Sample Solution from
Mathematical Statistics with Applications
7th Edition
ISBN: 9780495110811
Chapter 1
Problem 22SE
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Textbook Problem

Prove that the sum of the deviations of a set of measurements about their mean is equal to zero; that is,

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Expert Solution
To determine

Prove that the sum of deviations about the mean of a set of observations is zero, that is, ∑i=1n(yi−y¯)=0.

Explanation of Solution

The formula for the mean of a set of n observations is, y¯=1n∑i=1nyi.

Now, consider the following calculation, starting from the left-hand-side of the given expression:

∑i=1n(yi−y¯)=∑i=1nyi−∑i=1ny¯=n(1n∑i=1nyi)−ny¯=ny¯−ny¯=0.

Thus, the left-hand-side is 0, which is the right-hand-side of the given expression.

Hence, it is proved that ∑i=1n(yi−y¯)=0.

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