The Mean

The mean — the everyday "average" — is the total shared out equally. Add up all n values and divide by how many there are:

\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i = \frac{x_1 + x_2 + \cdots + x_n}{n}.

The bar in \bar{x} ("x-bar") is the standard symbol for the mean of a set of numbers. If five people scored 4, 6, 6, 8 and 6, the mean is (4+6+6+8+6)/5 = 6.

The balance point

There is a vivid way to picture the mean. Lay the data out as equal weights on a ruler. The mean is the exact spot where the ruler balances — the values pulling from the left and the values pulling from the right cancel out. That is why a single far-away value tips the balance so much: the mean slides toward it to keep the ruler level.

Drag the orange point below. Watch the balance line — the mean — chase it. Push the point to an extreme and the mean follows, because every value gets an equal vote.

\bar{x} = \frac{1}{n}\sum_i x_i — the sum of the values divided by their count n.
It is the data's balance point: the deviations above and below it sum to zero.
Every value counts equally, so a single extreme value pulls the mean toward it.