Summary statistics

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 of information 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]

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.

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

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.

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. A value of zero for the distance correlation implies independence.

References

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

Statistics

Descriptive statistics

Continuous data

Location	Mean arithmetic geometric harmonic Median Mode

Dispersion	Range Standard deviation Coefficient of variation Percentile Interquartile range

Shape	Variance Skewness Kurtosis Moments L-moments

Count data

Index of dispersion

Summary tables

Dependence

Statistical graphics

Data collection

Study design	Effect size Standard error Statistical power Sample size determination

Survey methodology	Sampling stratified cluster Opinion poll Questionnaire

Controlled experiments	Design control optimal Controlled trial Randomized Random assignment Replication Blocking Factorial experiment

Uncontrolled studies	Observational study Natural experiment Quasi-experiment

Statistical inference

Statistical theory

Frequentist inference

Confidence interval Testing hypotheses Power

Unbiased estimators	Mean unbiased minimum-variance Median unbiased

Biased estimators	Maximum likelihood Method of moments Minimum distance Density estimation

Parametric tests	Likelihood-ratio Wald Score

Specific tests

Z (normal) Student's t-test F Shapiro–Wilk Kolmogorov–Smirnov

Goodness of fit	Chi-squared G Sample source (Anderson–Darling) Sample normality (Shapiro–Wilk) Skewness / kurtosis normality (Jarque-Bera) Model comparison (Likelihood-ratio) Model quality (Akaike criterion)

Signed-rank	1-sample (Wilcoxon) 2-sample (Mann–Whitney U) 1-way anova (Kruskal–Wallis)

Bayesian inference

Correlation	Pearson product–moment Partial correlation Confounding variable Coefficient of determination

Regression analysis	Errors and residuals Regression model validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS)

Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression

Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity

Generalized linear model	Exponential families Logistic (Bernoulli) / Binomial / Poisson regressions

Partition of variance	Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / Multivariate / Time-series / Survival analysis

Categorical

Multivariate

Time-series

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality

Specific tests	Dickey–Fuller Johansen Q-statistic (Ljung–Box) Durbin–Watson Breusch–Godfrey

Time domain	Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model (Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR)

Frequency domain	Spectral density estimation Fourier analysis Wavelet

Survival

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time

Hazard function	Nelson–Aalen estimator

Test	Log-rank test

Applications

Biostatistics	Bioinformatics Clinical trials / studies Epidemiology Medical statistics

Engineering statistics	Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification

Social statistics	Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population Psychometrics

Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

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Summary statistics

Examples of summary statistics

Location

Spread

Shape

Dependence

See also

References