Understanding the Standard Deviation and Variance of the First n Natural Numbers

In statistical analysis, understanding the standard deviation and of a set of numbers is crucial. This article will guide you through the steps to calculate these values for the first 21 natural numbers. We'll explore the formulas and provide detailed calculations for both the standard deviation and variance, along with an explanation of the underlying mathematical concepts.

Calculating the Standard Deviation of the First 21 Natural Numbers

The standard deviation of the first n natural numbers can be calculated using the formula:

std_dev sqrt(n^2 - frac{1}{12})

For n 21, the calculation is as follows:

std_dev  sqrt(21^2 - frac{1}{12})          sqrt(441 - 0.08333333333333333)          sqrt(440.91666666666665)          6.055

This result shows that the standard deviation of the first 21 natural numbers is approximately 6.055.

Calculating the Variance of the First 21 Natural Numbers

Variance is the square of the standard deviation. To calculate the variance, we first need to find the mean (mu) and the sum of the squares of the first 21 natural numbers. The mean is:

mean (mu)  frac{1   2   3   ...   n}{n}             frac{n(n   1)}{2n}             frac{21(21   1)}{2(21)}             frac{462}{42}             11

The sum of the squares of the first 21 natural numbers is:

sum of squares  frac{n(n   1)(2n   1)}{6}                 frac{21(21   1)(2(21)   1)}{6}                 frac{21 times; 22 times; 43}{6}                 frac{19734}{6}                 3289

Using these values, we can calculate the variance:

variance  frac{frac{3289}{21} - 11^2}{21 - 1}          frac{156.66666666666669 - 121}{20}          frac{35.66666666666669}{20}          1.7833333333333337          36.66667

The standard deviation is then the square root of the variance:

std_dev  sqrt(36.66667)          6.055301

Thus, the variance of the first 21 natural numbers is 36.66667, and the standard deviation is approximately 6.055301.

Calculating the Variance of the First 20 Natural Numbers

For further exploration, let's calculate the variance of the first 20 natural numbers using the general formula:

E[X^2] - (E[X])^2

First, we calculate the mean (E[X]):

E[X]  frac{1   2   3   ...   20}{20}      frac{20 times; 21}{2 times; 20}      10.5

Next, we calculate the sum of the squares of the first 20 natural numbers (E[X^2]):

E[X^2]  frac{20 times; 21 times; 41}{6}        frac{17220}{6}        2870

The variance is then:

variance  frac{2870}{20} - 10.5^2          143.5 - 110.25          33.25

Thus, the variance of the first 20 natural numbers is 33.25, and the standard deviation is approximately 5.765.

Conclusion

Understanding the standard deviation and variance of a set of numbers is essential in statistical analysis. By applying the correct formulas and following the steps outlined in this article, you can accurately calculate these values for the first 21 and 20 natural numbers. This knowledge can be applied to a wide range of data sets and scenarios, providing valuable insights into the spread and distribution of numerical data.

Remember to always double-check your calculations and ensure that you are using the most appropriate formulas for your specific data set. With practice, you'll become more proficient in these calculations, making you a more effective data analyst or mathematician.