JEE Main 2023 — Statistics Question with Solution

1. Calculate the sum of the original observations:

$$\frac{x_1+x_2+\dots +x_9+50}{10}=20$$

$$x_1+x_2+\dots +x_9=150$$

2. Calculate the sum of the squares of the original observations using the original variance:

$${\sigma }^2=8^2=64$$

$$64=\frac{x_1^2+x_2^2+\dots +x_9^2+2500}{10}-400$$

$$x_1^2+x_2^2+\dots +x_9^2=2140$$

3. Calculate the new mean after correcting the error:

$$\text{New mean}=\frac{150+40}{10}=19$$

4. Calculate the new variance using the corrected sum of observations and the corrected sum of squares of observations:

$$\text{New }{\sigma }^2=\frac{2140+1600}{10}-(19{)}^2$$

$$\Rightarrow$$ $${\sigma }^2=13$$

The correct variance after correcting the error is 13 (Option C).