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.
How Standard Deviation Calculator Works
Calculate standard deviation, variance, and deviation from the mean for any data set. Enter numbers and get instant statistical analysis. Enter your values into the form above and the calculator processes them instantly in your browser — no data is sent to any server.
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.