Arithmetic Mean vs Median: Which Average Should You Use

The Two Definitions Side by Side

The arithmetic mean is the sum of all values divided by the count. The median is the middle value once the list is sorted, averaging the two middle values when the count is even. The mean uses every single number. The median ignores the size of everything except its position in the order.

When the Two Agree

If your data is perfectly symmetric, with the low values mirroring the high values, the mean and the median are identical. Heights of a symmetric group, or scores on a well balanced test, often behave this way. When you see the two figures match, there is probably nothing unusual in the list.

When One Extreme Value Changes Everything

Take six salaries: 32,000, 34,000, 35,000, 36,000, 37,000 and 250,000. The mean is 424,000 divided by 6, which is about 70,667. The median is the average of 35,000 and 36,000, which is 35,500. Not one person in that list earns the mean. The single outlier doubled it. This is the classic failure of the mean and the reason the calculator flags a large gap between the two.

Reading the Gap as a Diagnostic

When the mean sits above the median, high values are pulling it up. When the mean sits below the median, low values are pulling it down. Either way the gap between them is a signal that the distribution is lopsided and deserves an explanation before you quote the number to anyone.

The Berkeley Admissions Case Study

In 1975 three researchers at Berkeley published an examination of 12,763 graduate applications showing that pooled into a single average, male applicants appeared 1.8 times more likely to be admitted than female applicants. When the data were separated by the 85 individual departments, that advantage disappeared entirely and about as many departments favoured women as men. The pooled average was an artifact of averaging across groups that were not comparable.

When the Median Is the Better Choice

Report the median when values include a genuine outlier, when the distribution is skewed, when you want the typical case rather than the total per item, or when a single value could distort the result. Salary surveys, house prices, response times and exam scores all fall into this category.

When the Mean Is the Better Choice

Report the mean when every value genuinely belongs to the same group, when you need a figure that can be added and divided again later, when the data is symmetric, or when the mean is what the downstream calculation expects. Averages of averages are only valid when they are means and the groups are equal in size.

Reporting Both Is Usually Best

When the two figures differ, reporting both is the most honest option and takes one extra sentence. Something like an average of 70,667 with a median of 35,500 tells the reader immediately that the distribution is heavily skewed. Quoting only the mean invites them to draw the wrong conclusion.