Standard Deviation Calculator
Measure how spread out your data is. Paste a list of numbers and get the standard deviation and variance — both the sample version (for data that's a subset) and the population version (when you have every value).
How to use this calculator
- Paste your numbers, separated by commas or spaces.
- Choose Sample if your data is a subset of something larger (the usual case), or Population if it's everything.
- Press Calculate — both versions are shown so you can compare.
Formula used
Subtract the mean from each value, square the differences, average them (variance), then take the square root. Sample SD divides by n−1 (Bessel's correction) because using the sample's own mean slightly understates the true spread.
Example calculation
For 10, 12, 23, 23, 16, 23, 21, 16: mean = 144 ÷ 8 = 18.
Squared deviations sum to 192 → sample variance = 192 ÷ 7 ≈ 27.43, sample SD ≈ 5.24. (Population SD = √(192 ÷ 8) ≈ 4.90.)
What standard deviation tells you
Standard deviation measures typical distance from the mean — how spread out the data is, in the same units as the data itself. Two classes can both average 70% on a test, but an SD of 5 means everyone scored near 70, while an SD of 20 means scores ranged wildly.
For roughly bell-shaped data, the empirical rule applies: about 68% of values fall within one SD of the mean, 95% within two, 99.7% within three. That's why SD underpins everything from quality control limits to investment risk to grading curves.
Why use this calculator?
- See sample and population SD side by side — and stop guessing which one your homework wants.
- Quantify consistency: test scores, process quality, investment volatility.
- Get the intermediate values (mean, squared deviations) to follow the calculation by hand.
Frequently asked questions
Should I use sample or population standard deviation?
Use sample (÷ n−1) when your data is a subset drawn from something bigger — survey responses, measurements, experiment results. Use population (÷ n) only when you truly have every member — e.g. the test scores of one specific class, analyzed as that class.
What is the difference between variance and standard deviation?
Variance is the average squared deviation; standard deviation is its square root. SD is usually preferred because it's in the original units — an SD of 5 cm is meaningful, a variance of 25 cm² is not intuitive.
What is a "good" standard deviation?
There's no universal threshold — it depends on the scale and context. Compare it to the mean: the coefficient of variation (SD ÷ mean) lets you compare spread across datasets with different units or magnitudes.