population vs sample standard deviation

3 min read 15-03-2025

Understanding the difference between population and sample standard deviation is crucial in statistics. Both measure the spread or dispersion of a dataset, but they apply to different contexts and have slightly different formulas. This article will clarify the distinction, explain the formulas, and illustrate their applications.

What is Standard Deviation?

Standard deviation quantifies the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean (average) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.

Population Standard Deviation

The population standard deviation describes the spread of an entire population. This means you have data for every member of the group you're studying. For example, if you're studying the height of every student at a particular school, and you have the height of each student, you can calculate the population standard deviation.

Formula for Population Standard Deviation (σ)

The formula for population standard deviation (represented by the Greek letter sigma, σ) is:

σ = √[ Σ(xi - μ)² / N ]

Where:

σ: Population standard deviation
Σ: Summation (add up all the values)
xi: Each individual data point
μ: Population mean (average)
N: Total number of data points in the population

Sample Standard Deviation

The sample standard deviation estimates the spread of a larger population based on a smaller, randomly selected sample. You use this when it's impossible or impractical to collect data from the entire population. For instance, if you want to know the average income of all adults in a country, you wouldn't be able to survey everyone. Instead, you'd take a representative sample and calculate the sample standard deviation.

Formula for Sample Standard Deviation (s)

The formula for sample standard deviation (represented by the lowercase letter 's') is slightly different than the population standard deviation:

s = √[ Σ(xi - x̄)² / (n - 1) ]

Where:

s: Sample standard deviation
Σ: Summation (add up all the values)
xi: Each individual data point in the sample
x̄: Sample mean (average)
n: Total number of data points in the sample

Why (n-1) in the Sample Standard Deviation Formula?

The denominator (n-1) is used instead of 'n' in the sample standard deviation formula because it provides a better, less biased estimate of the population standard deviation. Using 'n' would underestimate the population standard deviation. This adjustment is known as Bessel's correction.

When to Use Which?

Population Standard Deviation (σ): Use this when you have data for the entire population you're interested in. This is rare in practice.
Sample Standard Deviation (s): Use this when you have data from a sample and want to estimate the standard deviation of the larger population the sample represents. This is the most common scenario.

Example: Calculating Standard Deviation

Let's say we have a sample of five exam scores: 85, 90, 78, 88, 92.

Calculate the sample mean (x̄): (85 + 90 + 78 + 88 + 92) / 5 = 86.6
Calculate the deviations from the mean (xi - x̄): -1.6, 3.4, -8.6, 1.4, 5.4
Square the deviations: 2.56, 11.56, 73.96, 1.96, 29.16
Sum the squared deviations: 2.56 + 11.56 + 73.96 + 1.96 + 29.16 = 119.2
Divide by (n-1): 119.2 / (5 - 1) = 29.8
Take the square root: √29.8 ≈ 5.46

Therefore, the sample standard deviation (s) is approximately 5.46.

Conclusion

Understanding the difference between population and sample standard deviation is fundamental for interpreting statistical data correctly. While both measure data spread, the sample standard deviation provides an estimate of the population's variability, while the population standard deviation describes the actual variability of the entire population. Remember to use the appropriate formula depending on whether you're working with the entire population or a sample.