For a statistical data x_1,x_2, ,x_10 of 10 values, a student obtained the mean as 5.5 and…

Question

For a statistical data ${x}_{1},{x}_{2},\ldots ,{x}_{10}$ of 10 values, a student obtained the mean as 5.5 and $\sum _{10}^{i=1}{x}_{2}^{i}=371$. He later found that he had noted two values in the data incorrectly as 4 and 5 , instead of the correct values 6 and 8 , respectively. The variance of the corrected data is

Accepted Answer

$$\begin{aligned}
& \text { Mean } \overline{\mathrm{x}}=5.5 \
& =\sum_{\mathrm{i}=1}^{10} \mathrm{x}_{\mathrm{i}}=5.5 \times 10=55 \
& =\sum_{\mathrm{i}=1}^{10} \mathrm{x}_{\mathrm{i}}^2=371 \
& \left(\sum \mathrm{x}_{\mathrm{i}}\right)_{\text {new }}=55-(4+5)+(6+8)=60 \
& \left(\sum \mathrm{x}_{\mathrm{i}}\right)_{\text {new }}=371-\left(4^2+5^2\right)+\left(6^2+8^2\right)=430 \
& \
& \text { Variance } \sigma^2=\frac{\sum \mathrm{x}_{\mathrm{i}}^2}{10}-\left(\frac{\sum \mathrm{x}_{\mathrm{i}}}{10}\right)^2 \
& \
& \sigma^2=\frac{430}{10}-\left(\frac{60}{10}\right)^2 \
& \sigma^2=43-36 \
& \sigma^2=7
\end{aligned}$$

Solutions · 1