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Sum of Deviations from Mean (Simple Explanation)

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The arithmetic mean has the important property that the sum of deviations of data values from the mean is equal to zero. In this article, we first explain this property by looking at an example and then give a proof as to why the sum of deviations from the mean is always zero.

Statement of the Property:

Let x_1, x_2,\ldots, x_n be a set of data values. Let \bar{x} denote their arithmetic mean obtained by summing up the values and then dividing by n. Then \bar{x} satisfies the following property:

\sum_{i=1}^{n}(x_i - \bar{x})=0\text{ that is,} \\ (x_1 - \bar{x}) + (x_2 - \bar{x}) +\ldots + (x_n - \bar{x}) = 0.

Example:

Consider the following data values: 1, 2, 3, 4, 5, 6. Then their arithmetic mean is given as: \bar{x} = \frac{1+ 2+ 3+ 4+ 5+ 6}{6} = \frac{21}{6} = 3.5 Then we have that the sum of the deviations is equal to \begin{align*}\sum_{i=1}^{6}(x_i - \bar{x}) &= (x_1 - \bar{x}) + (x_2 - \bar{x}) +\ldots + (x_6 - \bar{x}) \\ &= (1-3.5)+(2-3.5)+(3-3.5)+(4-3.5)+(5-3.5)+(6-3.5) \\ &= (-2.5)+(-1.5)+(-0.5)+(0.5)+(1.5)+(2.5) = 0 \end{align*} Thus we see that the sum of the differences from the mean is equal to zero. This means that the mean lies at the “center” of the data in some sense.

Why is the sum of Deviations from the Mean equal to Zero?

We now give the proof of the above property. Consider the sum of deviations,

\begin{align*}\sum_{i=1}^{n}(x_i - \bar{x}) &= (x_1 - \bar{x}) + (x_2 - \bar{x}) +\ldots + (x_n - \bar{x}) \\ &= (x_1 + x_2 + \ldots + x_n) - (n \times \bar{x}) \\ &= (\sum_{i=1}^{n}x_i) - (\bcancel{n} \times \frac{\sum_{i=1}^{n}x_i}{\bcancel{n}}) \\ &= (\sum_{i=1}^{n}x_i) - (\sum_{i=1}^{n}x_i) = 0\end{align*}

This completes our proof.

What happens if the sum of deviations from the mean doesn’t equal zero?

It is a property of the mean that the sum of deviations of the mean is ALWAYS equal to zero. If the sum is not equal to zero then it means that the either value of the mean has not been calculated correctly or there is a rounding error.

For example, if the value of the mean is 1/3, which is approximately taken to be equal to 0.3334 then the sum of the deviations would not equal zero. This is because we use the approximate value of 0.3334 when performing the calculation in a calculator rather than the actual value 1/3.

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