What the standard deviation calculator does
Paste a list of numbers — separated by commas, spaces or new lines — and the calculator returns the count, mean, variance and standard deviation, for both the population and sample definitions. Standard deviation measures how spread out the values are around the mean: a small value means the data cluster tightly, a large one means they are scattered. It is the most common measure of variability in statistics, finance, quality control and science.
The formulas
Mean μ = Σx / n
Population variance σ² = Σ(x − μ)² / n
Population SD σ = √σ²
Sample variance s² = Σ(x − x̄)² / (n − 1)
Sample SD s = √s²The only difference is dividing by n or by n − 1. Use the population formula when your numbers are the entire group you care about (every student in one class). Use the sample formula when they are a sample drawn from a larger group and you want to estimate that group's spread (30 customers out of thousands). Dividing by n − 1 — Bessel's correction — compensates for the sample mean being closer to the sample points than the true mean is, which would otherwise underestimate the spread.
Worked example
| x | x − mean | (x − mean)² |
|---|---|---|
| 4 | −4 | 16 |
| 6 | −2 | 4 |
| 8 | 0 | 0 |
| 10 | 2 | 4 |
| 12 | 4 | 16 |
| Σ = 40, n = 5, mean = 8 | Σ = 40 |
Population: σ² = 40 / 5 = 8 σ = √8 ≈ 2.828
Sample: s² = 40 / 4 = 10 s = √10 ≈ 3.162How to read the number
For roughly bell-shaped data, the empirical rule applies: about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three. If exam scores have a mean of 70 and SD of 8, most students scored between 62 and 78, and a score of 90 (2.5 SD above) is unusual. For skewed data — incomes, house prices, response times — the rule does not hold and the median with quartiles is often more informative.
Where it is used
- Finance: the SD of returns is the standard measure of an investment's volatility (risk). A fund with 15% annual SD is expected to swing within ±15% of its average return about two years in three.
- Quality control: Six Sigma aims for the specification limits to sit six SDs from the process mean, giving 3.4 defects per million.
- Science: error bars on charts are often ±1 SD (or the standard error, SD ÷ √n).
- Grading: z-scores — (score − mean) ÷ SD — put results from different tests on a common scale.
- Weather and sport: comparing consistency, not just averages — two runners with the same mean time but different SDs are different athletes.
Related measures
| Measure | What it is | When to prefer it |
|---|---|---|
| Variance | SD squared; in squared units | Mathematical work, combining independent variances |
| Range | Max − min | Quick look; very sensitive to outliers |
| Interquartile range | 75th − 25th percentile | Skewed data, outliers present |
| Mean absolute deviation | Average of |x − mean| | Intuitive; less common |
| Standard error | SD ÷ √n | Uncertainty of the mean, not spread of data |
| Coefficient of variation | SD ÷ mean | Comparing spread across different scales |