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Descriptive statistics, mean, median, standard deviation

Mean, median, range, variance and standard deviation of a series of values, with the sample / population distinction.

Parameters

Separate values with spaces, line breaks or semicolons. A comma is read as a thousands separator (1,234) when unambiguous, and rejected otherwise.

Result

Awaiting the quantities required for the calculation.

Calculation, assumptions and sources

OverviewMeasures of central tendency (mean, median, minimum, maximum) and dispersion (range, variance, standard deviation) of a series of values, in sample and population versions.

Measures of central tendency (mean, median, minimum, maximum) and dispersion (range, variance, standard deviation) of a series of values, in sample and population versions.

Method & formulasx̄ = (Σ xᵢ) / n
x̄ = (Σ xᵢ) / ns² = Σ (xᵢ − x̄)² / (n − 1) (sample)σ² = Σ (xᵢ − x̄)² / n (population)

The standard deviation is the square root of the variance and is expressed in the unit of the data.

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, dividing by n − 1, for a sample from a larger population.
σ
population standard deviation, dividing by n, when the series is the entire population.
Validity domainOn a small sample, the two standard deviations differ by more than 10%, and the population formula applied to a sample underestimates the dispersion.

On a small sample, the two standard deviations differ by more than 10%, and the population formula applied to a sample underestimates the dispersion. With a single value, the sample standard deviation is undefined (division by zero).

ExampleFor 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.

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.