VARIANCE

Question

The mean of 100 observation is 50 and there standard deviation is 5. The sum of all squares of all the observation is

1) 50000

2) 250000

3) 252500

4) 25500

Comments

we kNow THAT SD = squAre root (Variance ) 

therefore variAnce is SD^2 = 25

 

SD = (SUM OF SQAURES OF OBSERVATION/ N) - (MEAN )^2

25= X/100 - 2500

SO x = 252500 . IS IT RYT ?

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yes..

Answer 1

All items are equally likely.

Let $\sigma$ be variance and $\mu$ be the mean.

$ \sigma =\Sigma_{i=1}^n \frac{(x_i - \mu)^2}{n} \\= \Sigma_{i=1}^n \frac{{x_i}^2}{n} - \Sigma_{i=1}^n \frac{2x_i \mu}{n} + \mu^2
\\=\Sigma_{i=1}^n \frac{{x_i}^2}{n} -2\mu^2 + \mu^2
\\=\Sigma_{i=1}^n \frac{{x_i}^2}{n} - \mu^2$

So, $\Sigma_{i=1}^n {{x_i}^2} = (\sigma + \mu^2) n \\= (5^2 + 2500) 100 \\=252500$