arithmetic mean

If $a_1,\,a_2,\,\ldots,\,a_n$ are real numbers, their arithmetic mean is defined as $$A.M. \;=\; \frac{a_1+a_2+\ldots+a_n}{n}.$$

The arithmetic mean is what is commonly called the average of the numbers. The value of $A.M.$ is always between the least and the greatest of the numbers $a_j$ . If the numbers $a_j$ are all positive, then $A.M. \,>\, \frac{a_j}{n}$ for all $j$ .

A generalization of this concept is that of weighted mean, also known as weighted average. Let $w_1, \ldots, w_n$ be numbers whose sum is not zero, which will be known as weights. (Typically, these will be strictly positive numbers, so their sum will automatically differ from zero.) Then the weighted mean of $a_1,a_2,\ldots,a_n$ is defined to be $$W.M. \;=\; \frac{w_1 a_1 + w_2 a_2 + \ldots + w_n a_n}{w_1\!+\!w_2\!+\!\ldots+\!w_n}.$$ In the special case where all the weights are equal to each other, the weighted mean equals the arithmetic mean.