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When the mean is greater than the median, the usual reading is that the distribution has a long tail on the right: a few large values pull the mean up while the median barely moves. That reading is a tendency, not a law. For continuous, single-peaked data it is usually right; for discrete data it can fail, and this guide shows both the standard case and a counterexample with the arithmetic.
What it means when the mean is greater than the median
In a perfectly symmetrical distribution the mean and the median are the same. In a right-skewed distribution, OpenStax Introductory Statistics 2e (section 2.6) states that the mean is the largest of the three averages and the mode the smallest, and that when the distribution is skewed to the right the mean is often greater than the median. “Mean is greater than median” is therefore a signal to look for right skew, not proof of it.
The reason is mechanical. Every value contributes its full size to the mean, so one very large observation shifts it. The median depends only on the middle position in the sorted data, so the same observation changes it by at most one rank. Large values on the right raise the mean more than the median.
Definition in one paragraph
The mean is the sum of all values divided by how many there are. The median is the middle value of the sorted data, or the average of the two middle values when the count is even. The gap between them, mean minus median, is positive when the mean is greater than the median and negative in the opposite case, where the median is higher than the mean.
Formula, with every symbol explained
Mean: x̄ = (x1 + x2 + … + xn) / n
- xi is the i-th observation, and n is the number of observations.
- Median for odd n: the value at sorted position (n + 1) / 2. For even n: the average of the values at positions n / 2 and n / 2 + 1.
- Gap = x̄ − median. Gap > 0 means the mean is greater than the median.
- SD, the standard deviation, is used below only to put the gap in context: gap / SD says how large the gap is relative to the spread.
Worked example: why the mean is pulled above the median
Ten response times in seconds, already sorted: 1, 2, 3, 4, 5, 6, 7, 8, 9, 100.
- Sum = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 100 = 145, so mean = 145 / 10 = 14.5.
- n = 10 is even, so the median is the average of the 5th and 6th values: (5 + 6) / 2 = 5.5.
- Gap = 14.5 − 5.5 = 9.0. The sample standard deviation is 30.15, so the gap is about 0.30 SD.
- Remove the 100 and the mean of the remaining nine values is 45 / 9 = 5, close to the median of 5.5. One value created almost the whole gap.
A second check from OpenStax (section 2.6): for the data 6, 7, 7, 7, 7, 8, 8, 8, 9, 10, the sum is 77, the mean is 7.7 and the median is 7.5. Here the gap is small (0.2) and the distribution is only mildly right-skewed.
Median higher than mean: the opposite case
When the median is higher than the mean, the typical reading is left skew: a long tail of small values drags the mean down. OpenStax states that in a left-skewed distribution the mean is less than the median, and both are less than the mode. The same caution applies in reverse, and the next section shows why.
When the rule fails: discrete data
A textbook rule of thumb says the mean lies in the longer tail. Moore and McCabe (2003, p. 43) put it as: “In a skewed distribution, the mean is farther out in the long tail than is the median.” Paul von Hippel (2005, Journal of Statistics Education) shows that this rule fails often, most commonly in discrete distributions. His example is the Poisson distribution: its skewness is always positive, yet for more than 30% of parameter values the mean is below the median. At μ = 0.75, 47% of the probability lies left of the median and 17% right of it.
A small dataset shows the same thing. Twenty counts: nine 0s, seven 1s, three 2s and one 3.
- Sum = 9×0 + 7×1 + 3×2 + 1×3 = 0 + 7 + 6 + 3 = 16, so the mean = 16 / 20 = 0.8.
- Sorted positions 1 to 9 are 0 and positions 10 to 16 are 1. The 10th and 11th values are both 1, so the median = 1.
- The skewness (population formula, third standardised moment) is about +0.85, a right skew, yet the mean (0.8) is below the median (1).
The long right tail exists, but the median sits on a value that holds many observations, so the mean falls just under it. Von Hippel also reports continuous violations, which are rarer; the practical point is that the sign of the gap and the sign of the skew can disagree.
Interpretation: reading the gap
| Pattern | What it usually suggests | Source and caveat |
|---|---|---|
| Mean and median about equal | Roughly symmetrical distribution | OpenStax 2.6: identical in a perfectly symmetrical distribution |
| Mean greater than median | Right skew, often from a few large values | OpenStax 2.6 says “often”; von Hippel 2005 shows exceptions, mainly discrete data |
| Median higher than mean | Left skew, often from a few small values | OpenStax 2.6; the same exceptions apply |
| Mean and median disagree with the sign of the skewness statistic | Check whether the data are discrete or have many tied values | von Hippel 2005 (Poisson and survey examples) |
To judge how large a gap is, divide it by the standard deviation. In the example above, one extreme value produced a gap of 9.0, yet it is only 0.30 SD because the same value inflated the SD. This ratio is an illustration for reading your own data, not a published threshold.
Assumptions and common mistakes
- Treating the gap as proof of skew. It is a prompt to plot the data. A right-skewed histogram or a right-skewed box plot shows the shape directly.
- Applying the rule to discrete or tied data. Counts with many repeated values, such as the example above, can reverse the order.
- Reporting only the mean for skewed data. One extreme observation can make the mean describe almost no one in the sample.
- Dropping the outlier to close the gap. Remove a value only for a documented reason, not to make the mean and median agree.
- Assuming the cause is always an outlier. A naturally skewed process, such as a lognormal distribution, has a mean above its median without any erroneous values.
How to report it (APA example)
Report both measures of centre when they differ and say why. Using the response-time example, following the APA convention of italic M and SD:
Response times were strongly right-skewed (M = 14.5 s, Mdn = 5.5 s, SD = 30.15 s); the median is therefore the better description of a typical response.
See also the guide to left-skewed versus right-skewed distributions and the average versus median comparison.
How to calculate it in Excel, R or Python
Excel (values in A1:A10):
=AVERAGE(A1:A10)
=MEDIAN(A1:A10)
=AVERAGE(A1:A10)-MEDIAN(A1:A10)R:
x <- c(1,2,3,4,5,6,7,8,9,100)
mean(x) # 14.5
median(x) # 5.5
mean(x) - median(x) # 9Python:
import statistics as s
x = [1,2,3,4,5,6,7,8,9,100]
print(s.mean(x), s.median(x), s.mean(x) - s.median(x))
# 14.5 5.5 9.0FAQ
Is mean greater than median always right skewed?
No. It is often the case for continuous, single-peaked data, but not always. Von Hippel (2005) shows that discrete distributions can have positive skew with the mean below the median, and the reverse pattern is also possible.
Why is the mean greater than the median?
Because the mean uses the size of every value, so a few large values raise it, while the median depends only on the middle position in the sorted data.
What does it mean if the median is higher than the mean?
It usually points to left skew, where a few small values pull the mean down, with the same exceptions for discrete data.
Which should I report when the mean is greater than the median?
Report both and state the skew. When a few extreme values drive the gap, the median describes a typical case better, as in the response-time example above.
Sources
- von Hippel, P. T. (2005). Mean, median, and skew: Correcting a textbook rule. Journal of Statistics Education, 13(2). https://jse.amstat.org/v13n2/vonhippel.html (checked 2026-10-29).
- OpenStax. Introductory Statistics 2e, section 2.6, Skewness and the mean, median, and mode. https://openstax.org/books/introductory-statistics-2e/pages/2-6-skewness-and-the-mean-median-and-mode (checked 2026-10-29).
