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

The correct option is C

252500

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

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

Mean $\overline{x}=50$

$\Rightarrow \frac{\sum x_i}{100}=50$

$\Rightarrow \sum x_i=5000$ ....(1)

Now,

Standard deviation , $\sigma =5$

$\Rightarrow \sqrt{\frac{\sum x_i^2}{100}-{\left(\frac{\sum x_i}{100}\right)}^2}=5$

$\Rightarrow \frac{\sum x_i^2}{100}-{\left(\frac{5000}{100}\right)}^2=25$ [From (1)]

$\Rightarrow \frac{\sum x_i^2}{100}=25+2500=2525$

$\Rightarrow \sum x_i^2=252500$

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

Hence, the correct answer is option (c).