The mean and variance of a set of 15 numbers are 12 and 14...

Given that,

For set $1:$

Mean $= 12$ and Variance $= 14$

For set $2:$

Mean $= 14$ and Variance $= {\sigma}^{2}$

Now, For set $1$ we have,

$⇒ \dfrac{\mathop{∑}_{i = 1}^{i = 15}{x}_{i}}{15} = 12$

$⇒ \mathop{∑}_{i = 1}^{i = 15}{x}_{i} = 15 \times 12$

Also, $\dfrac{\mathop{∑}_{i = 1}^{i = 15}{{x}_{i}}^{2}}{15} - {\left(\dfrac{\mathop{∑}_{i = 1}^{i = 15}{x}_{i}}{15}\right)}^{2} = 14$

$⇒ \dfrac{\mathop{∑}_{i = 1}^{i = 15}{{x}_{i}}^{2}}{15} - {12}^{2} = 14$

$⇒ \mathop{∑}_{i = 1}^{i = 15}{{x}_{i}}^{2} = \left(14 + 144\right) \times 15$

Now for set $2:$

$⇒ \dfrac{\mathop{∑}_{i = 1}^{i = 15}{y}_{i}}{15} = 14$

and $⇒ \mathop{∑}_{i = 1}^{i = 15}{y}_{i} = 15 \times 14$

Also, $\dfrac{\mathop{∑}_{i = 1}^{i = 15}{{y}_{i}}^{2}}{15} - {\left(\dfrac{\mathop{∑}_{i = 1}^{i = 15}{y}_{i}}{15}\right)}^{2} = {\sigma}^{2}$

$⇒ \dfrac{\mathop{∑}_{i = 1}^{i = 15}{{y}_{i}}^{2}}{15} - {14}^{2} = {\sigma}^{2}$

$⇒ \mathop{∑}_{i = 1}^{i = 15}{{y}_{i}}^{2} = \left({\sigma}^{2} + 196\right) \times 15$

Now,

The combined variance will be

$⇒ 13 = \dfrac{\left(\mathop{∑}_{i = 1}^{i = 15}{{x}_{i}}^{2}\right) \times 15 + \left(\mathop{∑}_{i = 1}^{i = 15}{{y}_{i}}^{2}\right) \times 15}{15 + 15} - {\left(\dfrac{\mathop{∑}_{i = 1}^{i = 15}{x}_{i} + \mathop{∑}_{i = 1}^{i = 15}{y}_{i}}{15 + 15}\right)}^{2}$

$⇒ 13 = \dfrac{\left(14 + 144\right) \times 15 + \left({\sigma}^{2} + 196\right) \times 15}{30} - {13}^{2}$

$∴ {\sigma}^{2} = 10$

Hence, ${\sigma}^{2} = 10$