JEE MAIN - Mathematics (2023 - 30th January Morning Shift - No. 17)
The mean and variance of 7 observations are 8 and 16 respectively. If one observation 14 is omitted and a and b are respectively mean and variance of remaining 6 observation, then $$\mathrm{a+3 b-5}$$ is equal to ___________.
Answer
37
Explanation
$$\sum {{x_i} = 7 \times 8 = 56} $$
$${{\sum {x_i^2} } \over n} - {\left( {{{\sum {{x_i}} } \over n}} \right)^2} = 16$$
$${{\sum {x_i^2} } \over 7} - 64 = 16$$
$$\sum {x_i^2 = 560} $$
when 14 is omitted
$$\sum {{x_i} = 56 - 14 = 42} $$
New mean $$ = a = {{\sum {{x_i}} } \over 6} = 7$$
$$\sum {x_i^2 = 560 - 196 = 364} $$
new variance, $$b = {{\sum {x_i^2} } \over 6} - {\left( {{{\sum {{x_i}} } \over 6}} \right)^2}$$
$$ = {{364} \over 6} - 49 = {{35} \over 3}$$
$$3b = 35$$
$$a + 3b - 5 = 7 + 35 - 5 = 37$$
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