Descriptive statistics, mean, median, standard deviation
x̄ = (Σ xᵢ) / n
Paste or type a series of values: the tool computes measures of central tendency (mean, median, minimum, maximum) and dispersion (range, variance, standard deviation), explicitly distinguishing sample statistics from population statistics.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulasx̄ = (Σ xᵢ) / n
x̄ = (Σ xᵢ) / n
s² = Σ (xᵢ − x̄)² / (n − 1) (sample)
σ² = Σ (xᵢ − x̄)² / n (population)
The mean summarizes the position of the series, the variance its dispersion. The standard deviation is the square root of the variance: it is expressed in the same unit as the data, which makes it directly interpretable.
- n
- · number of values in the series.
- x̄
- · arithmetic mean.
- Median
- · the central value of the sorted series; for an even count, the mean of the two central values.
- Range
- · the difference between the largest and smallest value.
- s
- · sample standard deviation, divided by n − 1: use it when the data is a sample from a larger population.
- σ
- · population standard deviation, divided by n: use it only when the series is the entire population.
Validity domainChoosing n or n − 1 is not a minor detail: on a small sample, the gap between the two standard deviations exceeds 10%, and using the population formula on a sample systematically underestimates the true dispersion.
Choosing n or n − 1 is not a minor detail: on a small sample, the gap between the two standard deviations exceeds 10%, and using the population formula on a sample systematically underestimates the true dispersion. With a single value, the sample standard deviation doesn't exist (division by zero): it isn't zero. Accepted separators are space, line break and semicolon; a comma is read as a thousands separator when unambiguous (1,234), and rejected otherwise. Always check the reported count.
ExampleFor the series 2, 4, 4, 4, 5, 5, 7, 9: the mean is 5, the median 4.
For the series 2, 4, 4, 4, 5, 5, 7, 9: the mean is 5, the median 4.5, the population standard deviation exactly 2, and the sample standard deviation about 2.138. The gap between the two values narrows as n grows.