If the mean and variance of six observations 7, 10, 11, 15, a, b...

Given:

Mean$= 10$

$⇒ \dfrac{7 + 10 + 11 + 15 + a + b}{6} = 10$

$⇒ a + b = 17 \ldots \left(i\right)$

And,

Variance$= \dfrac{20}{3}$

$⇒ \dfrac{{7}^{2} + {10}^{2} + {11}^{2} + {15}^{2} + {a}^{2} + {b}^{2}}{6} - {10}^{2} = \dfrac{20}{3}$

$⇒ \dfrac{495 + {a}^{2} + {b}^{2}}{6} = \dfrac{320}{3}$

$⇒ 495 + {a}^{2} + {b}^{2} = 640$

$⇒ {a}^{2} + {b}^{2} = 145 \ldots \left(ii\right)$

Solve \left(i\right)$and \left(ii\right)$, we get

$a = 9, b = 8$ or $a = 8, b = 9$

$∴ \left|a - b\right| = 1$