The mean and variance of $$10$$ observations were calculated as $$15$$ and $$15$$ respectively by a student who took by mistake $$25$$ instead of $$15$$ for one observation. Then, the correct standard deviation is______.

The incorrect mean is

$$15$$

for

$$10$$

observations.

Therefore, the incorrect sum is

$$10\times 15=150$$

Since

$$25$$

was taken instead of

$$15$$,

the sum is excess by

$$10$$.

Hence the correct sum is

$$150-10=140$$

Therefore, the correct mean is

$$\bar x=\frac{140}{10}=14$$

Now use the variance formula

$$\sigma^2=\frac{\sum x_i^2}{n}-\bar x^2$$

The incorrect variance is

$$15$$

with incorrect mean

$$15$$.

Hence

$$15=\frac{\sum x_i^2}{10}-15^2$$

$$\frac{\sum x_i^2}{10}=240$$

$$\sum x_i^2=2400$$

Since

$$25$$

was used instead of

$$15$$,

the correct value of

$$\sum x_i^2$$

is

$$2400-25^2+15^2$$

$$=2400-625+225$$

$$=2000$$

Therefore, the correct variance is

$$\sigma^2=\frac{2000}{10}-14^2$$

$$=200-196$$

$$=4$$

Hence the correct standard deviation is

$$\sqrt{4}$$

$$=2$$

Final Answer :

$$2$$