sample standard deviation vs population standard deviation

3 min read 19-03-2025

sample standard deviation vs population standard deviation

Understanding the difference between sample and population standard deviation is crucial in statistics. Both measure the spread or dispersion of a dataset, but they apply to different contexts and use slightly different formulas. This article will clarify the distinction, explaining when to use each and highlighting the key differences.

Understanding 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), while a high standard deviation indicates that the data points are spread out over a wider range of values.

Population Standard Deviation: Measuring the Entire Group

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. The formula uses the Greek letter sigma (σ):

σ = √[Σ(xi - μ)² / N]

Where:

σ represents the population standard deviation.
Σ represents the sum.
xi represents each individual data point.
μ represents the population mean.
N represents the total number of data points in the population.

Think of it like this: you're measuring the height of every single student in a specific school. You'd use the population standard deviation to describe the variability in height within that entire school.

Sample Standard Deviation: Estimating from a Subset

More often, we don't have data for the entire population. Instead, we work with a sample – a smaller subset representing the larger population. The sample standard deviation estimates the population standard deviation based on this sample. The formula uses the letter 's':

s = √[Σ(xi - x̄)² / (n - 1)]

Where:

s represents the sample standard deviation.
Σ represents the sum.
xi represents each individual data point in the sample.
x̄ represents the sample mean.
n represents the total number of data points in the sample.

Notice the key difference: the denominator is (n-1) instead of N. This is called Bessel's correction. It accounts for the fact that a sample is less likely to capture the full range of variability present in the larger population. Using (n-1) provides a more accurate and unbiased estimate of the population standard deviation.

For example, imagine you're studying the average income of people in a large city. You can't survey everyone, so you take a sample of 1000 residents. You would use the sample standard deviation to estimate the income variability across the entire city's population.

Why the (n-1) correction?

The (n-1) correction, also known as Bessel's correction, is crucial for unbiased estimation. If you used 'n' in the sample standard deviation formula, your estimate would consistently underestimate the population standard deviation. This is because samples tend to have less variability than the population they represent. The (n-1) adjustment helps correct this bias.

When to Use Which?

Population Standard Deviation: Use this when you have data for the entire population you are studying. This is rare in practice.
Sample Standard Deviation: Use this when you're working with a sample and trying to estimate the standard deviation of the larger population that the sample represents. This is far more common.

Example: Calculating Sample Standard Deviation

Let's say we have a sample of exam scores: {70, 75, 80, 85, 90}.

Calculate the sample mean (x̄): (70 + 75 + 80 + 85 + 90) / 5 = 80
Calculate the deviations from the mean (xi - x̄): {-10, -5, 0, 5, 10}
Square the deviations: {100, 25, 0, 25, 100}
Sum the squared deviations: 100 + 25 + 0 + 25 + 100 = 250
Divide by (n-1): 250 / (5 - 1) = 62.5
Take the square root: √62.5 ≈ 7.9

Therefore, the sample standard deviation (s) is approximately 7.9.

Conclusion: Choosing the Right Measure

Choosing between sample and population standard deviation depends entirely on whether you have data for the entire population or just a sample. In most real-world applications, you'll be dealing with samples and therefore using the sample standard deviation. Understanding the subtle but critical difference between these two measures is essential for accurate statistical analysis and interpretation. Remember that the sample standard deviation provides an estimate of the true population variability.