Arithmetic Mean — Individual and Discrete Series

The Arithmetic Mean (AM), usually written $\bar{x}$, is the sum of all observations divided by the number of observations. It is the most widely used measure of central tendency and the starting point of this chapter.

Individual series (each observation listed separately, with no repetition or frequency attached):

where $\Sigma x$ is the sum of all the observations and $n$ is the number of observations. This is the direct method — Worked Example 1 applies it and then cross-checks the result by the assumed-mean method below.

Discrete series (a set of distinct values $x$, each occurring with a stated frequency $f$):

Direct method:

Assumed-mean method — useful when the values are large or inconvenient to multiply directly: choose any convenient value $A$ (an "assumed mean", ideally close to the middle of the data) and compute the deviation $d = x - A$ for every value:

Step-deviation method — a further shortcut when the $x$ values are equally spaced by a common width $h$ (as class mid-points always are): divide each deviation by $h$ to get $d' = \dfrac{x-A}{h}$, then

$\bar{x} = A + \dfrac{\Sigma fd'}{N} \times h$ …