This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Mean

### From Mathematics Is A Science

Suppose $a_1,\,a_2,\,\ldots,\,a_n$ are real numbers, then their *mean* is defined as
$$\mu =\frac{a_1+a_2+\ldots+a_n}{n}.$$

The mean is one of several possible interpretations of "average" of the numbers.

The value of $\mu$ is always between the least and the greatest of these numbers.

A related concept is that of *weighted mean* or weighted average. Let $w_1, \ldots, w_n$ be positive numbers whose sum is not zero, the "weights". Then the weighted mean of $a_1,a_2,\ldots,a_n$ is defined to be
$$\frac{w_1 a_1 + w_2 a_2 + \ldots + w_n a_n}{w_1\!+\!w_2\!+\!\ldots+\!w_n}.$$

If all the weights are equal to each other, the weighted mean equals the mean.