Standard Deviation Calculator
Calculate population and sample standard deviation, variance, mean, and more from a dataset.
| # | Value | Dev (x−μ) | (x−μ)² |
|---|
Enter any list of numbers to get the population or sample standard deviation, variance, mean, median, and range, along with a full deviation table for every value.
How Standard Deviation Calculator Works
The calculator first finds the mean (sum of all values ÷ count), then finds each value's deviation from that mean (x − mean) and squares it — squaring ensures negative and positive deviations don't cancel each other out when totaled.
Variance is the average of those squared deviations. For population variance, the total is divided by n (the count); for sample variance, it's divided by n−1 instead — a correction called Bessel's correction that compensates for the fact that a sample tends to underestimate the true spread of the full population.
Standard deviation is simply the square root of variance, bringing the units back in line with the original data (rather than squared units). Alongside it, the calculator reports the median, minimum, maximum, and range as additional ways of describing how spread out the data is.
See It In Action
Who Uses Standard Deviation Calculator and Why
- Judging how consistent a set of lab measurements or manufacturing tolerances are around their average.
- Comparing the spread of test scores between two classes that share the same mean.
- Deciding whether a dataset represents an entire population (like every employee at a small company) or just a sample drawn from a larger group.
- Getting a full deviation table showing exactly how far each data point sits from the mean.
Mistakes to Avoid
- Selecting Population when the data is actually just a sample of a larger group (or the reverse) — this changes whether the calculator divides by n or by n−1, and gives a different final answer.
- Comparing a variance value directly to the original data's scale — variance is in squared units, so only the standard deviation (its square root) is directly comparable to the original numbers.
- Assuming standard deviation alone tells you which dataset is 'better' — it measures spread, not quality; a tight cluster around a bad mean is still a bad result.
Tips for Best Results
- Default to Sample standard deviation unless your dataset genuinely is the entire population you care about — this is the more common real-world situation.
- Use the reported median and range alongside standard deviation for a fuller picture of spread, especially if your data might have outliers skewing the mean.
Fixing Common Problems
My result doesn't match an answer key or another tool. — Check whether Population or Sample mode is selected — sample standard deviation (dividing by n−1) is always slightly larger than population standard deviation (dividing by n) for the same data.
Terms Explained
Bessel's correction: Dividing by n−1 instead of n when calculating sample variance, correcting for the tendency of a limited sample to underestimate the true population's spread.
Variance: The average of the squared deviations from the mean — standard deviation is its square root, restoring the original units.