classic right-skewed case, where most people cluster at modest values while a few very high earners stretch the right tail. Scores on an easy test are the classic left-skewed case, where most students bunch near the top and a few low scores stretch a tail to the left. Fix the pattern in your head with the income and easy-test pictures, and the naming stops tripping you up.

What skew does to the mean

The mean, the average of all the values, gets pulled toward the longer tail. The median, the middle value once the data are sorted, barely moves, because it only tracks position. In a right-skewed distribution the high values in the tail drag the mean up, so the mean usually lands above the median. In a left-skewed distribution the low values in the tail drag the mean down, so the mean usually lands below the median.

When the shape is roughly symmetric the mean and median sit close together. Read the direction of the gap as a shape clue: a mean well above the median points to right skew or a high outlier, and a mean well below points to left skew or a low outlier. This is a strong tendency, not a law. It holds for nearly every distribution in this course, but skewed distributions exist where the mean falls on the other side of the median, so confirm the shape with a graph when you can.

You can watch this happen in the descriptive statistics sandbox by dragging one point out into a tail and seeing the mean chase it while the median holds.

Why the tail pulls the mean

The reason the mean chases the tail is mechanical. Each value is weighted equally in the mean, a $\frac{1}{n}$ share where $n$ is the number of values, so a value far out in a tail adds a large amount to the sum and pulls the average toward it. The median only tracks the middle position, so it ignores how far out that tail value sits and stays put near the bulk of the data. The bigger the pull from the tail, the wider the gap you see between the mean and the median.

This also explains the honest caution about the tendency. The pull depends on how the tail values are spread, and in unusual cases a tail can be long yet light enough that the mean does not cross the median. For the histograms and boxplots you will see on the AP exam, the mean-follows-the-tail read is a safe first move. For the full mechanism, see why the mean moves and the median holds.

Reading skew from a histogram or dotplot

Find the bulk of the data first, then see which way the thin tail stretches. In a histogram or dotplot the bars or dots pile up on one side and thin out into a tail on the other. The tail points in the direction of the skew: a tail trailing off to the right is right skew, and a tail trailing off to the left is left skew. If both sides fall away from a central peak at about the same rate, call it approximately symmetric.

While you are looking, describe the other features the AP course asks for. A shape with one clear peak is unimodal, one with two prominent peaks is bimodal, and one that is nearly flat with no real peak is approximately uniform. Note outliers, gaps, and clusters too, since a full description of a quantitative distribution covers shape, center, spread, and any unusual features in context.

Reading skew from a boxplot

A boxplot hides the individual points, so you read skew from the lengths of its pieces. The box spans the middle 50% of the data, from the first quartile to the third quartile, with a line at the median. Compare the two whiskers and the two halves of the box: the side with the longer whisker and the more stretched half is the tail. A long right whisker with the median pushed toward the left side of the box signals right skew, and the mirror image signals left skew.

One caution about boxplots. They show skew reliably through the box and the whiskers, but they cannot show peaks, so a boxplot will never tell you whether a distribution is unimodal or bimodal. If you need shape detail beyond the quartiles, go back to a histogram or dotplot. For a step-by-step build, see how to make a boxplot.

Which summaries to report when data are skewed

Under skew, report the median and the interquartile range, not the mean and standard deviation. The median and the IQR (the interquartile range, the distance from the first quartile to the third) are resistant, which means outliers and a long tail barely change them. The mean and the standard deviation are nonresistant, so the same tail that defines the skew also distorts them. That is why news reports quote median income and median home price rather than the mean.

When the AP exam asks you to justify a choice of summary, name the reason out loud. Point to the skew or an outlier when you choose the median and IQR, and point to a roughly symmetric shape with no extreme values when you choose the mean and standard deviation. Reporting both pairs is fine when you are unsure, and the gap between the mean and median doubles as a quick skew check.

Common mistakes

A handful of errors show up on nearly every skew question.

Naming the skew by the side with the peak. The skew is named for the tail, so a pile of data on the left with a tail to the right is right skew, not left.
Treating 'mean above median means right skew' as a guarantee. It is a strong tendency, so use it to reason but confirm with a graph.
Reporting the mean for skewed money data without checking the shape, when one large value can make it misleading.
Trying to count peaks in a boxplot. A boxplot cannot show modes, so a right-skewed set and a bimodal set can share the same box.
Forgetting the rest of the description. Shape is only one part; a complete answer also gives center, spread, and any outliers, gaps, or clusters in context.

Describe a right-skewed distribution

A cafe records how long, in minutes, nine customers waited in line: 1, 1, 2, 2, 3, 3, 4, 6, 14. Describe the distribution (shape, center, spread, and unusual features) and say what the mean and median reveal about the skew.

Sort the data (already sorted): 1, 1, 2, 2, 3, 3, 4, 6, 14. Here $n = 9$ is the number of values.
Median: with $n = 9$ (odd), the median is the middle (5th) value $= 3$ minutes.
Mean: the sum is $1 + 1 + 2 + 2 + 3 + 3 + 4 + 6 + 14 = 36$ , so the mean $\bar{x}$ (read x-bar) $= \frac{36}{9} = 4$ minutes.
Compare center: the mean (4) sits above the median (3). A mean above the median is the usual signature of right skew.
Quartiles (TI-84 median-excluded method): the lower half is the four values below the median, 1, 1, 2, 2, so the first quartile $Q_1 = \frac{1+2}{2} = 1.5$ . The upper half is 3, 4, 6, 14, so the third quartile $Q_3 = \frac{4+6}{2} = 5$ .
Spread: the interquartile range is $IQR = Q_3 - Q_1 = 5 - 1.5 = 3.5$ minutes, and the range is $14 - 1 = 13$ minutes.
Check for an outlier with the 1.5 IQR rule: $1.5 \times 3.5 = 5.25$ , and $Q_3 + 5.25 = 5 + 5.25 = 10.25$ . The value 14 is above 10.25, so it is a high outlier. The low fence is $Q_1 - 5.25 = 1.5 - 5.25 = -3.75$ , so there are no low outliers.
Shape: the data pile up at the low end and thin out toward the high values into a long right tail, so the distribution is skewed right and unimodal.

The wait times are skewed right and unimodal, with a high outlier at 14 minutes. Center: median 3 minutes (the mean is 4 minutes, pulled up by the tail). Spread: IQR 3.5 minutes, range 13 minutes. Because of the skew and the outlier, report the median and IQR rather than the mean and standard deviation.

Left skew pulls the mean below the median

Seven students take an easy quiz scored out of 10 and earn 4, 7, 8, 9, 9, 10, 10. Find the mean and median and say what the gap tells you about the skew.

Sort the data (already sorted): 4, 7, 8, 9, 9, 10, 10. Here $n = 7$ is the number of values.
Median: with $n = 7$ (odd), the median is the middle (4th) value $= 9$ points.
Mean: the sum is $4 + 7 + 8 + 9 + 9 + 10 + 10 = 57$ , so the mean $\bar{x} = \frac{57}{7} \approx 8.14$ points.
Compare: the mean (about 8.14) is below the median (9), a gap of about 0.86 point.
The single low score of 4 sits in a thin left tail and drags the mean down, while the median stays up near the bunched high scores.

Mean about 8.14 points, median 9 points. The mean sits below the median, the usual pattern for a left-skewed distribution with a long left tail. The low score of 4 pulls the mean down while the median holds at 9.

Frequently asked questions

Is skewed right the same as positively skewed?

Yes. Skewed right and positively skewed are two names for the same shape, a distribution with a longer tail toward the larger values. Skewed left and negatively skewed likewise mean the same thing, a longer tail toward the smaller values.

Does the peak or the tail decide the skew direction?

The tail decides. A distribution is named for the side its long thin tail stretches toward, not the side where the values pile up. Right-skewed data have most values on the left with a tail reaching right, which is why the name can feel backward at first.

In a skewed distribution, is the mean or the median bigger?

Usually the mean is bigger in a right-skewed distribution and smaller in a left-skewed one, because the long tail pulls the mean toward it. This is a strong tendency rather than a rule, so check a graph when you can. In a roughly symmetric distribution the two are close.

Can you tell skew from a boxplot?

Yes, from the lengths of the whiskers and the two halves of the box. A longer right whisker with the median pushed toward the left of the box points to right skew, and the mirror image points to left skew. A boxplot cannot show the number of peaks, so use a histogram if you need to know whether the shape is unimodal or bimodal.