The mean and standard deviation of 40 observations are 30 and 5 respectively. It was noticed that two of these observations 12 and 10 were wrongly recorded. If $\sigma$ is the standard deviation of the data after omitting the two wrong observations from the data, then $38 \sigma^{2}$ is equal to ______.

JEE Main · 2022 · 26 Jul 2022 · Shift 2 · Statistics

The mean and standard deviation of 40 observations are 30 and 5 respectively. It was noticed that two of these observations 12 and 10 were wrongly recorded. If

σ

is the standard deviation of the data after omitting the two wrong observations from the data, then

38 σ^{2}

is equal to ______.

Answer: 238

Solution

From the wrong data:

\sum x_{i} = 40 \times 30 = 1200

and

σ_{1}^{2} = 25 = \frac{\sum x _{i}^{2}}{40} - 3 0^{2}

, so

\sum x_{i}^{2} = 37000

.

After removing the two wrong observations (10 and 12):

New sum: $\sum x_{i}^{'} = 1200 - 10 - 12 = 1178$
New mean: $μ^{'} = \frac{1178}{38} = 31$
New sum of squares: $\sum (x_{i}^{'})^{2} = 37000 - 100 - 144 = 36756$

The new variance is

σ^{2} = \frac{36756}{38} - 3 1^{2}

.

Therefore:

38 σ^{2} = 36756 - 38 \times 961 = 36756 - 36518 = 238

JEE Main · 2022 · 26 Jul 2022 · Shift 2 · Integer answer

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