Question

A sample of N=9 scores has a mean of M=20. One of the scores is changed and the new mean is found to be 22. If the score was originally X=7, what is its new value.

Answer 1

We have

n = 9

$\mu_{old}=20$

$\mu_{new}=22$

Lets denote the changing score $x_{old}=7$ , so we need to find $x_{new}$

$$\mu=\frac{1}{n}\sum\limits_{i=1}^9 x_i$$

$\mu_{old}=\frac{1}{n}(\sum\limits_{i=1}^8 x_i + x_{old})$

$\mu_{new}=\frac{1}{n}(\sum\limits_{i=1}^8 x_i + x_{new})$

Thus we have

$\begin{cases} 20=\frac{1}{9}(\sum\limits_{i=1}^8+7) \\ 22=\frac{1}{9}(\sum\limits_{i=1}^8+x_{new}) \end{cases} \implies \begin{cases} \sum\limits_{i=1}^8+7=180 \\ \sum\limits_{i=1}^8+ x_{new}=198 \end{cases}\implies x_{new}=25 ​$

Answer: 25.