The mean and variance of the marks obtained by the students in a test are 10 and 4 respect…

Question

The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12. If the new mean of the marks is 10.2, then their new variance is equal to:

Accepted Answer

$\bar{x}=10 ~\&~ \sigma^{2}=4$, No. of students $=N$ (let)

$$
\therefore \quad \frac{\sum x_{i}}{N}=10 ~\&~ \frac{\sum x_{i}^{2}}{N}-(10)^{2}=4
$$

Now if one of $x_{i}$ is changed from 8 to 12 we have

New mean $\frac{\sum x_{i}+4}{N}=10+\frac{4}{N}=10.2$

$\Rightarrow N=20$

and $\sigma_{\text {new }}^{2}=\frac{\sum x_{i}^{2}-(8)^{2}+(12)^{2}}{20}-(10 \cdot 2)^{2}$

$$
\begin{aligned}
& =\frac{\sum x_{i}^{2}}{20}+\frac{144-64}{20}-(10 \cdot 2)^{2} \\
& =104+4-(10 \cdot 2)^{2} \\
& =108-104.04=3.96
\end{aligned}
$$

Solutions · 1