Calculating Standard Deviation for a Frequency Distribution: A Comprehensive Guide

Understanding how to calculate the standard deviation for a frequency distribution is crucial for statistical analysis. This guide will walk you through the steps to calculate the standard deviation using a given data set. We'll cover the determination of midpoints, calculation of the mean, variance, and finally, the standard deviation.

1. Determining the Midpoints of Class Intervals

Midpoints are the central values of each class interval in a frequency distribution. For the given data:

Class Interval (C.I.) Frequency (f) Midpoint (x) Frequency times; Midpoint (f times; x) 0 - 10 5 5 25 10 - 20 8 15 120 20 - 30 15 25 375 30 - 40 7 35 245 40 - 50 3 45 135 Total 38 900

2. Calculating the Mean

The mean ((bar{x})) is calculated by dividing the sum of the frequency multiplied by the midpoint by the total frequency:

(bar{x}) (frac{sum f cdot x}{sum f}) (frac{900}{38}) ≈ 23.68

3. Calculating the Variance

The variance is calculated using the formula:

(sigma^2 frac{sum f(x - bar{x})^2}{sum f})

The first step is to calculate ((x - bar{x})^2) and then (f(x - bar{x})^2):

Midpoint (x) Midpoint - Mean (x - (bar{x})) (x - (bar{x}))^2 Frequency (f) f(x - (bar{x}))^2 5 5 - 23.68 -18.68 349.5424 5 1747.712 15 15 - 23.68 -8.68 75.4624 8 603.6992 25 25 - 23.68 1.32 1.7424 15 26.136 35 35 - 23.68 11.32 128.3584 7 898.5088 45 45 - 23.68 21.32 454.3584 3 1363.0752 Total 38 4139.1304

The variance is then calculated as:

(sigma^2 frac{4139.1304}{38}) ≈ 109.8

4. Calculating the Standard Deviation

The standard deviation ((sigma)) is the square root of the variance:

(sigma sqrt{109.8}) ≈ 10.48

Conclusion

The standard deviation of the given data is approximately 10.48, which indicates the spread of the data points from the mean.

For more detailed insights and further applications in statistical analysis, refer to the Handbook of Mathematics and Statistics or the internet for additional resources.