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).
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