First slide

Introduction to statistics

Question

$Let a, b, c, d and e be the observations with mean m and standard deviation s . The standard deviation of the$ $observations a + k, b + k, c + k, d + k and e + k is$

Moderate

A

s
B

ks
C

s + k
D

$\frac{s}{k}$

Solution

$\begin{array}{l} Given observations are a, b, c, d and e . \\ Mean = m = \frac{a + b + c + d + e}{5} \\ Σ x_{i} = a + b + c + d + e = 5 m \end{array}$
Mean of new observations
$\begin{array}{l} = \frac{a + k + b + k + c + k + d + k + e + k}{5} \\ = \frac{(a + b + c + d + e) + 5 k}{5} = m + k \\ ∴ S.D. = \sqrt{\frac{\sum (x_{i}^{2} + k^{2} + 2 k x_{i})}{n} - (m^{2} + k^{2} + 2 m k)} \\ = \sqrt{\frac{Σ x_{i}^{2}}{n} - m^{2} + \frac{2 k Σ x_{i}}{n} - 2 m k} \end{array}$
$\begin{array}{l} = \sqrt{\frac{Σ x_{i}^{2}}{n} - m^{2} + 2 k m - 2 m k} [∵ \frac{Σ x_{i}}{n} = m] \\ = \sqrt{\frac{Σ x_{i}^{2}}{n} - m^{2}} \\ = s \end{array}$

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