Definition
The mean — formally the arithmetic mean — is the sum of all observed values divided by the number of observations. It is the most common summary number used to describe where the centre of a dataset lies, and it has the property that the deviations of all observations from the mean sum to exactly zero. The mean uses every value in the data, which makes it informative when the distribution is well-behaved and misleading when it is not.
In statistics the population mean is usually written with the Greek letter mu, and the sample mean is written with a bar over the variable name. Both are point estimates of the same underlying centre, but the sample mean carries uncertainty that decreases as the sample grows.
Why it matters
How it works
Computing the mean is the simplest operation in statistics: add every observation together and divide by the number of observations. The result occupies the same units as the original data, which is convenient for interpretation — a mean income is in dollars, a mean height is in centimetres. The mean also has a useful algebraic property: it is the value that minimizes the sum of squared distances from itself to every data point. That property is what links the mean to least squares and to linear regression downstream.
The honest use of the mean depends on the shape of the underlying distribution. When the data is roughly symmetric and bell-shaped, the mean sits at the visual peak of the histogram and matches the median closely. When the distribution has a long tail — as with income, response times, or wealth — the mean is pulled toward the tail and overstates where a typical observation lies. The standard remedy is to report the median alongside, to inspect the histogram, and where appropriate to use a trimmed mean that discards a small percentage of extreme values before averaging.