When can one unusual value pull the mean away from most observations?

When a value is far above or below the rest, it can pull the mean in its direction. The mean is the sum of the values divided by how many values there are, so an unusually large value adds a lot to the sum, while an unusually small value can pull it down. The mean uses all the observations and can be affected by extreme values. [S1]

For a simple high-value example, suppose five made-up observations are 4, 5, 5, 6, and 5. Their sum is 25, so their mean is 25 ÷ 5 = 5. Add one unusually high value, 30: the new sum is 55, and the mean is 55 ÷ 6 ≈ 9.17. Five of the six observations are still between 4 and 6, but the mean is now much higher than most of them.

Yes. Imagine five more invented observations: 8, 9, 10, 11, and 12. Their sum is 50, so the mean is 50 ÷ 5 = 10. Now add an unusually low value, −20. The sum becomes 30, and the mean becomes 30 ÷ 6 = 5. The five original observations are still between 8 and 12, but the mean has dropped below all of them.

These examples are deliberately small so you can check the arithmetic. They show how a single extreme value can move a mean; they do not tell us why that value occurred or whether it should be excluded. Don’t remove a value just because it changes the result. First check what the observation represents and whether it belongs in the data for the question you are asking.

Look at the observations themselves, not just the calculated summary. Check whether one or more values stand apart from the rest, and compare the mean with the individual values. Ask what each value represents and whether the set is being summarized fairly by one average. The median can be a useful comparison because an outlier’s precise numerical value does not affect the median in the same way it affects the mean. [S1]

For the first example, the mean after adding 30 is about 9.17, while the middle observation in the sorted list 4, 5, 5, 5, 6, 30 is 5. The contrast signals that the high value matters to the mean. It does not, by itself, establish whether 30 is an error or an important observation. A particular data set needs to be interpreted in context.

Use this copyable template, filling in the brackets with your own values:

“Most observations are around [range or typical values]. The mean is [mean] before including [unusual value] and [mean] after including it. This shows that the mean is [higher/lower] when that value is included. I checked the observation’s meaning and [briefly state what I found]. I will [keep it and explain its effect / investigate it further], rather than deciding based only on its effect on the mean.”

Keep the calculation visible: add the values, divide by the number of values, and state whether the unusual observation is included. If you also report a median, label it separately. That lets a reader see how the summary changes without mistaking the mean for a description of every individual observation.