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Reporting results in APA style
Choose the population standard deviation only if your data covers the entire population; otherwise use the sample standard deviation.
Solution
Squared differences
The pink square's area is the average of the blue squares' areas, so the length of its side is the standard deviation.
Deviations from the mean
above the mean below the mean the sample standard deviation (S) the value itself
Standard deviation calculator
Free standard deviation calculator — instantly computes the sample standard deviation and population standard deviation, with a full step-by-step solution, interactive visualizations and results in APA format. Also computes the confidence interval, population variance, sample variance, sum of squares, sample size, sum and mean.
What is standard deviation?
Standard deviation measures the variability of data. It is the square root of the average squared difference between each value and the mean. A larger standard deviation means more variability.
Population standard deviation formula
To calculate the population standard deviation you need the entire dataset.
What is sample standard deviation? (S)
When you do not have the entire population, you calculate the sample standard deviation from the sampled data. Unlike the population version, you divide by (n − 1), which makes the resulting statistic less biased.
Sample standard deviation formula
Which one should I use?
Use the population standard deviation (σ) if your data is the entire population; otherwise use the sample standard deviation (S).
Why divide by (n − 1)?
Usually you cannot reach the entire population, so you estimate from a random sample. That means using the sample average (x̄) instead of the population average (µ), and the sample average sits slightly closer to the centre of the sample than the population average does. Dividing by n would therefore underestimate the variance on average. Dividing by (n − 1) corrects the biased estimate of the variance, and partially corrects the one for the standard deviation (Bessel's correction). The sample standard deviation is still biased, but this correction makes it the best simple formula.
Simulation example
Normal(µ=10, σ=4), sample size n=20, 1,000,000 repetitions. The averages of the statistics:
| Statistic | Dividing by (n−1) | Dividing by n |
|---|---|---|
| Standard deviation | 3.958 | 3.849 |
| Variance | 16.008 | 15.208 |
For an unbiased statistic we expect 4 and 16. Dividing by (n−1) gets much closer: the variance is essentially unbiased at 16.008, while the standard deviation remains slightly biased.
Glossary
n — sample size, the number of values.
Mean — the average.
S — sample standard deviation; the best estimate of the population standard deviation when your values are only a sample.
σ — population standard deviation; exact when your values are the entire population.