JEE Mathematics 2024 Question with Solution

Given: Mean of $20$ observations is $10$ and standard deviation is $2$. One observation was taken as $8$ instead of $12$.

Find: The corrected standard deviation.

From the given mean,

So,

After replacing the incorrect observation $8$ by $12$,

Hence the corrected mean is

The given standard deviation is $2$, so the original variance is

Using

we get

Therefore,

and

Now correct the squared sum by replacing $8^2$ with $12^2$:

The corrected variance is

Substituting the corrected values,

that is,

Hence the corrected standard deviation is

Therefore, the correct option is C.

Given: Original mean $=10$, original standard deviation $=2$, number of observations $=20$.

Find: Corrected standard deviation after changing $8$ to $12$.

First find the original total:

After correction,

Next use the variance relation:

Since $\sigma = 2$ and $\bar{x}=10$,

So,

and hence,

Now update the sum of squares:

Also,

Therefore the corrected variance becomes

So the corrected standard deviation is $\sqrt{3.96}$.