Standard Deviation Calculator

How to use this standard deviation calculator

What this calculator does

This calculator measures the spread of up to 30 ungrouped numeric observations around their arithmetic mean. It reports the standard deviation, variance, mean, sum of squared deviations, observation count, and – when requested – the contribution of every observation to the variance calculation. Standard deviation is expressed in the same units as the original data, while variance is expressed in squared units. The tool supports both the sample formula and the population formula; it does not determine whether your data should be treated as a sample or an entire population, and it does not test normality or remove outliers automatically. For a concise statistical definition and both formulas, see the OpenStax explanation of standard deviation and variance.

When to use it

Use the calculator when you need to compare the consistency of repeated measurements, summarize the spread of a small research dataset, check the variability of scores or process readings, or verify a hand calculation before using the result in later statistical work. A smaller standard deviation means the observations lie more tightly around their mean; a larger value means they are more dispersed. Because standard deviation squares deviations before averaging them, unusually distant observations can have a strong effect.

How to calculate

1. The calculator opens with the sample dataset 2, 4, 5, 6, 6, 9, 10, so the first screen already contains a complete demonstration and the Excel workbook is immediately available. Replace those values in Data values with your own observations.
2. Set Dataset type to Sample when your observations represent a subset used to estimate a wider population, or to Population when the values constitute the complete population you want to describe. Use Steps to show to keep the detailed deviation table visible or switch to the compact summary.
3. Read Standard deviation first, then use the mean, variance, sum of squares, and observation count to audit the calculation. Choose Download Excel to export the current validated model. Reset deliberately clears the demonstration data rather than restoring it; after Reset, Excel export remains disabled until a complete valid dataset is entered again.

Input guide

Data values is required and accepts one to 30 finite ordinary numbers separated by commas, spaces, semicolons, or line breaks. Decimals use a period, for example 12.5; scientific notation such as 1e3 is rejected. Negative and zero observations are valid because standard deviation describes spread regardless of sign. For a sample, at least two observations are required; a population may contain one observation, for which the standard deviation is exactly zero. A common mistake is entering a decimal comma such as 1,5 while intending 1.5: this calculator follows a dot-decimal convention and treats commas as value separators.

Dataset type is required. Sample mode uses Bessel's correction and divides the sum of squared deviations by n – 1; population mode divides by N. With the same nonconstant values, the sample standard deviation is therefore slightly larger. Example: the startup data in Sample mode produces 2.7689, while the same seven values in Population mode produce 2.5635. The mistake to avoid is choosing Sample merely because the dataset is small – the decision depends on what the observations represent, not their count.

Steps to show is a display control, not a formula input. All steps shows the row-level calculation table; Summary only hides that table while preserving the same canonical calculation and workbook contents. Use All steps when teaching, reviewing, or checking arithmetic. Use Summary only when you only need the final metrics.

Output guide

Standard deviation is the main result and is shown in the same units as the data. Zero means every observation is identical. A larger result means greater spread, but “high” or “low” is meaningful only relative to the scale and context of the data. Mean is the arithmetic center used to form every deviation. Variance is the average squared deviation using the selected denominator and therefore has squared units. Sum of squares (SS) is the total of all squared deviations before division. Observations is the number of parsed values. The summary pills repeat Dataset type, count, and mean from the same model. In the detailed table, Value (xᵢ) is the original observation, Deviation from mean is xᵢ minus the mean, and Squared deviation is that deviation squared; the squared-deviation column adds to SS.

Worked example

For the startup sample 2, 4, 5, 6, 6, 9, 10, the mean is 6. The deviations are – 4, – 2, – 1, 0, 0, 3, and 4; their squares are 16, 4, 1, 0, 0, 9, and 16, which sum to SS = 46. Sample variance is therefore 46 ÷ (7 – 1) = 7.6667. Taking the positive square root gives the displayed sample standard deviation 2.7689. The workbook exports those typed values, the seven input observations, and the same row-level deviations rather than copying rounded screen text.