# The S.D of a variate x is S. The S.D of the variate $\frac{ax+b}{c}$  where a,b c are constant is Option 1) $\left ( \frac{a}{c} \right )\sigma$ Option 2) $|\frac{a}{c}|\sigma$ Option 3) $\left ( \frac{a^{2}}{c^{2}} \right )\sigma$ Option 4) None of These

Use the concept

Standard Deviation -

If x1, x2...xn are n observations then square root of the arithmetic mean of

$\dpi{100} \sigma = \sqrt{\frac{\sum \left ( x_{i}-\bar{x} \right )^{2}}{n}}$

$\dpi{100} \bar{}$

- wherein

where $\bar{x}$ is mean

Let $y= \frac{ax+b}{c}$

Then $\bar{y} = \frac{1}{c}(a\bar{x}+b)$

$\Rightarrow y - \bar{y}=\frac{a}{c} (x-\bar{x})$

$\frac{1}{n}\sum (y-\bar{y})^{2} = \frac{a^{2}}{c^{2}}\cdot \frac{1}{n}\sum (x-\bar{x})^{2}$

S.D of $y=\sqrt{\frac{a^{2}}{c^{2}}\frac{1}{n}\sum (x-\bar{x})^{2}}$

$=\sqrt{\frac{a^{2}}{c^{2}}\sigma ^{2}} =$$\left | \frac{a}{c} \right | \sigma$

Option 1)

$\left ( \frac{a}{c} \right )\sigma$

This solution is incorrect.

Option 2)

$|\frac{a}{c}|\sigma$

This solution is correct.

Option 3)

$\left ( \frac{a^{2}}{c^{2}} \right )\sigma$

This solution is incorrect.

Option 4)

None of These

This solution is incorrect.

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