Standard Deviation Calculator

Enter a list of numbers to calculate standard deviation, variance, mean, median, mode, and more. Results update instantly.

Last reviewed: June 2026
Built and maintained by · Last updated June 2026

Separate numbers with commas or spaces. Example: 2 4 4 4 5 5 7 9

Enter at least two numbers to see statistics.

Quick Answer

Standard deviation measures how spread out a set of numbers is. Find the mean, then the average of the squared differences from the mean (the variance), then take its square root. For 2, 4, 6, 8 the mean is 5 and the population standard deviation is about 2.24. Enter your data set above for the mean, variance, and standard deviation.

Important Disclaimer: This calculator is for education, study, and quick statistical checks. For regulated reporting, research, finance, quality-control decisions, or medical analysis, validate the data source and model assumptions with a qualified analyst.

What this calculator returns

This calculator parses a list of numbers and returns the main one-variable summaries: count, sum, mean, median, mode, range, minimum, maximum, population variance, sample variance, population standard deviation, sample standard deviation, and standard error of the mean. Commas, spaces, tabs, and line breaks all work as separators, so you can paste a column of values from a spreadsheet.

Population variance divides the squared deviations by N. Sample variance divides by n - 1, which is Bessel's correction for estimating a larger population from a sample. Standard deviation is the square root of variance, so it comes back to the original units of the data while variance remains in squared units.

Sample vs population standard deviation

Use population standard deviation when the numbers are the complete set you care about: every item in a batch, every score in a small class, or every value in a closed list. Use sample standard deviation when the values are a sample from a bigger process or population, such as ten inspected parts from a long production run or a survey sample from a larger audience.

The difference is small for large n but visible for small lists. For the classic dataset 2, 4, 4, 4, 5, 5, 7, 9, the mean is 5, population variance is 4, population SD is 2, sample variance is about 4.571429, and sample SD is about 2.13809.

How to interpret standard deviation

Standard deviation is meaningful for any numeric dataset where the spread is finite, but the familiar normal-distribution shortcuts are not universal. If a distribution is roughly bell-shaped, one standard deviation is often a useful practical band around the mean. If the data are skewed, bounded, clumped, or heavy-tailed, the same number may hide important structure. In those cases, compare SD with the median, interquartile range, or a histogram before making decisions.

A higher standard deviation means larger typical distances from the mean, not automatically bad data. In finance it can describe return volatility; in quality control it can show process spread; in experiments it can describe measurement variability. It does not prove that most values fall inside mean +/- one SD unless the shape of the data supports that interpretation.

Common mistakes

Using sample SD for a complete population: if you have every value in the set, population SD is the direct descriptive statistic. Sample SD is intentionally a little larger because it estimates a population from incomplete data.

Calling a high SD unreliable by itself: high spread can be real variation rather than a bad measurement process. Look at the context, units, and distribution shape.

Ignoring outliers: because squared deviations give extra weight to extreme values, one unusual value can inflate SD. When outliers are possible, inspect the raw values and compare against robust summaries.

Worked examples

Quality-control weights: if 100 parts average 100 grams with a sample SD of 10 grams, the result describes spread in grams. It does not by itself prove that most parts are between 90 and 110 grams; that statement needs a distribution assumption or direct count.

Investment returns: a portfolio with a 15% annual return SD has return volatility around its mean return. It should be compared with average return, drawdowns, time horizon, and downside-risk measures before making investment decisions.

Repeated measurements: if ten temperature readings average 25°C with sample SD 2°C, the readings vary by about two degrees in the original temperature units. If you want uncertainty in the average reading, use the standard error of the mean instead.

Sources

For deeper background on spread statistics and robust alternatives, see the NIST/SEMATECH e-Handbook section on measures of scale.

Frequently Asked Questions

Use population standard deviation only when the numbers are the complete population you want to describe. Use sample standard deviation when the numbers are a subset used to estimate a larger population; the n - 1 denominator applies Bessel's correction.
No. Standard deviation can describe spread for many numeric datasets, but normality affects interpretation. Rules such as about 68 percent of values falling within one standard deviation are normal-distribution approximations, not guarantees for skewed, bounded, or heavy-tailed data.
Standard deviation squares each distance from the mean, so extreme values can inflate it substantially. For skewed or heavy-tailed data, compare it with the median, interquartile range, or median absolute deviation before drawing conclusions.
Variance is the average squared distance from the mean, so its units are squared. Standard deviation is the square root of variance, which brings the spread back into the original units of the data.
The standard error of the mean is s divided by the square root of n. It estimates how much a sample mean would vary across repeated samples, not how spread out individual observations are.