The mean of 8 variates is 10.5. If seven of them are 3, 15, 7, 19, 2, 17 and 8, then find the 8th variate.

By formula,

Mean = $\dfrac{\text{Sum of observations}}{\text{No. of observations}}$

Given,

Mean of 8 variates = 10.5

$\therefore 10.5 = \dfrac{\text{Sum of eight variates}}{8} \\[1em] \text{Sum of 8 variates} = 10.5 \times 8 = 84.$

Given,

Seven variates = 3, 15, 7, 19, 2, 17 and 8.

Sum of seven variates = (3 + 15 + 7 + 19 + 2 + 17 + 8) = 71.

8th variate = Sum of 8 variates - Sum of 7 variates = 84 - 71 = 13.

Hence, 8th variate = 13.