sample and population standard deviation

Daniel Parker

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10-03-2025

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Understanding the difference between sample and population standard deviation is crucial in statistics. Both measures describe the spread or dispersion of a dataset, but they're calculated differently and used in different contexts. This article will clarify the distinction, providing clear explanations and examples.

What is Standard Deviation?

Standard deviation measures how spread out a dataset is from its mean (average). A low standard deviation indicates data points cluster closely around the mean, while a high standard deviation shows data points are more widely dispersed. It's a vital statistic for understanding data variability.

Population Standard Deviation

The population standard deviation calculates the spread of an entire population. This means you have data for every single member of the group you're studying. The formula uses the Greek letter sigma (σ) to represent the population standard deviation:

σ = √[ Σ(xi - μ)² / N ]

Where:

• σ is the population standard deviation
• Σ denotes summation (adding up all values)
• xi represents each individual data point
• μ is the population mean
• N is the total number of data points in the population

Example: Population Standard Deviation

Let's say we have data on the height of every student in a small school (our entire population): [160, 165, 170, 175, 180] cm.

1. Calculate the mean (μ): (160 + 165 + 170 + 175 + 180) / 5 = 170 cm
2. Calculate the deviations from the mean (xi - μ): [-10, -5, 0, 5, 10]
3. Square the deviations: [100, 25, 0, 25, 100]
4. Sum the squared deviations: 100 + 25 + 0 + 25 + 100 = 250
5. Divide by N (5): 250 / 5 = 50
6. Take the square root: √50 ≈ 7.07 cm

Therefore, the population standard deviation (σ) is approximately 7.07 cm.

Sample Standard Deviation

Often, it's impractical or impossible to collect data from an entire population. Instead, we use a sample—a smaller, representative subset of the population. The sample standard deviation (s) estimates the population standard deviation based on this sample. The formula is slightly different:

s = √[ Σ(xi - x̄)² / (n - 1) ]

Where:

• s is the sample standard deviation
• Σ denotes summation
• xi represents each individual data point in the sample
• x̄ is the sample mean
• n is the number of data points in the sample

Why (n-1)? Using (n-1) instead of n in the denominator is called Bessel's correction. It provides a less biased estimate of the population standard deviation, particularly when the sample size is small.

Example: Sample Standard Deviation

Let's say we take a sample of four students from the same school: [165, 170, 175, 180] cm.

1. Calculate the sample mean (x̄): (165 + 170 + 175 + 180) / 4 = 172.5 cm
2. Calculate the deviations from the mean (xi - x̄): [-7.5, -2.5, 2.5, 7.5]
3. Square the deviations: [56.25, 6.25, 6.25, 56.25]
4. Sum the squared deviations: 56.25 + 6.25 + 6.25 + 56.25 = 125
5. Divide by (n - 1) (3): 125 / 3 ≈ 41.67
6. Take the square root: √41.67 ≈ 6.46 cm

The sample standard deviation (s) is approximately 6.46 cm. Notice it's slightly different from the population standard deviation.

When to Use Which?

• Population Standard Deviation (σ): Use when you have data for the entire population. This is rare in practice.
• Sample Standard Deviation (s): Use when you have data from a sample of the population. This is the far more common scenario.

Key Differences Summarized

Understanding the nuances between sample and population standard deviation is essential for accurate data interpretation and statistical inference. Remember to choose the appropriate calculation based on whether you're working with an entire population or a sample.