Standard Deviation Calculator
Enter your data set to calculate both population and sample standard deviation, variance, and mean. See the full step-by-step working: mean calculation, deviations, squared deviations, and final results — plus a data table showing each value's deviation from the mean.
Understanding Standard Deviation
Standard deviation is a measure of how spread out numbers are from their average (mean). A small standard deviation means the data points are clustered close to the mean, while a large standard deviation indicates the data is spread over a wider range. It is the most commonly used measure of variability in statistics and is essential for understanding data distributions, quality control, finance, and scientific research.
Standard Deviation Formulas
Population: σ = √[ Σ(xi − μ)² / N ]
Sample: s = √[ Σ(xi − x̄)² / (N − 1) ]
Variance = σ² (population) or s² (sample)
Z-score: z = (x − mean) / σ
Population vs. Sample
When you have data for an entire population (every student in a school, every product manufactured), use population standard deviation (σ), which divides by N. When you have a sample (a subset of data representing a larger group), use sample standard deviation (s), which divides by N−1. The N−1 correction (called Bessel's correction) compensates for the tendency of a sample to underestimate population variance.
The 68-95-99.7 Rule (Empirical Rule)
For data that follows a normal (bell-curve) distribution: approximately 68% of values fall within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This rule is widely used in quality control (Six Sigma), grading on a curve, and identifying statistical outliers.
Z-Scores
A z-score tells you how many standard deviations a specific value is from the mean. It is calculated as z = (x − mean) / σ. A z-score of 0 means the value equals the mean. A z-score of +1.5 means the value is 1.5 standard deviations above the mean. Z-scores are used to compare values from different distributions and to calculate probabilities.
Applications
Finance: Standard deviation of stock returns measures investment risk (volatility). A stock with σ = 20% is twice as volatile as one with σ = 10%. Quality control: Manufacturing processes use 6σ (six sigma) to ensure defect rates below 3.4 per million. Science: Error bars on graphs typically show ±1 standard deviation to indicate measurement uncertainty.