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Standard deviation measures how spread out a set of numbers is from the mean (average). A low standard deviation means data points cluster close to the mean. A high standard deviation means they are spread over a wider range. It is one of the most fundamental concepts in statistics and is used in science, finance, quality control, and everyday data analysis.
Enter your data set as comma-separated numbers. The calculator shows the mean, population standard deviation, sample standard deviation, variance, range, and count. It also displays a visual distribution of your data. Use sample standard deviation (n-1) when your data is a subset of a larger population. Use population standard deviation (n) when your data includes the entire population.
The difference is in the denominator: population SD divides by n; sample SD divides by (n-1). The (n-1) correction (called Bessel's correction) adjusts for the fact that a sample tends to underestimate population variability. For large data sets (n > 30), the difference is small. For small samples, it matters. If you measured the heights of every student in a class, that is population data. If you measured 30 students from a school of 1,000, that is a sample.
Standard deviation measures how spread out data points are from the average. The empirical rule (68-95-99.7) states that for normally distributed data: 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. This has practical applications: if a class's test scores have a mean of 75 and standard deviation of 10, roughly 68% of students scored between 65 and 85. In quality control, the "six sigma" methodology aims for processes where defects occur more than 6 standard deviations from the target, translating to 3.4 defects per million opportunities. In investing, standard deviation measures volatility: a stock with 20% annual returns and 15% standard deviation will see returns between 5% and 35% about 68% of the time.