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How These Statistics Are Calculated

Paste any list of numbers — separated by commas, spaces, or line breaks — and this tool computes the full descriptive picture at once. The mean (arithmetic average) is the sum of all values divided by the count, n. The median is the middle value once the data is sorted: for an odd count it is the single center value, for an even count it is the average of the two middle values. The mode is the value (or values) that appear most often; a data set can have one mode, several tied modes (bimodal or multimodal), or no meaningful mode if every value appears exactly once. The range is simply the maximum value minus the minimum.

Variance and standard deviation measure spread. Population variance is the average of each value's squared distance from the mean, Σ(x - mean)² / n, and population standard deviation is its square root, restoring the original units. For example, the set 2, 4, 4, 4, 5, 5, 7, 9 has a mean of 5 and a mode of 4 (it appears three times, more than any other value); its squared deviations sum to 32, giving a population variance of 32/8 = 4 and a population standard deviation of exactly 2. Sample variance divides that same sum by n - 1 instead of n, and sample standard deviation is its square root — here that's 32/7 ≈ 4.57 and about 2.14, both larger than the population figures from identical numbers.

Why Divide by n - 1? Bessel's Correction

That gap is not a rounding quirk — it is a deliberate fix known as Bessel's correction, named after 19th-century German mathematician and astronomer Friedrich Bessel. A sample mean is, by construction, the specific value that minimizes squared distances to that sample's own data points. Because of that, measuring spread around it and dividing by n systematically underestimates the true population variance — a biased estimator. Dividing by n - 1 instead inflates the result just enough to correct that downward bias, producing an unbiased estimate from sample data. This is why the sample formula, not the population one, is the standard choice whenever your numbers are a subset standing in for a larger, not fully measured group.

The mean and median can each be the more useful summary depending on the data. The median is far more resistant to outliers: in an income distribution, a handful of extremely high earners pull the mean well above what a typical person earns, while the median — the actual middle value — stays anchored to where most of the data really sits. That resistance is exactly why income and house-price statistics are usually reported as medians. The mode behaves differently again: it can identify more than one peak (a bimodal data set might represent two distinct underlying groups mixed together), or be undefined when every value in the set is unique.