Free Standard Deviation Calculator
Calculate population and sample standard deviation, variance, mean, and range from any set of numbers. Supports both σ (population) and s (sample) formulas. Free, private — all calculations run in your browser.
Sample vs Population Comparison
| Metric | Sample (÷n−1) | Population (÷N) | Difference |
|---|---|---|---|
| Std Deviation | 6.239658 | 5.919459 | 0.320199 |
| Variance | 38.933333 | 35.040000 | 3.893333 |
Formula Reference
About This Standard Deviation Calculator
The Standard Deviation Calculator computes the complete statistical summary of a dataset: population standard deviation (σ), sample standard deviation (s), variance (σ² or s²), mean, range, and count. Enter your numbers separated by commas, spaces, or new lines and get instant results with both population and sample formulas shown.
The Formulas
The key difference between the two formulas is the denominator. Population SD divides by N (all members present). Sample SD divides by N−1 — Bessel's correction — which compensates for the fact that a sample tends to underestimate the spread of the full population. For almost all real-world work, use sample SD.
The Empirical Rule (68-95-99.7)
For data that follows a normal (bell-curve) distribution, standard deviation defines three key probability zones: approximately 68% of values fall within ±1 SD of the mean; 95% fall within ±2 SD; and 99.7% fall within ±3 SD. A data point more than 3 SDs from the mean is considered a statistical outlier and occurs in fewer than 0.3% of cases by chance.
Privacy Notice
All calculations run in your browser. No data is transmitted or stored. See our Privacy Policy.
When to Use This Calculator
Calculate SD and variance for a dataset as part of homework, research papers, or data analysis projects.
Monitor process consistency using control charts. A rising SD signals the process is becoming less consistent before defects occur.
Measure volatility of investment returns. Higher SD = higher risk. Portfolio managers use SD to quantify and compare investment risk.
Report variability in patient measurements, clinical trial results, and medical test accuracy alongside mean values.
Analyse score distributions to understand whether a test was appropriately difficult, identify outlier performance, and normalise grades.
💡 Pro Tips
Always clarify whether you need population SD (σ) or sample SD (s) before reporting results. In almost all real-world research, surveys, and quality control applications, you are working with a sample — so sample SD (divide by N−1) is correct. Population SD is only appropriate when your dataset contains every member of the population, which is rare outside of census data.
The coefficient of variation (CV = SD/mean × 100%) is a more useful comparison metric than raw SD when comparing variability across datasets with different units or scales. A manufacturing process with SD = 5 mm and mean = 100 mm (CV = 5%) is far more consistent than one with SD = 5 mm and mean = 10 mm (CV = 50%), even though the SD is identical.
Outliers have a disproportionately large effect on standard deviation because each deviation is squared before summing. A single extreme value can dramatically inflate the SD. Always examine your dataset for outliers before reporting SD, and consider reporting the interquartile range (IQR) alongside SD when outliers are present.
Standard deviation is the foundation of many statistical tests: t-tests, ANOVA, confidence intervals, and control charts all rely on SD. Understanding that the standard error of the mean = SD/√n is critical for interpreting confidence intervals — a larger sample size reduces the standard error but not the underlying SD of the data.
Frequently Asked Questions
Related Calculators
Your input is processed locally in your browser and is never stored, transmitted, or shared with any server. See our Privacy Policy.