Standard Deviation Calculator
Enter a list of numbers and calculate the standard deviation with a complete step-by-step breakdown. Toggle between population (σ) and sample (s) standard deviation.
The 68-95-99.7 Rule (Empirical Rule)
For data that follows a normal distribution, the standard deviation defines predictable intervals:
| Interval | Coverage | Example (mean=100, SD=15) |
|---|---|---|
| μ ± 1σ | ~68.27% | 85 – 115 |
| μ ± 2σ | ~95.45% | 70 – 130 |
| μ ± 3σ | ~99.73% | 55 – 145 |
Population vs Sample Formulas
| Property | Population | Sample |
|---|---|---|
| Symbol | σ | s |
| Divisor | n | n − 1 |
| Use when | You have ALL data | You have a subset |
| Mean symbol | μ | x̄ |
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Frequently Asked Questions
How to calculate standard deviation?
1) Calculate the mean. 2) Subtract the mean from each value to get deviations. 3) Square each deviation. 4) Sum the squared deviations. 5) Divide by n (population) or n−1 (sample). 6) Take the square root. The calculator above shows every step.
Population vs sample standard deviation?
Use population (σ) when you have data for the entire group. Use sample (s) when working with a subset — this is more common. The sample formula divides by n−1 (Bessel’s correction) to give an unbiased estimate.
What does standard deviation tell you?
Standard deviation measures how spread out data is from the mean. For a normal distribution: ~68% of values fall within 1 SD, ~95% within 2 SD, and ~99.7% within 3 SD (the 68-95-99.7 rule). A small SD means data is tightly clustered around the mean.
What is a good standard deviation?
There is no universal “good” value — it depends on context. Use the Coefficient of Variation (CV = SD ÷ Mean × 100%) to compare variability. A CV under 15% generally indicates low variability.
Standard deviation uses the same units as the original data, making it more intuitive than variance for interpretation.
