Mean and median: why one unusual value changes the story
Original small datasets show how to calculate the mean and median and compare their behaviour.
The arithmetic mean uses the sum of all observations, while the median uses the middle position after the observations are ordered.
Two different questions about the centre
NIST’s statistical handbook defines the mean as the total divided by the number of observations. The median is the middle ordered value for an odd count, or the average of the two middle values for an even count.
Because these definitions use the data differently, they need not give the same answer. A central summary should be chosen with the question and distribution in mind.
Original worked example with five values
Take the invented values 2, 4, 5, 7 and 12. Their total is 30, so the mean is 30 / 5 = 6. The middle ordered value is 5, making the median 5. Neither calculation is an approximation of the other.
Now replace 12 with 62. The new total is 80 and the mean is 16. The median remains 5 because the middle position has not changed. This illustrates sensitivity to one high observation, not a rule that the median can never change.
An even-numbered dataset
For the ordered values 3, 6, 8 and 15, take the two middle observations: 6 and 8. The median is (6 + 8) / 2 = 7. It need not equal an observed value.
The mean is (3 + 6 + 8 + 15) / 4 = 8. If the values are presented out of order, sort them before selecting the middle positions. Sorting is essential for this median method but does not change the sum used for the mean.
Interpret before choosing a headline number
A mean can be useful when every observation’s contribution matters; a median can help describe the middle position when extreme observations pull the mean away. A complete report may show both.
Check the count, total and ordering, then state which summary you calculated. “Average” without a definition can hide a real difference in what a reported number means.
Publication disclosure
This article was researched and published with AI assistance. No human prepublication review is claimed. Report factual errors through our contact page.
Frequently asked questions
Must the median be one of the observed values?
No. For an even count it can lie between the two middle observations.
Does changing only the highest observation always change the median?
No. It may leave the middle ordered position unchanged.