Central Tendency & DispersionPYQ Sep 24Question 3147 of 473

If each observation of a set is divided by 10, then the Standard Deviation of the new observation is:

\displaystyle \frac{1}{10}

of SD of original observation

\displaystyle 10^{\text{th}}

of SD of original observation

\displaystyle 100

of SD of original observation

\displaystyle 10

of SD of original observation

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Correct Answer

✅ Option a — $\displaystyle \frac{1}{10}$ of SD of original observation

**Concept:** When every observation is multiplied (or divided) by a constant

\displaystyle k

, the Standard Deviation is also multiplied (or divided) by

\displaystyle |k|

. **Proof:** If

\displaystyle x_i' = \frac{x_i}{10}

, then

\displaystyle \bar{x}' = \frac{\bar{x}}{10}

\sigma' = \sqrt{\frac{\sum\left(\frac{x_i}{10} - \frac{\bar{x}}{10}\right)^2}{n}} = \sqrt{\frac{\sum\left(\frac{x_i - \bar{x}}{10}\right)^2}{n}} = \frac{1}{10}\sqrt{\frac{\sum(x_i - \bar{x})^2}{n}} = \frac{\sigma}{10}

Therefore, the new SD =

\displaystyle \frac{1}{10}

of the original SD. Hence, **Option A** is the correct answer.

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