Statistics
Standard deviation calculator
Paste a list of numbers. Returns both the sample standard deviation (n−1 in denominator — what you'll usually want) and the population standard deviation (n), plus variances.
Formula
Standard deviation measures how spread out a dataset is around its mean. A small σ means most values cluster tightly around the average; a large σ means they're widely scattered.
There are two flavors: sample σ uses (n−1) in the denominator (Bessel's correction) and is what you use when your data is a sample from a larger population. Population σ uses n and is for when your data IS the entire population. In real life you almost always want sample σ — even in 'population' studies, your data is usually still a sample.
For normally distributed data, ~68% of values fall within ±1σ of the mean, ~95% within ±2σ, and ~99.7% within ±3σ (the '68-95-99.7 rule'). This is why σ is the natural scale for describing outliers and confidence intervals.
Examples
- 0110 test scores: 12, 15, 18, 22, 25, 29, 30, 31, 33, 40→ Mean 25.5, sample σ ≈ 8.83, population σ ≈ 8.38
- 02Identical values: 50, 50, 50, 50→ σ = 0 (no spread)
- 03Two values: 100, 200→ Mean 150, sample σ ≈ 70.71, population σ = 50
FAQ
- Almost always sample (n−1). Use population (n) only when you literally have every member of the population — every employee at a 12-person company, every shot in a 5-round match. For survey data, polling, scientific experiments, A/B tests — use sample.