How To Calculate The Median

The median is the middle value of a set of numbers once they have been arranged in order. For the three values twelve, five and nine, the sorted list is five, nine, twelve, and the median is nine.

Ordering is the essential first step and the one most often skipped. Reading a value off an unsorted list gives a number that looks reasonable but is not the median at all, and there is no way to tell from the result that it is wrong.

With an odd number of values there is a single middle one. With an even number there is no single middle, so the median is defined as the average of the two central values. Four values have their median halfway between the second and the third.

The median is a positional statistic. It depends on where a value sits in the order, not on how large it is. Changing the largest value from twelve to a thousand does not move the median at all, which is exactly the property that makes it useful.

That robustness is why income and house prices are usually reported as medians rather than averages. A handful of extremely large values drags a mean upward, while the median stays where the typical case actually is.

The mean and the median answer different questions. The mean is the total divided by the count, the value each would get if everything were shared equally. The median is the value that half the data lies below and half above.

When the two differ substantially, the distribution is skewed. A mean well above the median says a few large values are pulling the total up, which is the signature of income, wealth and property price distributions.

The range is a related but separate measure: the largest value minus the smallest. It is easy to compute and extremely sensitive to outliers, which is the opposite of the median's behaviour.

For three values there is a neat shortcut. The median equals the sum of all three minus the smallest minus the largest. It gives the same answer as sorting and is convenient when you are working with a formula rather than a list.

The median is well defined for any numeric data, but not for categories. You can order shoe sizes and find a median, but you cannot order colours. For categorical data the mode is the appropriate summary instead.

Finally, be careful with an even count. Forgetting to average the two middle values and simply picking one of them is the most common error, and on small data sets it can shift the answer noticeably.