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Let \(x_1\), \(x_2\) and \(x_3\) be three numbers. Their mean is \(\frac{x_1 + x_2 + x_3}{3}\). Determine the sum of the deviations. (Here, deviations is the difference of each number from the arithmetic mean.)
NumberDeviation from the mean
\(x_1\)\(x_1 - \frac{x_1 + x_2 + x_3}{3} = \frac{2x_1 - x_2 - x_3}{3}\)
\(x_2\)\(x_2 - \frac{x_1 + x_2 + x_3}{3} = \frac{2x_2 - x_1 - x_3}{3}\)
\(x_3\)\(x_3 - \frac{x_1 + x_2 + x_3}{3} = \frac{2x_3 - x_1 - x_2}{3}\)
Total\(\frac{2x_1 - x_2 - x_3}{3} + \frac{2x_2 - x_1 - x_3}{3} + \frac{2x_3 - x_1 - x_2}{3} = 0\)
Similarly, if \(\overline X\) is the arithmetic mean of \(n\) number of observations \(x_1\), \(x_2\), …, \(x_n\), then \((x_1 - \overline X) + (x_2 - \overline X) + … + (x_n - \overline X) = 0\). That is, \(\sum_{i=1}^{n} (x_i - \overline X) = 0\).
This can be generalised as "The sum of the deviations of the entries from the arithmetic mean is always zero".
The other properties of arithmetic mean are:
1. If each observation is increased or decreased by \(k\), then the arithmetic mean is also increased or decreased by \(k\).
2. If each observation is multiplied or divided by \(k\), then the arithmetic mean is also multiplied or divided by \(k\).