Question

If the standard deviation of first n natural numbers is 2, then the value of n is

• A. 6
• B. 7
• C. 5
• D. 4

Answer 1

The Correct Option is B

Concept:
The standard deviation (SD) is a measure of the dispersion or spread of a set of values. For the first n natural numbers, the formula for standard deviation is derived from the variance formula for an arithmetic series.

The standard deviation of the first n natural numbers (1, 2, 3, ..., n) is given by:
SD = \(\sqrt{\frac{n^2 - 1}{12}}\)

Given: Standard deviation = 2
So we set up the equation:
\(2 = \sqrt{\frac{n^2 - 1}{12}}\)

Step-by-step Calculation:
1. Square both sides:
\(4 = \frac{n^2 - 1}{12}\)

2. Multiply both sides by 12:
\(48 = n^2 - 1\)

3. Add 1 to both sides:
\(n^2 = 49\)

4. Take square root:
\(n = \sqrt{49} = 7\)

Conclusion:
The value of n for which the standard deviation of the first n natural numbers is 2 is 7.

Therefore, the correct option is: (B) 7.