Question

The mean of 100 observations is 50 and their standard deviation is 5. The sum of all squares of all the observations is

(a) 50,000 (b) 250,000 (c) 252500 (d) 255000

Answer 1

Let $\overline{x}$ and $\sigma$ be the mean and standard deviation of 100 observations, respectively.

$\therefore \overline{x}=50,\sigma =5$ and n = 100

Mean, $\overline{x}=50$

$$\begin{gathered} \Rightarrow \frac{\sum _{}{x}_i}{100}=50 \\ \Rightarrow \sum _{}{x}_i=5000.....\left(1\right) \end{gathered}$$

Now,

Standard deviation, $\sigma =5$

$$\begin{gathered} \Rightarrow \sqrt{\frac{\sum _{}{x}_i^2}{100}-{\left(\frac{\sum _{}{x}_i}{100}\right)}^2}=5 \\ \Rightarrow \frac{\sum _{}{x}_i^2}{100}-{\left(\frac{5000}{100}\right)}^2=25\left[\mathrm{From}\left(1\right)\right] \\ \Rightarrow \frac{\sum _{}{x}_i^2}{100}=25+2500=2525 \\ \Rightarrow \sum _{}{x}_i^2=252500 \end{gathered}$$

Thus, the sum of all squares of all the observations is 252500.

Hence, the correct answer is option (c).