The standard deviation calculator computes mean, variance, population SD (σ), and sample SD (s) for any dataset. Enter comma-separated values and get a full descriptive statistics summary instantly.
Standard Deviation Calculator
How to Use the Standard Deviation Calculator
Standard deviation measures how spread out data is around the mean. A low SD means data points cluster near the mean; a high SD means they're spread widely.
Population vs Sample SD
Use population SD (σ) when you have data for the entire population. Use sample SD (s) when your data is a sample from a larger population — this is most common. The difference: sample SD divides by (N−1) instead of N, giving a slightly larger (more conservative) value.
The Empirical Rule
For normally distributed data: 68% of values fall within 1 SD of the mean, 95% within 2 SD, 99.7% within 3 SD. If test scores have mean=80 and SD=10: 68% scored 70-90, 95% scored 60-100. Values more than 3 SDs from the mean are extremely rare outliers.
Reading the Results
The classic dataset [2,4,4,4,5,5,7,9] has mean=5, population SD=2. This is a textbook example from Wikipedia demonstrating how SD captures the spread around the mean.
FAQ
What is standard deviation?
Standard deviation measures how spread out data is from the mean. Low SD means data clusters near the mean; high SD means it's spread out. For test scores with mean=75 and SD=10: 68% of students scored 65-85 (within 1 SD), 95% scored 55-95 (within 2 SD) — from the empirical rule.
What is the difference between population and sample standard deviation?
Population SD divides by N (use when you have all data). Sample SD divides by N-1 (Bessel's correction, use for a sample). Sample SD is unbiased when estimating population SD. For large datasets the difference is tiny, but for small samples (N<30) it matters. Most calculators default to sample SD.
Is this calculator free?
Yes, completely free with no signup required. All calculations run in your browser.
Is my data private?
Yes. All calculations run locally. Nothing is transmitted.
What is variance?
Variance = (standard deviation)². Population variance = Σ(xi − μ)²/N. It's in squared units, making it hard to interpret directly. Standard deviation is the square root of variance, so it's in the original units of the data and more intuitive.