JAMIA 2022 — Mathematics PYQ

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Question

JAMIA 2022 — Mathematics PYQ

JAMIA | Mathematics | 2022

For two datasets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4 respectively. The variance of combined dataset is

Choose the correct answer:

Explanation

The formula for combined variance is:

\sigma^2 = \frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}

Given:

$n_1 = 5, \sigma_1^2 = 4, d_1^2 = 1$
$n_2 = 5, \sigma_2^2 = 5, d_2^2 = 1$

Substituting the values:

\sigma^2 = \frac{5(4 + 1) + 5(5 + 1)}{5 + 5}

\sigma^2 = \frac{5(5) + 5(6)}{10}

\sigma^2 = \frac{25 + 30}{10}

\sigma^2 = \frac{55}{10}

Reducing the fraction by dividing both numerator and denominator by $5$ :

\sigma^2 = \frac{11}{2}

Final Answer:

The variance of the combined dataset is $\frac{11}{2}$ .