The mean and variance of 8 observations are 10 and 13 . 5,...

Let remaining two observations are $a and b$

$\dfrac{Σ{x}_{i}}{n} = \dfrac{5 + 7 + 12 + 10 + 15 + 14 + a + b}{8} = 10$

$⇒ 63 + a + b = 80$

$⇒ a + b = 17$

${\sigma}^{2} = \dfrac{Σ{x}_{i}^{2}}{n} - {\left(\dfrac{Σ{x}_{i}}{n}\right)}^{2}$

$⇒ 13 .5 = \dfrac{25 + 49 + 144 + 100 + 225 + 196 + {a}^{2} + {b}^{2}}{8} - 100$

$⇒ 908 = {a}^{2} + {b}^{2} + 739$

$⇒ {a}^{2} + {b}^{2} = 169$

$⇒ {\left(a + b\right)}^{2} - 2ab = 169$

$⇒ 289 - 169 = 2ab$

$⇒ ab = 60$

$∴ {\left|a - b\right|}^{2} = {\left(a + b\right)}^{2} - 4ab = 289 - 240 = 49$

$∴ \left|a - b\right| = 7$.