Standard Deviation & Variance Calculator

Paste your values, then choose the result that matches your purpose. The calculator reports population and sample standard deviation, both variances, the mean, sum, range, minimum, maximum, and sum of squared deviations.

Open the calculation details to see each value, its deviation from the mean, and its squared deviation. Those squared values add to SS, the quantity divided by n or n − 1 before taking the square root.

Showing the work matters because two calculators can display different answers simply because one used the population denominator and the other used the sample denominator. The arithmetic should make that difference visible rather than leaving you to guess.

The choice is not determined by how many values you entered, and it is not determined merely by whether more observations could exist. It depends on what you intend the result to describe.

Use the population calculation when the entered values are the complete group you care about. All scores in one completed class can be treated as a population when the goal is only to describe that class, even though other classes exist.

Use the sample calculation when the observations are being used to estimate variation in a broader process or population. Ten measurements from an ongoing manufacturing process are normally a sample when the conclusion is meant to extend beyond those ten measurements.

The sample mean is calculated from the same observations whose spread is being measured. Once the sample mean is fixed, only n − 1 deviations can vary independently because all deviations must add to zero.

Dividing the squared-deviation total by n − 1 is Bessel’s correction. Under the standard assumptions of independent observations from the same population with finite variance, it makes sample variance an unbiased estimator of population variance.

That statement applies to variance. Taking the square root is nonlinear, so the conventional sample standard deviation remains slightly biased, especially for small samples. The n − 1 result is still the standard practical estimator, but it should not be described as perfectly unbiased standard deviation.

If the data are measured in seconds, the standard deviation is also measured in seconds. Variance is measured in seconds squared because each deviation is squared before averaging.

This is why standard deviation is generally easier to interpret beside the mean. A mean of 20 seconds with a standard deviation of 2 seconds remains on one understandable scale. A variance of 4 seconds squared contains the same spread information but is less direct.

Standard deviation is scale-dependent. Converting metres to centimetres multiplies the standard deviation by 100 and the variance by 10,000. Values calculated in different units should not be compared without first putting them on a common scale.

Squaring gives distant values disproportionate influence. A value two units from the mean contributes four squared units, while a value ten units away contributes 100.

That sensitivity is useful when large departures matter, but a single extreme observation can dominate the result. Standard deviation does not decide whether the value is an error, a legitimate rare event, or evidence that the distribution has a long tail.

When the data are strongly skewed or contain important extreme values, inspect the distribution and consider a robust measure such as the interquartile range or median absolute deviation. The appropriate measure depends on which part of the spread matters for the decision.

The calculator shows mean ± one selected standard deviation as a descriptive interval. It is not a confidence interval and does not guarantee that a particular percentage of the entered values falls inside it.

The familiar 68–95–99.7 rule describes an approximately normal distribution: about 68% within one standard deviation, 95% within two, and 99.7% within three. It should not be applied automatically to skewed, multi-peaked, bounded, seasonal, or heavy-tailed data.

A dataset can have a perfectly valid standard deviation without being normal. Standard deviation measures spread around the mean; it does not test the shape of the distribution.

A common shortcut calculates variance from the average of the squared values minus the square of the average. Although mathematically equivalent in exact arithmetic, it can lose substantial precision when the values share a large offset and their actual spread is small.

This calculator updates the running mean and corrected sum of squares as each value is read. That approach is commonly associated with Welford’s method and is more reliable for datasets such as 1,000,000,001; 1,000,000,002; and 1,000,000,003.

The final formulas shown on the page remain the ordinary statistical definitions. The running update changes how the computer reaches the result, not which population or sample statistic is being calculated.

Enter finite numerical values separated by commas, spaces, semicolons, or line breaks. Decimals and negative values are accepted. The numbers are treated as one unweighted dataset in the order entered.

A single value has population variance and population standard deviation equal to zero because there is no spread within that one-value group. Sample variance and sample standard deviation are unavailable because n − 1 equals zero.

The calculator does not know the units, sampling design, data source, measurement quality, or intended population. It calculates the statistics accurately from the entered numbers, but the meaning and appropriateness of those statistics still depend on the data context.