First slide

Introduction to statistics

Question

Mean and standard deviation of 100 observations were found to be 40 and 10, respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27 , respectively, find the correct standard deviation.

Moderate

A

10.20
B

10.24
C

10.29
D

10.27

Solution

$∴$ New mean, $\begin{array}{l} ∣ \bar{x} = \frac{100 \times 40 + 3 + 27 - 30 - 70}{100} \\ = \frac{4000 - 70}{100} = \frac{3930}{100} \end{array}$
=39.3
$\begin{array}{l} ∵ {Σx}^{2} = N (σ^{2} + {\bar{x}}^{2}) \\ ∴ = 100 (100 + 1600) \\ = 170000 \\ New, {Σx}^{2} = 170000 - (30)^{2} - (70)^{2} + (3)^{2} + (27)^{2} \\ = 170000 - 900 - 4900 + 9 + 729 \\ = 164938 \\ ∴ New, SD = \sqrt{\frac{New {Σx}^{2}}{N} - (New \bar{x})^{2}} \\ = \sqrt{\frac{164938}{100} - (39 .3)^{2}} \\ = \sqrt{1649 .38 - 1544 .49} = \sqrt{104 .89} \\ = 10 .24 \end{array}$

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