# Arithmetic Mean

Arithmetic Mean is one of the most common ways of calculating a sample’s average. The method of calculating the Arithmetic Mean is to sum the values in the sample set and divide the result by the sample size. In Mathematical Notation this is written as:

$$Arithmetic\ Mean\ (\mu)\ \stackrel{\text{def}}{=}\ \left.\frac{\sum_{i=1}^{n}x_i}{n}\right|\ x\ \in\ S\ \land\ n\ =\ \left|S\right|,\ where\ S\ is\ the\ sample$$

The mathematical formula above with the sample examples can be found implemented in 5 different programming languages in our Github repository.

## Examples

### Example 1:

Consider the following simple set $S = {1, 2, 3, 4, 5, 6, 7, 8, 9}$. Using the formula above, as shown below in 3 simple steps we find the Arithmetic mean.

1. Add the values in the set$$1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45$$
2. Find the sample size$$n = 9$$
3. Find the arithmetic mean$$\mu = \frac{45}{9} = 5$$

### Example 2:

$S = \left\{\begin{matrix}0.4963474212, 0.1976482578, 0.9688106204,\\ 0.6871861999, 0.0291216746, 0.9002658844,\\0.5500229849, 0.7883546496, 0.9563570268,\\0.5221225236\end{matrix}\right\}$

Workings:

1. Add the values in the set$$\begin{matrix} \ \ \ 0.4963474212 + 0.1976482578\\+ 0.9688106204 + 0.6871861999\\+ 0.0291216746 + 0.9002658844\\+ 0.5500229849 + 0.7883546496\\+ 0.9563570268 + 0.5221225236\end{matrix} = 6.0962372432$$
2. Find the sample size$$n = 10$$
3. Find the arithmetic mean$$\mu = \frac{6.0962372432}{10} = 0.60962372432$$

### Example 3:

$S = \left\{\begin{matrix}2434328.636, 2165063.314, 672636.861, 1577901.667, 1036641.384,\\1308041.737, 1394538.174, 1156174.874, 2070147.199, 2325744.443,\\2418455.663, 1837957.274, 435248.174, 1190710.265, 1723196.454,\\192363.666, 2611679.169, 2401823.254, 300617.99, 1492514.693,\\1782919.758, 824565.687, 1271028.593, 2015003.283, 1813062.483\end{matrix}\right\}$

Workings:

1. Add the values in the set$$\begin{matrix}\ \ \ 2434328.636 + 2165063.314 + \ \ 672636.861\\+ 1577901.667+ 1036641.384 + 1308041.737\\+ 1394538.174 + 1156174.874 + 2070147.199\\+ 2325744.443 + 2418455.663 + 1837957.274\\+ \ \ 435248.174 + 1190710.265 +1723196.454\\ + \ \ 192363.666 + 2611679.169 + 2401823.254\\ + \ \ 300617.990 + 1492514.693 + 1782919.758\\ + \ \ 824565.687 + 1271028.593 + 2015003.283\\ + 1813062.483\hspace{13em}\end{matrix} = 38452364.695$$
2. Find the sample size$$n = 25$$
3. Find the arithmetic mean$$\mu = \frac{38452364.695 }{25} = 0.60962372432$$

### Example 4

$S = \left\{\begin{matrix}51,97,43,20,48,48,96,63,10,35,16,\\4,42,80,18,1,67,75,46,92,38,44,\\87,69,54,91,6,8,60,64,53,23,86\end{matrix}\right\}$

Workings:

1. Add the values in the set$$\begin{matrix} \ \ \ 51 + 97 + 43 + 20 + 48 + 48 + 96 + 63 + 10 + 35\\+ 16 + \ \ 4+ 42+ 80 + 18 + \ \ 1+ 67+ 75+ 46+ 92\\+ 38 + 44 + 87 + 69 + 54 + 91 + \ \ 6 + \ \ 8 + 60 + 64\\+ 53 + 23 + 86\hspace{15.5em}\end{matrix} = 1635$$
2. Find the sample size$$n = 33$$
3. Find the arithmetic mean$$\mu = \frac{1635}{33} = 49.54545455$$