Summary statistic

Box plot of the Michelson–Morley experiment, showing several summary statistics.

In descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate the largest amount as simply as possible. Statisticians commonly try to describe the observations in

• a measure of location, or central tendency, such as the arithmetic mean
• a measure of statistical dispersion like the standard deviation
• a measure of the shape of the distribution like skewness or kurtosis
• if more than one variable is measured, a measure of statistical dependence such as a correlation coefficient

A common collection of order statistics used as summary statistics are the five-number summary, sometimes extended to a seven-number summary, and the associated box plot.

Entries in an analysis of variance table can also be regarded as summary statistics.^[1]

Example

The following example using R is the standard summary statistics of a randomly sampled normal distribution, with a mean of 0, standard deviation of 1, and a population of 50:

> x <- rnorm(n=50, mean=0, sd=1)
> summary(x)
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
-1.72700 -0.49650 -0.05157  0.07981  0.67640  2.46700

Examples of summary statistics

Location

Common measures of location, or central tendency, are the arithmetic mean, median, mode, and interquartile mean.

Spread

Common measures of statistical dispersion are the standard deviation, variance, range, interquartile range, absolute deviation and the distance standard deviation. Measures that assess spread in comparison to the typical size of data values include the coefficient of variation.

The Gini coefficient was originally developed to measure income inequality and is equivalent to one of the L-moments.

Shape

Common measures of the shape of a distribution are skewness or kurtosis, while alternatives can be based on L-moments. A different measure is the Distance skewness, for which a value of zero implies central symmetry.

Percentiles

A simple summary of a dataset is sometimes given by quoting particular order statistics as approximations to selected percentiles of a distribution.

Dependence

The common measure of dependence between paired random variables is the Pearson product-moment correlation coefficient, while a common alternative summary statistic is Spearman's rank correlation coefficient. Distance correlation equals zero implies independence.

See also

• Descriptive statistics
• Sufficient statistic

References

1. ^ Upton, G., Cook, I. (2006). Oxford Dictionary of Statistics, OUP. ISBN 978-0-19-954145-4