Statistical dispersion - wikipedia Statistical dispersion This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. ( December 2010 ) Example of samples from two populations with the same mean but different dispersion. The blue population is much more dispersed than the red population. In statistics, dispersion ( also called variability, scatter, or spread ) is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range. Dispersion is contrasted with location or central tendency, and together they are the most used properties of distributions. Contents 1 Measures 2 Sources 3 A partial ordering of dispersion 4 See also 5 References Measures A measure of statistical dispersion is a nonnegative real number that is zero if all the data are the same and increases as the data become more diverse. Most measures of dispersion have the same units as the quantity being measured. In other words, if the measurements are in metres or seconds, so is the measure of dispersion. Examples of dispersion measures include : Standard deviation Interquartile range ( IQR ) Range Mean absolute difference ( also known as Gini mean absolute difference ) Median absolute deviation ( MAD ) Average absolute deviation ( or simply called average deviation ) Distance standard deviation These are frequently used ( together with scale factors ) as estimators of scale parameters, in which capacity they are called estimates of scale. Robust measures of scale are those unaffected by a small number of outliers, and include the IQR and MAD. All the above measures of statistical dispersion have the useful property that they are location - invariant and linear in scale. This means that if a random variable X has a dispersion of S then a linear transformation Y = aX + b for real a and b should have dispersion S = a S, where a is the absolute value of a, that is, ignores a preceding negative sign --. Other measures of dispersion are dimensionless. In other words, they have no units even if the variable itself has units. These include : Coefficient of variation Quartile coefficient of dispersion Relative mean difference, equal to twice the Gini coefficient Entropy : While the entropy of a discrete variable is location - invariant and scale - independent, and therefore not a measure of dispersion in the above sense, the entropy of a continuous variable is location invariant and additive in scale : If Hz is the entropy of continuous variable z and y = ax + b, then Hy = Hx + log ( a ). There are other measures of dispersion : Variance ( the square of the standard deviation ) -- location - invariant but not linear in scale. Variance - to - mean ratio -- mostly used for count data when the term coefficient of dispersion is used and when this ratio is dimensionless, as count data are themselves dimensionless, not otherwise. Some measures of dispersion have specialized purposes, among them the Allan variance and the Hadamard variance. For categorical variables, it is less common to measure dispersion by a single number ; see qualitative variation. One measure that does so is the discrete entropy. Sources In the physical sciences, such variability may result from random measurement errors : instrument measurements are often not perfectly precise, i.e., reproducible, and there is additional inter-rater variability in interpreting and reporting the measured results. One may assume that the quantity being measured is stable, and that the variation between measurements is due to observational error. A system of a large number of particles is characterized by the mean values of a relatively few number of macroscopic quantities such as temperature, energy, and density. The standard deviation is an important measure in Fluctuation theory, which explains many physical phenomena, including why the sky is blue. In the biological sciences, the quantity being measured is seldom unchanging and stable, and the variation observed might additionally be intrinsic to the phenomenon : It may be due to inter-individual variability, that is, distinct members of a population differing from each other. Also, it may be due to intra-individual variability, that is, one and the same subject differing in tests taken at different times or in other differing conditions. Such types of variability are also seen in the arena of manufactured products ; even there, the meticulous scientist finds variation. In economics, finance, and other disciplines, regression analysis attempts to explain the dispersion of a dependent variable, generally measured by its variance, using one or more independent variables each of which itself has positive dispersion. The fraction of variance explained is called the coefficient of determination. A partial ordering of dispersion A mean - preserving spread ( MPS ) is a change from one probability distribution A to another probability distribution B, where B is formed by spreading out one or more portions of A 's probability density function while leaving the mean ( the expected value ) unchanged. The concept of a mean - preserving spread provides a partial ordering of probability distributions according to their dispersions : of two probability distributions, one may be ranked as having more dispersion than the other, or alternatively neither may be ranked as having more dispersion. See also Wikimedia Commons has media related to Dispersion ( statistics ). Average Summary statistics Qualitative variation Robust measures of scale Measurement uncertainty References Jump up ^ NIST / SEMATECH e-Handbook of Statistical Methods. `` 1.3. 6.4. Location and Scale Parameters ''. www.itl.nist.gov. U.S. Department of Commerce. Jump up ^ McQuarrie, Donald A. ( 1976 ). Statistical Mechanics. NY : Harper & Row. ISBN 0 - 06 - 044366 - 9. Jump up ^ Rothschild, Michael ; Stiglitz, Joseph ( 1970 ). `` Increasing risk I : A definition ''. Journal of Economic Theory. 2 ( 3 ) : 225 -- 243. doi : 10.1016 / 0022 - 0531 ( 70 ) 90038 - 4. 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 Data collection Study design Population Statistic Effect size Statistical power Sample size determination Missing data Survey methodology Sampling stratified cluster Standard error Opinion poll Questionnaire Controlled experiments Design control optimal Controlled trial Randomized Random assignment Replication Blocking Interaction Factorial experiment Uncontrolled studies Observational study Natural experiment Quasi-experiment Statistical inference Statistical theory Population Statistic Probability distribution Sampling distribution Order statistic Empirical distribution Density estimation Statistical model L 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 Wald Score Specific tests Z ( normal ) Student 's t - test Goodness of fit Chi - squared 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 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 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 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 statistics Psychometrics Spatial statistics Cartography Environmental statistics Geographic information system Geostatistics Kriging Portal Commons WikiProject Retrieved from `` https://en.wikipedia.org/w/index.php?title=Statistical_dispersion&oldid=814304932 '' Categories : Statistical deviation and dispersion Summary statistics Accuracy and precision Hidden categories : Articles needing additional references from December 2010 All articles needing additional references Talk Contents Interaction About Wikipedia Беларуская Български Català Deutsch Español Euskara فارسی Français Íslenska Italiano Latviešu Македонски Bahasa Melayu Nederlands 日本 語 Norsk Norsk nynorsk Oʻzbekcha / ўзбекча Polski Português Română Русский Slovenščina Basa Sunda Suomi Svenska Türkçe Українська 中文 Edit links This page was last edited on 8 December 2017, at 00 : 45. 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Question: measures of indicate how scores are spread out in a distribution
Answer: Standard deviation