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The mean and variance of 10 observations are 12 and 9 respectively. Two observations, 6 and 18, are removed. If σ^2 is the variance of the remaining…

Numerical answer

The mean and variance of

10

observations are

12

and

9

respectively. Two observations,

6

and

18

, are removed. If

\sigma^2

is the variance of the remaining

8

observations, then

8\sigma^2

equals

Show answer and solution

Answer: 18

Original sum of squares:

\sum x_i^2=10\,(\sigma^2+\bar x^2)=10(9+144)=1530

, and

\sum x_i=120

. Removing

6

and

18

: the sum becomes

120-24=96

, so the new mean is

12

; the sum of squares becomes

1530-(36+324)=1170

. The new variance is

\sigma^2=\dfrac{1170}{8}-12^2=146.25-144=2.25

. Therefore

8\sigma^2=18

. Equivalently,

8\sigma^2=\sum x^2-8\bar x^2=1170-1152=18

. The mean stays

12

because the removed values are symmetric about it.

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