Standard Deviation Calculator

Standard deviation measures how spread out values are around their mean. A small standard deviation means observations cluster close to the average. A large standard deviation means observations vary widely.

The measure is expressed in the same units as the original data, which makes it easier to interpret than variance. If delivery times have a mean of 30 minutes and a standard deviation of 5 minutes, the spread is directly about minutes, not squared minutes.

Standard deviation begins by measuring each value's deviation from the mean. Squaring those deviations prevents positive and negative differences from canceling and gives larger deviations more weight. The average squared deviation is variance, and the square root returns the measure to the original units.

This structure makes standard deviation mathematically convenient, but it also means outliers can influence it strongly. A few extreme values can inflate the spread and change conclusions about consistency.

Population standard deviation divides by the number of values when the data include the entire population of interest. Sample standard deviation usually divides by n - 1 to correct bias when estimating population variability from a sample.

The distinction matters most for small samples. With large samples, the difference becomes minor. Still, the choice should match the question: are you describing the data you have, or estimating the variability of a larger population?

Standard deviation is most intuitive for roughly symmetric distributions. In a normal distribution, about 68 percent of values fall within one standard deviation of the mean, and about 95 percent within two.

For skewed or heavy-tailed distributions, those rules may not hold. A histogram or box plot can show whether standard deviation is the right summary. Variability is not just a number; it is a feature of the distribution's shape.