Average Deviations: Why is it Always Zero?

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SUMMARY

The average of the deviations of a set of measurement values is always zero due to the properties of the mean. Specifically, for any distribution X, the equation \(\langle X - \langle X \rangle \rangle = 0\) holds true. This is a direct consequence of the linearity of expectations, which states that the expected value of a sum is the sum of the expected values. Therefore, the average deviation from the mean will always equate to zero.

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  • Familiarity with linearity of expectations in probability theory
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Why is the average of the deviations of a set of measurement values always zero?
 
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Deviations from what?
 
You probably mean, why is:

[itex]\langle X-\langle X\rangle \rangle =0[/itex]

for any distribution X. It's in the nature of the average. Use the linearity of expectations to show it.
 

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