Arithmetic mean

The arithmetic mean is often known simply as the mean. It is an average, a measure of the centre of a set of data. The arithmetic mean is calculated by adding up all the values and dividing the sum by the total number of values.

For example, the mean of $7$, $4$, $5$ and $8$ is $\frac{7+4+5+8}{4}=6$.

If the data values are $x_1$, $x_2$, …, $x_n$, then we have $\bar{x}=\frac{1}{n}\sum_{i=1}^n x_i$, where $\bar{x}$ is a symbol representing the mean of the $x_i$ values.

This rearranges to give the useful result $n\bar x = \sum_{i=1}^n x_i,$ that is, the arithmetic mean is the number $\bar x$ for which having $n$ copies of this number gives the same sum as the original data. So the sum of a set of numbers in some sense “averages” them.

If the data are grouped, with $f_i$ occurrences of the value $x_i$ for $i=1$, $2$, …, $n$, then their mean is given by $\bar{x}=\frac{\sum_{i=1}^n f_ix_i}{\sum_{i=1}^n f_i},$ where the numerator is the sum of all of the $x_i$ values and the denominator is the total number of values.

The arithmetic mean is sensitive to outlier values.

The mean value of a function $f(x)$ over the interval $a\le x\le b$ is likewise the value $M$ for which the constant function $f(x)=M$ has the same “sum” as the original function. The “sum” of a function over an interval is the integral of the function, as shown in this sketch:

Thus the mean $M$ is given by $M(b-a)=\int_a^b f(x)\,dx$, so $M=\frac{\int_a^b f(x)\,dx}{b-a}.$ The integral therefore “averages” the function.