Geometric Mean

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Geometric Mean is another way of finding the average of a sample. While the Arithmetic Mean takes the sum of the sample divided by the sample size, the geometric mean takes the n^{th} root of the sample items product. In Mathematical Notation this is written as:

Geometric\ Mean\ (g\mu)\ \stackrel{\text{def}}{=}\ \left.\sqrt[n]{\prod_{i=1}^{n}x_i}\right|\ x\ \in\ S\ \land\ n\ =\ \left|S\right|,\ where\ S\ is\ the\ sample

Note: The Geometric Mean symbol used in the expression above, g\mu, and in the articles on this blog, is not an official notation. It is used by the author to distinguish between the different formulæ in view that \mu is the standard symbol for all type of averages.

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

Note: The code in Github uses the exponential expression notation for n^{th} roots.\sqrt[n]{x} \equiv x^{\frac{1}{n}}

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 Geometric mean.

  1. Multiply the values in the set1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 = 362880
  2. Find the sample sizen = 9
  3. Find the geometric meang\mu = \sqrt[9]{362880} = 4.1472

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. Multiply the values in the set\begin{matrix} \ \ \ 0.4963474212 \times 0.1976482578\\\times 0.9688106204 \times 0.6871861999\\\times 0.0291216746 \times 0.9002658844\\\times 0.5500229849 \times 0.7883546496\\\times 0.9563570268 \times 0.5221225236\end{matrix} = 0.0003707
  2. Find the sample sizen = 10
  3. Find the arithmetic meang\mu = \sqrt[10]{0.0003707} = 0.4538449

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 \times 2165063.314 \times \ \ 672636.861\\\times 1577901.667\times 1036641.384 \times 1308041.737\\\times 1394538.174 \times 1156174.874 \times 2070147.199\\\times 2325744.443 \times 2418455.663 \times 1837957.274\\\times \ \ 435248.174 \times 1190710.265 \times 1723196.454\\ \times \ \ 192363.666 \times 2611679.169 \times 2401823.254\\ \times \ \ 300617.990 \times 1492514.693 \times 1782919.758\\ \times \ \ 824565.687 \times 1271028.593 \times 2015003.283\\ \times 1813062.483\hspace{13em}\end{matrix} = 8.63868 \times 10^{152}
  2. Find the sample sizen = 25
  3. Find the arithmetic meang\mu = \sqrt[25]{8.63868 \times 10^{152}} = 1310562.9914397

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 \times 97 \times 43 \times 20 \times 48 \times 48 \times 96 \times 63 \times 10 \times 35\\\times 16 \times \ \ 4\times 42\times 80 \times 18 \times \ \ 1\times 67\times 75\times 46\times 92\\\times 38 \times 44 \times 87 \times 69 \times 54 \times 91 \times \ \ 6 \times \ \ 8 \times 60 \times 64\\\times 53 \times 23 \times 86\hspace{15.5em}\end{matrix} = 1.6277606 \times 10^{51}
  2. Find the sample sizen = 33
  3. Find the arithmetic meang\mu = \sqrt[33]{1.6277606 \times 10^{51}} = 35.6341480