First slide

Introduction to statistics

Question

The SD of a variate x is $σ$ . The SD of the variate (ax + b)/c, where a, b, c ate constants, is

Moderate

A

$(\frac{a}{c}) σ$
B

$|\frac{a}{c}| σ$
C

$(\frac{a^{2}}{c^{2}}) σ$
D

None of these

Solution

$\begin{array}{l} Let y = \frac{α x + b}{c}, i.e., y = \frac{a}{c} x + \frac{b}{c} \\ i.e, y = \frac{a}{c} x + \frac{b}{c} \\ ∴ \bar{y} = A \bar{x} + B \\ ∴ y - \bar{y} = A (x - \bar{x}) \Rightarrow (y - \bar{y})^{2} = A^{2} (x - \bar{x})^{2} \\ \Rightarrow Σ (y - \bar{y})^{2} = A^{2} Σ (x - \bar{x})^{2} \\ \Rightarrow n \cdot σ_{y}^{2} = A^{2} \cdot n σ_{x}^{2} \\ \Rightarrow σ_{y}^{2} = A^{2} σ_{x}^{2} \\ \Rightarrow σ_{y} = | A | σ_{x} \Rightarrow σ_{y} = |\frac{a}{c}| σ_{x} \\ Thus, new SD = |\frac{a}{c}| σ \end{array}$

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