Question

Find the sum of the deviations of the variate values 3, 4, 6, 7, 8, 14 from their mean.

Answer 1

Ifbe the mean of the n observations, then we have

The given 6 variate values are 3, 4, 6, 7, 8 and 14. Let us denote the values by $X_1,X_2,...,X_6$. Their mean value is given by

$$\begin{gathered} \overline{X}=\frac{1}{6}\sum _{i=1}^6{X}_i \\ =\frac{1}{6}\left(3+4+6+7+8+14\right) \\ =\frac{1}{6}\times 42=7 \end{gathered}$$

The sum of the deviations of the variate values from their mean is

$$\begin{gathered} \sum _{i=1}^6\left(X_i-\overline{X}\right) \\ =\sum _{i=1}^6\left(X_i-7\right) \\ =\sum _{i=1}^6{X}_i-\sum _{i=1}^{6}7 \\ =6\overline{X}-6\times 7 \\ =6\times 7-6\times 7 \\ =0 \end{gathered}$$

Hence the proof is complete.