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

Question

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:

TestHub · Accepted Answer

Let the number of observations of the data set is $n$ .

So,

$\sum_{} x_{i} = 10 n . . . . (i)$

Also,

$\sum_{} x_{i} - 8 + 12 = 10.2 n$

$\Rightarrow 10 n + 4 = 10.2 n$

$\Rightarrow 0.2 n = 4$

$\Rightarrow n = 20$

Also,

$σ^{2} = 4 = \frac{\sum_{} x_{i}^{2}}{20} - 100$

$\Rightarrow \sum_{} x_{i}^{2} = 104 \times 20 = 2080$

Now, new variance is

$= \frac{\sum_{} x_{i}^{2} - 8^{2} + 12^{2}}{20} - {(10.2)}^{2}$

$= \frac{2080 + 4 \times 20}{20} - 104.04$

$= 108 - 104.04$

$= 3.96$

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