Standard Deviation & Variance Calculator
Calculate standard deviation and variance for any data set — population or sample.
What this calculator does
Calculates variance and standard deviation for any data set you enter, letting you choose between population and sample formulas depending on whether your data represents the whole group or a sample from it.
Who this is for
Students checking statistics homework, anyone analyzing how spread out a data set is (test scores, measurements, survey results), or researchers needing a quick calculation without opening a full statistics package.
Methodology
Variance = average of the squared differences from the mean. Population variance divides by n (the number of data points); sample variance divides by n−1 (Bessel's correction, which reduces bias when your data is a sample rather than the whole population). Standard deviation is simply the square root of variance, bringing the units back in line with the original data.
Worked example
For the data set 12, 15, 18, 20, 22, 25, 30 (n=7): mean = 142 ÷ 7 = 20.29. Summing each value's squared difference from the mean gives approximately 221.4. Population variance = 221.4 ÷ 7 ≈ 31.6, so population standard deviation = √31.6 ≈ 5.62. Sample variance = 221.4 ÷ 6 ≈ 36.9, so sample standard deviation = √36.9 ≈ 6.07 — notice the sample figure is always slightly larger, since dividing by the smaller n−1 inflates the result to correct for sample bias.
Interpretation
Standard deviation tells you how spread out your data is around the mean. A small standard deviation relative to the mean means most values cluster tightly together; a large one means values are widely dispersed. This is why standard deviation is often more informative than the mean alone — two data sets can share the same average but look completely different in how consistent the individual values are.
Your data vs. the mean
Run the calculator above to see each data point compared to the mean.
Common mistakes
- Using population formula on sample data. If your numbers are a sample meant to represent a larger population, using n instead of n-1 will underestimate the true variability.
- Forgetting to square the differences before averaging. Simply averaging the raw differences from the mean always equals zero — squaring is what makes the measure meaningful.
- Confusing variance with standard deviation. Variance is in squared units (hard to interpret directly); standard deviation converts back to the original units, which is why it's the more commonly reported figure.
- Comparing standard deviations across data sets with very different means. A standard deviation of 10 means something different for a data set averaging 20 versus one averaging 1,000 — consider the coefficient of variation (standard deviation divided by the mean) for a fairer comparison across scales.
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Frequently Asked Questions
Why is sample standard deviation always slightly larger than population standard deviation for the same data?
Sample variance divides by n-1 instead of n, which produces a slightly larger number to correct for the tendency of a sample to underestimate the true population variance. This correction (Bessel's correction) is standard practice whenever your data is a sample rather than the complete population.
When should I use sample vs population standard deviation?
Use population standard deviation when your data represents the entire group you care about. Use sample standard deviation when your data is a sample drawn from a larger population.
Why does sample standard deviation divide by n-1 instead of n?
Dividing by n-1 (Bessel's correction) corrects for the fact that a sample tends to underestimate the true population variance.
What does a high standard deviation mean?
It means the data points are spread out further from the mean. A low standard deviation means the data points cluster closely around the mean.