Statistics Calculator (mean, median, mode, SD)

Paste a list of numbers and get the mean, median, mode, range, quartiles, variance, and both sample and population standard deviation.

Count (n)–

Sum–

Mean (average)–

Median–

Mode–

Range (min–max)–

Std dev (sample, n−1)–

Std dev (population, n)–

Variance (sample)–

Quartiles Q1 / Q3 (IQR)–

Descriptive statistics

Descriptive statistics summarise a set of numbers in two ways: where the middle is, and how spread out the values are. Paste any list — separated by commas, spaces, or new lines — and this calculator reports both.

Three kinds of "average"

The mean is the sum divided by the count. Themedian is the middle value once sorted, which is far more robust when a few extreme values would drag the mean around — it's why incomes are usually reported as medians. The mode is the most frequent value; a set with no repeats has no mode, and a set can have several.

Sample vs population standard deviation

Standard deviation measures typical distance from the mean, and there are two versions. Divide by n when your numbers are the entire population you care about. Divide by n − 1 when they're a sample being used to estimate a larger population — that smaller divisor corrects a bias that would otherwise make the spread look too small. Most statistics work uses the sample version, so if you're unsure, that's the one. Both are shown here.

σ = √( Σ(x − mean)² / n ) · s = √( Σ(x − mean)² / (n−1) )

Worked example

For 2, 4, 4, 4, 5, 5, 7, 9: the sum is 40 over 8 values, so the mean is 5. Sorted, the middle pair is 4 and 5, making the median 4.5, and 4 appears most often so it's the mode. The squared deviations total 32, giving a population standard deviation of √(32/8) = 2, and a sample standard deviation of √(32/7) ≈ 2.14 — the sample figure is always the larger of the two.

Sample or population makes a real difference

The most consequential choice here is which standard deviation you want. The population form divides by n; the sample form divides by n−1, because a sample systematically underestimates the spread of the population it came from. For small n the gap is large — with ten values the two differ by about 5% — and it shrinks as n grows. Use the sample form unless you truly have every member of the population. Beyond that, mean and standard deviation describe a symmetric distribution well and a skewed one poorly; for skewed data the median usually says more.