The mean and standard deviation of 15 observations are found to be...

The mean and standard deviation of

15

observations are found to be

8

and

3

respectively. On rechecking it was found that, in the observations,

20

was misread as

5

. Then, the correct variance is equal to _____.

Correct answer

Step-by-step solution

Old mean $= \dfrac{{x}_{1} + {x}_{2} + . . . {x}_{14} + 5}{15} = 8$ (given)

$⇒ {x}_{1} + {x}_{2} + . . . {x}_{14} = 115$

Correct mean $= \dfrac{{x}_{1} + {x}_{2} + . . . {x}_{14} + 20}{15} = \overset{¯}{x}new ⇒ \overset{¯}{x}new = 9$

Given, old variance $= 9 ⇒ \dfrac{{{x}_{1}}^{2} + {{x}_{2}}^{2} + . . . {{x}_{14}}^{2} + {5}^{2}}{15} - {8}^{2} = 9 ⇒ {{x}_{1}}^{2} + {{x}_{2}}^{2} + . . . {{x}_{14}}^{2} = 1070$

Now ${var}_{new} = \dfrac{{{x}_{1}}^{2} + {{x}_{2}}^{2} + . . . {{x}_{14}}^{2} + {20}^{2}}{15} - {9}^{2} = \dfrac{1070 + 400}{15} - 81 = 17$

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