If the mean and variance of the following data: 6, 10, 7, 13, a,...

Mean $= \dfrac{6 + 10 + 7 + 13 + a + 12 + b + 12}{8} = 9$

$⇒ 60 + a + b = 72$

$⇒ a + b = 12 . . . \left(1\right)$

variance $= \dfrac{∑ {x}_{i}^{2}}{n} - {\left(\dfrac{∑ {x}_{i}}{n}\right)}^{2} = \dfrac{37}{4}$

Here, $Σ{x}_{i}^{2} = {6}^{2} + {10}^{2} + {7}^{2} + {13}^{2} + {a}^{2} + {b}^{2} + {12}^{2} + {12}^{2}$

$= {a}^{2} + {b}^{2} + 642$

So, variance is $\dfrac{{a}^{2} + {b}^{2} + 642}{8} - {\left(9\right)}^{2} = \dfrac{37}{4}$

$⇒ \dfrac{{a}^{2} + {b}^{2}}{8} + \dfrac{321}{4} - 81 = \dfrac{37}{4}$

$⇒ \dfrac{{a}^{2} + {b}^{2}}{8} = 81 + \dfrac{37}{4} - \dfrac{321}{4}$

$⇒ \dfrac{{a}^{2} + {b}^{2}}{8} = 81 - 71$

$∴ {a}^{2} + {b}^{2} = 80$

From \left(1\right)${a}^{2} + {b}^{2} + 2ab = 144$

$⇒ 80 + 2ab = 144 ∴ 2ab = 64$

So, {\left(a - b\right)}^{2} = {a}^{2} + {b}^{2} - 2ab = 80 - 64 = 16 .