Standard Deviation Calculator

Calculate population standard deviation σ and sample standard deviation s from the same accepted finite values, with both formulas shown.

The Standard Deviation Calculator reports population standard deviation σ and sample standard deviation s from the same accepted numeric values. Both describe spread around a mean, but they use different denominators. The page presents the two calculations side by side so the choice is visible rather than silently assuming that the entered list is either a complete population or a sample.

Population σ divides the sum of squared deviations from the population mean by n before taking the square root. Sample s divides the corresponding sum by n − 1. The sample formula is the conventional estimate used when observations are treated as a sample from a wider group; it is not simply another label for σ.

The input accepts comma or whitespace separators and lists malformed or non-finite tokens that were ignored. A minimum of two accepted finite values is required. Both standard deviations use the same accepted values and are expressed in the original unit, unlike variance, which is expressed in squared units.

Enter at least two finite observations in one consistent unit.
Review the ignored-token message and confirm that every excluded token is truly invalid or unintended.
Decide whether the entered list is the complete group of interest or an observed sample from a larger group.
Use population σ for the complete group and sample s for a sample-based description or estimate, keeping the selected label with the reported number.
Compare the result with the measurement scale and the raw observations; the calculator does not decide which denominator is appropriate.

Population and sample standard-deviation formulas

For a complete population with mean μ, σ = √[Σ(xᵢ − μ)² / n]. The squared differences are averaged over the n population values, and the square root returns the result to the original measurement unit.

For a sample with mean x̄, s = √[Σ(xᵢ − x̄)² / (n − 1)]. The denominator n − 1 is Bessel’s correction in the sample variance; taking its square root gives the sample standard deviation.

The page calculates the mean from the same accepted values in each formula. It does not accept externally supplied population parameters, weights, or a frequency table. If those are required by the analysis, use a method that accepts them explicitly rather than substituting them into this result.

Reading the two standard deviations

Choose the denominator that matches the data

The population and sample outputs answer related but distinct questions. σ describes the spread of the complete entered population. s describes sample spread with the n − 1 denominator. The correct output is not determined by which number is larger; it is determined by how the entered values relate to the population of interest.

For the default values 4, 8, 15, 16, 23, 42, population σ is approximately 12.3153 and sample s is approximately 13.4907. Both are in the same units as the observations. The sample result is larger because the same squared deviations are divided by 5 rather than 6 before taking the square root.

A standard deviation summarizes squared distance from the mean. It does not show the direction of an observation, identify outliers, or guarantee that a particular proportion falls within a given number of deviations. Such interpretations require additional assumptions or calculations not performed on this page.

Statistics tips and best practices

State whether you used σ or s whenever you report the result; the labels are not interchangeable.
Check the original unit. Standard deviation should have the same unit as the input, not a squared unit.
Use the same accepted observations for both values when comparing population and sample formulas.
Inspect extreme values because squared deviations give distant observations substantial influence.
Do not infer a normal distribution or a fixed percentage of values from standard deviation alone.

Standard deviation worked example

For the six default values, the mean is 18 and the sum of squared deviations is 910.
Population variance is 910 / 6 = 151.6667, so population σ is √(910 / 6) ≈ 12.3153.
Sample variance is 910 / 5 = 182, so sample s is √182 ≈ 13.4907.

Common mistakes to avoid

Reporting one result without its population or sample label.
Using n for a sample standard deviation or n − 1 for the population calculation.
Calling standard deviation a range; it uses all observations’ squared deviations, not only the minimum and maximum.
Comparing results from datasets measured in different units without converting them first.
Treating a small standard deviation as proof that the data are accurate or representative.

These outputs are descriptive calculations from the accepted finite values. The page does not estimate a confidence interval, conduct a hypothesis test, check distribution assumptions, or determine whether the list is a sample or a complete population.

Choose and report the denominator based on the data-collection design. For consequential statistical analysis, verify the formula and assumptions against the study plan.