What Is a Standard Deviation Calculator?

A standard deviation calculator measures how spread out a dataset is around its mean. It computes all key descriptive statistics: mean, median, variance, standard deviation, range, and coefficient of variation — giving you a complete picture of your data's distribution in one step. This calculator supports both sample SD (divides by n−1, for data sampled from a larger population) and population SD (divides by n, for complete dataset analysis).

Standard deviation is the most widely used measure of statistical dispersion. It appears in finance (portfolio risk, volatility), quality control (Six Sigma), scientific research (margin of error), education (grading on a curve), weather forecasting, and sports analytics.

The Standard Deviation Formula

Sample SD: s = √[Σ(xᵢ − x̄)² / (n − 1)]

Population SD: σ = √[Σ(xᵢ − μ)² / n]

Steps: (1) Find the mean x̄ = Σxᵢ / n. (2) Compute each deviation xᵢ − x̄. (3) Square each deviation. (4) Sum the squared deviations. (5) Divide by n−1 (sample) or n (population). (6) Take the square root. For the default dataset [4,7,13,2,1,8,6,11,5,9]: mean = 6.6, sample variance ≈ 13.6, sample SD ≈ 3.69.

Sample vs Population Standard Deviation

Bessel's correction (dividing by n−1 instead of n) corrects for the bias that arises when estimating population variance from a sample. Because sample means tend to be closer to sample data than the true population mean, dividing by n underestimates the population variance. Dividing by n−1 corrects this, producing an unbiased estimator. Use sample SD in virtually all research and data analysis contexts.

The Empirical Rule (68-95-99.7 Rule)

For normally distributed data: approximately 68% of values fall within one SD of the mean (μ ± σ), 95% within two SDs (μ ± 2σ), and 99.7% within three SDs (μ ± 3σ). Values more than 3 SDs from the mean are statistical outliers. This rule is the foundation of z-scores, hypothesis testing, and quality control limits.

Applications of Standard Deviation

Finance: Stock volatility is measured by SD of daily returns. A high-SD stock (e.g., 3% daily SD) is riskier than a low-SD stock (0.5% daily SD). Portfolio theory uses SD to quantify and minimize risk. Quality control: Six Sigma aims for processes where the defect rate is beyond 6 standard deviations from the target — fewer than 3.4 defects per million. Research: Standard error = SD/√n; confidence intervals = mean ± 1.96 × SE (for 95% CI). Education: Standardized test scores (SAT, IQ) are scaled using mean and SD of the reference population.

Tips for Using This Calculator

Enter numbers separated by commas or spaces. Toggle between Sample and Population mode — the results update immediately. The bar chart shows each value with a red dashed line at the mean — values far from the line contribute most to the standard deviation. For z-score analysis, use the z-score calculator. For probability distributions, see the normal distribution calculator.