The Standard Deviation Calculator measures how far your numbers stray from their average. Type a list into the Numbers field, mark the input as Sample or Population under Type, press Calculate, and the SD appears alongside the mean and variance that support it.
Teachers checking test-score consistency, investors gauging volatility, and lab technicians tracking repeatability all rely on it. The tool also handles rate data: switch the Data type to proportion (rate) and enter the Sample size and Event prop. instead of a raw list of measurements.
The Formula and a Quick Example
Standard deviation is the square root of the variance. Find the mean, square each deviation from it, and average those squares, dividing by n – 1 for a sample or by n for a population. The square root then converts the variance back into your original units of measurement.
The familiar set 2, 4, 4, 4, 5, 5, 7, 9 has a mean of 5. Its squared deviations sum to 32, so treating the eight values as a population gives a variance of 4 and an SD of exactly 2. As a sample, dividing by 7 instead raises the result to about 2.14.
Using the Inputs and Reading Results
For continuous data, paste your values into the Numbers box and select Sample or Population before pressing Calculate. For a rate, set Data type to proportion (rate), then fill in the Sample size and the Event prop. so the correct variance formula kicks in automatically.
Interpret the SD next to the mean rather than alone: an SD of 2 around a mean of 5 signals far more relative spread than 2 around 50. In roughly normal data, close to two thirds of all values fall within one SD either side of the average.
Frequently Asked Questions
What separates sample from population SD?
The sample version divides the summed squared deviations by n – 1 to correct for missing data, while the population version divides by n. Sample results run slightly larger, which keeps the estimate honest.
Why divide by n – 1 for samples?
A sample underestimates spread because it misses information about extreme tail values. Dividing by n – 1, known as Bessel’s correction, inflates the variance just enough to make the estimate unbiased on average.
What does an SD of 2 mean?
It means typical values sit about 2 units from the mean. For the dataset 2, 4, 4, 4, 5, 5, 7, 9, most points lie between 3 and 7 around the mean of 5.
Can standard deviation be zero or negative?
It can be zero but never negative. Zero means every value equals the mean exactly, so the list holds one repeated number, while any genuine variation pushes the result above zero.
How does the proportion mode work?
For rates the variance is p times 1 – p. Enter the Sample size and Event prop., for example 0.5 across 100 trials, and the tool returns 0.05 as the standard deviation of that rate.