Standard Deviation Calculator
Enter a list of numbers to calculate the mean, variance, and standard deviation.
How this calculator works
Standard deviation measures how spread out a data set is from its mean. Sample standard deviation divides by n−1 (Bessel's correction), while population standard deviation divides by n.
Use 'sample' when your data is a subset of a larger population; use 'population' when it's the entire population.
Interpreting standard deviation
A small standard deviation means data points cluster tightly around the mean; a large standard deviation means they're more spread out. In a normal distribution, roughly 68% of data falls within one standard deviation of the mean, and about 95% falls within two standard deviations.
Standard deviation is widely used to measure consistency and risk — in finance, a stock with higher standard deviation in its returns is considered more volatile; in quality control, it measures how consistent a manufacturing process is.
A worked example
For the data set 4, 8, 6, 5, 3, 9, 7: the mean is 6, and the sample standard deviation (dividing by n minus 1, or 6) works out to approximately 2.16.
Using population standard deviation instead (dividing by n, or 7) gives approximately 2.0 — slightly lower, illustrating why choosing the correct type (sample vs. population) matters for the final result, especially with smaller data sets.
What standard deviation tells you
A low standard deviation means data points cluster tightly around the mean, while a high standard deviation means values are more spread out. It's one of the most widely used measures of variability across statistics, science, and finance — for example, in investing, standard deviation is commonly used as a measure of volatility or risk.
In a normal (bell-curve) distribution, roughly 68% of data falls within one standard deviation of the mean, and about 95% falls within two — a useful rule of thumb for interpreting how typical or unusual a given value is relative to a data set.
Frequently asked questions
Should I use sample or population standard deviation?
Use sample if your numbers are a subset drawn from a larger group (most common case). Use population only if your data represents the entire group you care about.
Why does sample standard deviation divide by n−1?
Dividing by n−1 (Bessel's correction) corrects for the bias that occurs when estimating a population's variability from a smaller sample.
What's considered a 'high' standard deviation?
It depends entirely on context and the scale of your data — there's no universal threshold, so standard deviation is most useful when compared relative to the mean or to another dataset's standard deviation.
Why does sample size matter for standard deviation?
Smaller samples produce less reliable standard deviation estimates, which is part of why sample standard deviation uses n−1 in its formula (Bessel's correction) to help correct for this bias.
What does a standard deviation of zero mean?
It means every value in the data set is identical — there's no variability at all.
How is standard deviation used in real life?
It's used to measure investment volatility, quality control in manufacturing, test score spread in education, and variability in scientific measurements, among many other applications.