Central Tendency & DispersionPYQ Jun 24Question 3144 of 473

For the first 20 natural numbers the standard deviation is ________.

\displaystyle 5.77

\displaystyle 7.75

\displaystyle 5.64

\displaystyle 6.54

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Correct Answer

✅ Option a — $\displaystyle 5.77$

**Formula for SD of first

\displaystyle n

natural numbers:**

\sigma = \sqrt{\frac{n^2 - 1}{12}}

**Given:**

\displaystyle n = 20

**Step 1:** Apply formula.

\sigma = \sqrt{\frac{20^2 - 1}{12}} = \sqrt{\frac{400 - 1}{12}} = \sqrt{\frac{399}{12}}

**Step 2:** Compute.

\sigma = \sqrt{33.25} \approx 5.766 \approx 5.77

**Verification:** - Mean of 1 to 20 =

\displaystyle \frac{20+1}{2} = 10.5

\displaystyle \sum_{i=1}^{20} i^2 = \frac{20 \times 21 \times 41}{6} = 2870

\displaystyle \sigma^2 = \frac{2870}{20} - (10.5)^2 = 143.5 - 110.25 = 33.25

\displaystyle \sigma = \sqrt{33.25} \approx 5.766 \approx 5.77

Hence, **Option A** is the correct answer.

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