Question

Steven Gerard, arguably one of the most consistent and skilled player in English premier league was given monthly ratings by a Football website. The ratings are given on monthly basis. The following are the ratings for the past 40 months.

$$\begin{array}{ccccccc}Rating& 6& 8& 8.5& 9& 7& 10\\ Numberofmonthswiththerating& 2& 8& 10& 7& 8& 5\end{array}$$

Calculate how his rating is deviating from his average performance in terms of mean deviation about mean.

Answer 1

Mean = $\frac{{\sum }_{i=1}^n{f}_i{x}_i}{{\sum }_{i=1}^n{f}_i}$
Here, n = 6
$\Rightarrow \frac{6\left(2\right)+8\left(8\right)+\left(8.5\right)\left(10\right)+9\left(7\right)+7\left(8\right)+10\left(5\right)}{2+8+10+7+8+5}=\frac{330}{40}=8.25=\overline{x}$
Now , mean deviation about mean would be ,

$\frac{{\sum }_{i=1}^n{f}_i|x_i-\overline{x}|}{{\sum }_{i=1}^n{f}_i}$ = $\frac{2\left(2.25\right)+8\left(0.25\right)+10\left(0.25\right)+7\left(0.75\right)+8\left(1.25\right)+5\left(1.75\right)}{40}=\frac{33}{40}=0.825$

$\therefore$ Mean deviation about mean for his performance across 40 months about his average performance rating of 8.25 is 0.825 which shows he is quite consistent.