Calculating Standard Deviation of First 20 Natural Numbers

Understanding the standard deviation of a set of numbers helps us measure how spread out the numbers are from their average. For the set of the first n natural numbers, which are $1, 2, 3, \dots, n$, there is a specific formula to calculate the population standard deviation ($\sigma$).

Formula for Standard Deviation

The population standard deviation ($\sigma$) of the first n natural numbers is given by the formula:

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

Calculating Standard Deviation for the First 20 Natural Numbers

We need to find the standard deviation for the first 20 natural numbers. Here, n = 20.

Let's substitute n = 20 into the formula:

\(\sigma_{20} = \sqrt{\frac{20^2 - 1}{12}}\)

First, calculate \(20^2\):

\(20^2 = 400\)

Now substitute this back into the formula:

\(\sigma_{20} = \sqrt{\frac{400 - 1}{12}}\)

\(\sigma_{20} = \sqrt{\frac{399}{12}}\)

Next, perform the division:

\(\frac{399}{12} = 33.25\)

So, the standard deviation is the square root of 33.25:

\(\sigma_{20} = \sqrt{33.25}\)

Calculating the square root:

\(\sqrt{33.25} \approx 5.766\)

Comparing with Options

Our calculated value for the standard deviation of the first 20 natural numbers using the standard formula is approximately 5.766.

Let's look at the given options:

• Option 1: 7.525
• Option 2: 5.845
• Option 3: 3.028 (This is the given SD for the first 10 natural numbers)
• Option 4: 9.325

Comparing our result \(5.766\) with the options, we see that Option 2 (\(5.845\)) is the closest value provided among the choices.

While our precise calculation using the standard population standard deviation formula for the first n natural numbers gives approximately 5.766, the option 5.845 is provided as the intended answer and is the closest choice.

Revision Table: Key Concepts in Standard Deviation

Additional Information: Understanding Standard Deviation

The standard deviation is a crucial measure of the dispersion or spread of a data set. A higher standard deviation indicates that the data points are more spread out from the mean, while a lower standard deviation indicates that the data points are clustered closer to the mean.

For the first n natural numbers, the data points are uniformly distributed integers. The formulas for mean, variance, and standard deviation reflect this specific distribution.

The formula \(\sigma = \sqrt{\frac{n^2 - 1}{12}}\) is derived from the definition of population variance \(\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}\) by substituting \(x_i = i\), \(\mu = \frac{n+1}{2}\), and performing the necessary summation and algebraic simplification. This formula applies specifically to the set of first natural numbers $\{1, 2, \dots, n\}$.

In statistics, there is also the sample standard deviation (s), which uses a slightly different denominator (\(n-1\)) in the variance calculation. However, for theoretical problems involving the set of the first n natural numbers as the entire population, the population standard deviation formula is typically used.