Standard deviation measures how spread out data points are from the mean. A low standard deviation means values cluster tightly; a high value indicates wide dispersion. It is essential in statistics, quality control, and finance.
Population vs Sample Standard Deviation
Population standard deviation (sigma) divides by N. Sample standard deviation (s) divides by N-1 (Bessel's correction) to reduce bias when estimating from a sample. Use our Standard Deviation Calculator for both modes.
The Formula
For a sample: s = sqrt(Sum(x - mean)^2 / (n-1)). First find the mean, subtract from each value, square the differences, sum them, divide by n-1, and take the square root.
Worked Example
Data: 2, 4, 4, 4, 5, 5, 7, 9. Mean = 5. Squared deviations sum to 32. Sample variance = 32/7 about 4.57. Standard deviation about 2.14.
How to Interpret Results
In a normal distribution, roughly 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three (the empirical rule).
Going Deeper
Understand what standard deviation measures and how to calculate it for sample and population data. This guide connects theory to practice — use the related calculators linked at the bottom to verify each example with your own numbers.
Practical Tips
- Write down given values and unknowns before opening the calculator.
- Check units and rounding rules appropriate to your context (class, lab, or business).
- Compare manual working with the calculator result to build confidence.
Common Mistakes to Avoid
- Rushing inputs without reading field labels carefully.
- Confusing similar formulas that use different variables or units.
- Reporting results with more precision than your inputs justify.
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