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Download now on Google PlayLet \(x_1\), \(x_2\) 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.)

Number | Deviation 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 \(k\).