Mean vs Median with Skewed Data

One extreme value added to the set 1, 2, 3, 4, 5, 6, 8, 9 shows the mean jumping while the median barely moves — the test for choosing between them.

Take the original set 1, 2, 3, 4, 5, 6, 8, 9 with mean 4.75 and median 4.5. Now append a single extreme value, 100. The new set has n = 9 and the sum 138, so the mean becomes 138 / 9 ≈ 15.33.

The median moves much less. With n = 9 the middle position is the 5th value of the sorted list 1, 2, 3, 4, 5, 6, 8, 9, 100 — that value is 5. The median has shifted from 4.5 to 5, a change of 0.5, while the mean has shifted by more than 10.

The table below sets the two figures side by side before and after the extreme value. It is the quickest visual argument for why skewed data is summarised with the median.

Report the median when extreme values carry reduced importance, typically because the distribution is skewed: salaries, prices, waiting times and any data with a natural floor but no natural ceiling.

Report the mean when the distribution is roughly symmetric and no single observation dominates. The mean uses every value and is the better basis for further analysis.

In practice, quote both. The gap between them is itself a rough signal of skewness: a mean well above the median suggests a long right tail.

• Skewed distribution → median + interquartile range.
• Symmetric distribution → mean + standard deviation.
• Always state n and whether the count is even or odd.
• A mean far above the median is a red flag for outliers.

Further reading

• The Measures: Formulas and Worked Examples
• Summarise a Data Set, Step by Step