Statistics Calculator

Calculate mean, median, mode, standard deviation, and variance free online from any data set. Real statistical formulas, both population and sample standard deviation.

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See all statistics instantlyMean, median, mode, std dev, variance

Compare population vs sampleBoth standard deviation types shown

The N vs N-1 Distinction That Actually Matters

Population standard deviation (dividing by N) and sample standard deviation (dividing by N-1) produce genuinely different numbers, and using the wrong one for your actual situation introduces real error. This calculator shows both, since which one applies depends on whether your data represents an entire group or a sample meant to estimate a larger population.

Why Sample Standard Deviation Needs the Correction

A sample's mean is calculated specifically to minimize that sample's own internal variance, which means using N directly would systematically understate the true variability of the broader population the sample is drawn from. Bessel's correction (dividing by N-1 instead) compensates for this statistical bias.

Mean Versus Median -- Different Sensitivity to Outliers

A single extreme value pulls the mean substantially but barely affects the median, which is exactly why median is often the more representative measure for skewed real-world data like income or home prices, where a few extreme values would otherwise distort the average.

Frequently Asked Questions

Population standard deviation divides by N (the total count), used when your data represents the entire group you care about. Sample standard deviation divides by N-1, used when your data is a sample meant to estimate a larger population's spread -- a subtle but mathematically important distinction called Bessel's correction.

Using N would systematically underestimate the true population variance when working from a sample, since a sample's own mean is calculated to minimize variance within that specific sample. Dividing by N-1 corrects for this bias, producing a more accurate estimate of the broader population's actual spread.

If multiple values tie for the highest frequency, this calculator reports all of them as the mode (a 'multimodal' result), rather than picking just one arbitrarily.

Median (the middle value) is resistant to extreme outliers, while mean (the average) is pulled by them -- a dataset with one extremely large value will show a much higher mean than median, which is exactly why median is often preferred for skewed data like income distributions.

Variance is the average squared deviation from the mean. Standard deviation is its square root, returning the measure to the original data's units -- which is why standard deviation is more commonly reported, since variance's squared units are less easy-to-use to interpret directly.