The key difference between sample and population standard deviation lies in what data they're calculated from: a population standard deviation reflects the spread within an entire group, while a sample standard deviation represents the spread within a subset of that group.
Here's a more detailed breakdown:
Understanding Population Standard Deviation
- Definition: Population standard deviation measures the dispersion of a dataset that includes every individual in a defined group.
- Calculation: This calculation uses all data points from the entire population. It's represented by the Greek letter sigma (σ).
- Type: According to the reference, the population standard deviation is a parameter, a fixed value for the population.
Understanding Sample Standard Deviation
- Definition: Sample standard deviation measures the dispersion of a dataset that includes only some individuals from a larger group.
- Calculation: This calculation uses data points from a representative sample of the population. It's usually denoted by 's' or 'sd'.
- Type: The reference states that the sample standard deviation is a statistic, meaning it is calculated from a portion (or sample) of the population.
Key Differences in a Table
| Feature | Population Standard Deviation | Sample Standard Deviation |
|---|---|---|
| Data Source | Entire population | A sample from the population |
| Symbol | σ (sigma) | s or sd |
| Type | Parameter | Statistic |
| Purpose | Describe the entire population | Estimate the population variability based on a sample |
Why the Difference Matters
- Accuracy: Population standard deviation provides a true picture of the population's variability, but it is only feasible to compute when all the data for a population is available, which is very rare in many real-world cases.
- Estimation: Sample standard deviation is often used to estimate the population standard deviation. Since we usually don't have data from the entire population, the sample standard deviation allows us to generalize the spread from a smaller sample.
Practical Insights
- Bias Correction: When calculating sample standard deviation, you'll notice a slight modification in the formula (using 'n-1' instead of 'n' in the denominator). This adjustment is called Bessel’s correction. It provides a less biased estimate of the population's standard deviation. Without it, the sample standard deviation would underestimate the population standard deviation, making it less reliable.
- Applications:
- Population: Calculating standard deviation when assessing an entire factory's output quality, if the data for all output is available.
- Sample: Estimating the variability in public opinion from a sample in a survey.
Example Scenario
Imagine you want to understand the average height variability of all adults in the US (population). If you could measure everyone, you would calculate the population standard deviation. However, that is impractical, so you would measure the heights of, say, 1000 adults. Using this sample, you would calculate the sample standard deviation to get an idea of the height variability of the population.
In short, while both are used to measure dispersion of data, they come from and represent different groups. The population standard deviation gives the exact spread of an entire population, but the sample standard deviation estimates the spread of an entire population based on a portion.
