Standard Deviation Calculator
Enter a set of numbers (up to 8) to compute the mean, variance and standard deviation (population/sample), measuring data dispersion.
Input Data
Results
At a glance:Standard deviation is the square root of variance, measuring dispersion around the mean. Steps: (1) mean μ = Σxᵢ ÷ n; (2) sum of squared deviations Σ(xᵢ − μ)²; (3) population variance = sum ÷ n, sample variance = sum ÷ (n−1); (4) standard deviation = √variance. Sample mode uses n−1 (Bessel's correction) for an unbiased estimate.
Formula
Mean = Σ values ÷ count.
Sum of squares = Σ(value − mean)².
Population variance = sum ÷ n; sample variance = sum ÷ (n−1).
Standard deviation = √variance.
Empirical rule: ~68% within ±1σ, ~95% within ±2σ (normal).
$$s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2}$$$$\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_i-\mu)^2}$$How to Use
- Choose population or sample mode.
- Enter values (up to 8; at least 2).
- The tool returns mean, variance and standard deviation.
- The note shows the mode and the empirical-rule range.
Sample vs Population
| Item | Sample | Population |
|---|---|---|
| Denominator | n−1 | n |
| Symbol | s | σ |
| Use | estimate overall | full data |
Use sample (÷n−1) for inference; population (÷n) when you have all data.
Case Studies
80, 85, 90, 95
Mean = 87.5.
Sum of squares = 56.25 + 6.25 + 6.25 + 56.25 = 125.
Sample variance = 125 ÷ 3 ≈ 41.67; sample SD ≈ 6.45.
10, 20, 30 (population vs sample)
Mean = 20; sum of squares = 100 + 0 + 100 = 200.
Sample: variance = 200 ÷ 2 = 100, SD = 10.
Population: variance = 200 ÷ 3 ≈ 66.67, SD ≈ 8.16 (smaller).
Grade spread
Two classes both average 70, but A has σ=5, B has σ=15.
Class B is more spread out and needs more remedial teaching.
Normal empirical rule
Height μ=170, σ=6: ~95% lie between 158–182 (μ±2σ).
Smaller σ means data is more concentrated.
Financial volatility
Two stocks same return; A volatility σ=2%, B σ=8%.
B is riskier; SD is a common risk metric.
Why n−1 for sample
Using the sample mean slightly underestimates the squared deviations; ÷(n−1) corrects it to an unbiased estimate.
The difference vanishes for large samples.
FAQ
Sample vs population standard deviation?
Population (÷ n) when you have the complete data; sample (÷ n−1, Bessel's correction) when you only have a sample and want to estimate the whole — slightly larger to reflect uncertainty. Inference usually uses sample.
What does SD = 0 mean?
All values are identical — no dispersion. Larger SD means more spread and more volatility.
SD vs variance?
Variance is the average squared deviation; SD is its square root. SD shares the original unit, so it is more intuitive and more commonly used.
Does SD give a 'normal range'?
For normal data, ~68% lie within mean ±1σ and ~95% within ±2σ. This underlies six-sigma QC and risk control.
How many data points needed?
At least 2 to measure spread; more points give a stabler estimate. This tool supports up to 8; use a spreadsheet for more.
SD vs mean absolute deviation?
SD is more sensitive to extremes (squaring); MAD is more robust to outliers. Both measure spread but differ in behavior.
Why divide sample by n−1?
Using the sample mean instead of the true mean makes the squared sum too small; ÷(n−1) corrects the bias for an unbiased variance estimate.
What unit is the SD?
SD has the same unit as the original values (cm, points); variance is the unit squared. That is why SD is preferred in practice.
Related Tools
References
Content review: Calculatorism Science Team. Standard deviation and variance formulas verified. Results are for reference only.