The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12. If the new mean of the marks is 10.2, then their new variance is equal to: [2023

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Solution

Q.
The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12. If the new mean of the marks is 10.2, then their new variance is equal to: [2023]

1 3.92

2 3.96

3 4.04

4 4.08

Ans.
(2)

$Mean, μ = 10$

$Variance, σ^{2} = 4$

$Let the number of observations be n$

$Sum of observations \sum x_{i} = 10 n$

$When marks of one student increased from 8 to 12$

$\sum x_{i} = 10 n - 8 + 12 = 10 n + 4$

$New mean = \frac{10 n + 4}{n} = 10.2$ $\Rightarrow n = 20;$ $4 = \frac{\sum x_{i}^{2}}{20} - 10^{2}$

$Incorrect \sum x_{i}^{2} = 104 \times 20 = 2080$ $⇒Correct \sum x_{i}^{2} = 2080 - 8^{2} + 12^{2} = 2160$

$New σ^{2} = \frac{2160}{20} - 10 . 2^{2} = 3.96$

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