Understanding Standard Deviation
To find the value of 'n' for which the standard deviation of the first 'n' natural numbers is 2, we start with the formula for standard deviation:
- Standard deviation (σ) is given by:
σ = √(Σ(x - μ)² / n)
Where:
- μ is the mean,
- n is the number of data points.
Calculating the Mean
For the first 'n' natural numbers, the mean (μ) can be calculated as follows:
- The sum of the first 'n' natural numbers:
Σx = n(n + 1)/2
- Therefore, the mean (μ) is:
μ = Σx / n = (n + 1) / 2
Calculating the Variance
Next, we calculate the variance (σ²):
- The variance is given by:
σ² = Σx² / n - μ²
The sum of squares of the first 'n' natural numbers is:
Σx² = n(n + 1)(2n + 1) / 6
Thus, the variance can be expressed as:
- σ² = [n(n + 1)(2n + 1) / (6n)] - [(n + 1) / 2]²
Setting up the Equation
Given that the standard deviation is 2, we have:
- σ = 2 ⇒ σ² = 4
Setting the variance equal to 4 gives us the equation:
- [n(n + 1)(2n + 1) / (6n)] - [(n + 1) / 2]² = 4
Simplifying the Equation
Solving this equation leads to a cubic expression in terms of 'n'. After simplifying, we find that:
- n = 8 satisfies the condition of a standard deviation of 2.
Conclusion
Thus, the value of 'n' for which the standard deviation of the first 'n' natural numbers equals 2 is:
- n = 8