Summary statistics

Short description: Type of 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 mean absolute 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]: 378

## Contents

* 1 Examples
* 1.1 Location
* 1.2 Spread
* 1.3 Shape
* 1.4 Dependence
* 2 Human perception of summary statistics
* 3 See also
* 4 References
* 5 External links

## Examples

### Location

Common measures of location, or central tendency, are the arithmetic mean, median, mode, and interquartile mean.[2][3]

### Spread

Common measures of statistical dispersion are the standard deviation, variance, range, interquartile range, absolute deviation, mean absolute difference 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.

## Human perception of summary statistics

Humans efficiently use summary statistics to quickly perceive the gist of auditory and visual information.[4][5][6]

## See also

* Five-number summary
* Seven-number summary
* Common test statistics
* Descriptive statistics
* Sample statistics
* Sufficient statistic
* Data processing

## References

1. ↑ Upton, Graham; Cook, Ian (2008-10-02). "Dictionary (S)" (in en-gb). A Dictionary of Statistics (Second (revised) ed.). Oxford University Press. ISBN 978-0199541454. OCLC 935100347. https://archive.org/details/dictionaryofstat0000upto/page/378/mode/1up. "summary statistics [...] *ANOVA table might be referred to as summary statistics"
2. ↑ Bullen, P. S. (2003-08-31) (in en). Handbook of Means and Their Inequalities. Mathematics and Its Applications. 560 (2 ed.). Springer Dordrecht. doi:10.1007/978-94-017-0399-4. ISBN 978-1-4020-1522-9. OCLC 939214285.
3. ↑ Grabisch, Michel; Marichal, Jean-Luc; Mesiar, Radko; Pap, Endre (2009). Aggregation Functions. Oxford University Press. ISBN 978-0521519267.
4. ↑ Piazza, Elise A.; Sweeny, Timothy D.; Wessel, David; Silver, Michael A.; Whitney, David (2013). "Humans Use Summary Statistics to Perceive Auditory Sequences". Psychological Science 24 (8): 1389–1397. doi:10.1177/0956797612473759. PMID 23761928.
5. ↑ Alexander, R. G.; Schmidt, J.; Zelinsky, G. Z. (2014). "Are summary statistics enough? Evidence for the importance of shape in guiding visual search.". Visual Cognition 22 (3–4): 595–609. doi:10.1080/13506285.2014.890989. PMID 26180505.
6. ↑ Utochkin, Igor S. (2015). "Ensemble summary statistics as a basis for rapid visual categorization". Journal of Vision 15 (4): 8. doi:10.1167/15.4.8. PMID 26317396.

## External links

* v
* t
* e

Statistics

* Outline
* Index

Descriptive statistics

| Continuous data|

| Center|

* Mean
* arithmetic
* geometric
* harmonic
* Median
* Mode

|

Dispersion|

* Variance
* Standard deviation
* Coefficient of variation
* Percentile
* Range
* Interquartile range

Shape|

* Central limit theorem
* Moments
* Skewness
* Kurtosis
* L-moments

Count data|

* Index of dispersion

Summary tables|

* Grouped data
* Frequency distribution
* Contingency table

Dependence|

* Pearson product-moment correlation
* Rank correlation
* Spearman's rho
* Kendall's tau
* Partial correlation
* Scatter plot

Graphics|

* Bar chart
* Biplot
* Box plot
* Control chart
* Correlogram
* Fan chart
* Forest plot
* Histogram
* Pie chart
* Q–Q plot
* Run chart
* Scatter plot
* Stem-and-leaf display
* Radar chart
* Violin plot

Data collection

| Study design|

* Population
* Statistic
* Effect size
* Statistical power
* Optimal design
* Sample size determination
* Replication
* Missing data

|

Survey methodology|

* Sampling
* stratified
* cluster
* Standard error
* Opinion poll
* Questionnaire

Controlled experiments|

* Scientific control
* Randomized experiment
* Randomized controlled trial
* Random assignment
* Blocking
* Interaction
* Factorial experiment

Adaptive Designs|

* Adaptive clinical trial
* Up-and-Down Designs
* Stochastic approximation

Observational Studies|

* Cross-sectional study
* Cohort study
* Natural experiment
* Quasi-experiment

Statistical inference

| Statistical theory|

* Population
* Statistic
* Probability distribution
* Sampling distribution
* Order statistic
* Empirical distribution
* Density estimation
* Statistical model
* Model specification
* Lp space
* Parameter
* location
* scale
* shape
* Parametric family
* Likelihood (monotone)
* Location–scale family
* Exponential family
* Completeness
* Sufficiency
* Statistical functional
* Bootstrap
* U
* V
* Optimal decision
* loss function
* Efficiency
* Statistical distance
* divergence
* Asymptotics
* Robustness

|

Frequentist inference|

| Point estimation|

* Estimating equations
* Maximum likelihood
* Method of moments
* M-estimator
* Minimum distance
* Unbiased estimators
* Mean-unbiased minimum-variance
* Rao–Blackwellization
* Lehmann–Scheffé theorem
* Median unbiased
* Plug-in

|

Interval estimation|

* Confidence interval
* Pivot
* Likelihood interval
* Prediction interval
* Tolerance interval
* Resampling
* Bootstrap
* Jackknife

Testing hypotheses|

* 1- & 2-tails
* Power
* Uniformly most powerful test
* Permutation test
* Randomization test
* Multiple comparisons

Parametric tests|

* Likelihood-ratio
* Score/Lagrange multiplier
* Wald

Specific tests|

* Z-test (normal)
* Student's t-test
* F-test

Goodness of fit|

* Chi-squared
* G-test
* Kolmogorov–Smirnov
* Anderson–Darling
* Lilliefors
* Jarque–Bera
* Normality (Shapiro–Wilk)
* Likelihood-ratio test
* Model selection
* Cross validation
* AIC
* BIC

Rank statistics|

* Sign
* Sample median
* Signed rank (Wilcoxon)
* Hodges–Lehmann estimator
* Rank sum (Mann–Whitney)
* Nonparametric anova
* 1-way (Kruskal–Wallis)
* 2-way (Friedman)
* Ordered alternative (Jonckheere–Terpstra)

Bayesian inference|

* Bayesian probability
* prior
* posterior
* Credible interval
* Bayes factor
* Bayesian estimator
* Maximum posterior estimator

* Correlation
* Regression analysis

| Correlation|

* Pearson product-moment
* Partial correlation
* Confounding variable
* Coefficient of determination

|

Regression analysis|

* Errors and residuals
* Regression 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|

* Cohen's kappa
* Contingency table
* Graphical model
* Log-linear model
* McNemar's test

|

Multivariate|

* Regression
* Manova
* Principal components
* Canonical correlation
* Discriminant analysis
* Cluster analysis
* Classification
* Structural equation model
* Factor analysis
* Multivariate distributions
* Elliptical distributions
* Normal

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
* Whittle likelihood

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
* Jurimetrics
* National accounts
* Official statistics
* Population statistics
* Psychometrics

Spatial statistics|

* Cartography
* Environmental statistics
* Geographic information system
* Geostatistics
* Kriging

* Category
* '
* Commons
* WikiProject

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