Standard Deviation Calculator
Compute mean, variance, and standard deviation of a list of numbers — both sample (n−1) and population (n) versions.
What is Standard Deviation Calculator?
The standard deviation calculator measures how spread out a set of numbers is. A small standard deviation means values cluster near the mean; a large one means they're spread out.
This calculator returns both versions: sample standard deviation (divides by n−1, used when your data is a sample of a larger population) and population standard deviation (divides by n, used when you have the whole population).
Formula
Mean: x̄ = Σx ÷ n
Sample variance: s² = Σ(x − x̄)² ÷ (n − 1)
Sample standard deviation: s = √s²
Population variance: σ² = Σ(x − x̄)² ÷ n
Population standard deviation: σ = √σ²
Worked example
Data: 4, 8, 15, 16, 23, 42 (n = 6)
- Mean ≈ 18.0
- Sample SD ≈ 13.65
- Population SD ≈ 12.46
Sample SD is always slightly larger than population SD on the same data because it divides by a smaller number (n−1).
How to use this calculator
- Paste or type your numbers, separated by commas, spaces, or new lines.
- The calculator returns count, mean, both standard deviations, range, and min/max.
Frequently asked questions
Sample vs population — which one should I use?
If your data represents the entire population, use population (σ, n). If it's a sample drawn from a larger population, use sample (s, n−1). Most real-world stats and AP Statistics homework use the sample version.
Why divide by n−1 for the sample?
It's called Bessel's correction. Dividing by n underestimates the true population variance because the sample mean tends to be closer to the sample's own values than the true population mean. The n−1 correction is unbiased.
What's a "high" standard deviation?
It's relative — high or low compared to the mean. The coefficient of variation (SD ÷ mean) gives a scale-free measure: above 0.3 is often called highly variable.
How does this connect to the normal distribution?
For a normal (bell-shaped) distribution: ~68% of values fall within 1 SD of the mean, ~95% within 2 SD, and ~99.7% within 3 SD. This is the 68–95–99.7 rule.