Central tendency - wikipedia Central tendency For the graph / network concept, see Centrality. In statistics, a central tendency ( or measure of central tendency ) is a central or typical value for a probability distribution. It may also be called a center or location of the distribution. Colloquially, measures of central tendency are often called averages. The term central tendency dates from the late 1920s. The most common measures of central tendency are the arithmetic mean, the median and the mode. A central tendency can be calculated for either a finite set of values or for a theoretical distribution, such as the normal distribution. Occasionally authors use central tendency to denote `` the tendency of quantitative data to cluster around some central value. '' The central tendency of a distribution is typically contrasted with its dispersion or variability ; dispersion and central tendency are the often characterized properties of distributions. Analysts may judge whether data has a strong or a weak central tendency based on its dispersion. Contents 1 Measures 2 Solutions to variational problems 2.1 Uniqueness 3 Relationships between the mean, median and mode 4 See also 5 References Measures The following may be applied to one - dimensional data. Depending on the circumstances, it may be appropriate to transform the data before calculating a central tendency. Examples are squaring the values or taking logarithms. Whether a transformation is appropriate and what it should be, depend heavily on the data being analyzed. Arithmetic mean or simply, mean the sum of all measurements divided by the number of observations in the data set. Median the middle value that separates the higher half from the lower half of the data set. The median and the mode are the only measures of central tendency that can be used for ordinal data, in which values are ranked relative to each other but are not measured absolutely. Mode the most frequent value in the data set. This is the only central tendency measure that can be used with nominal data, which have purely qualitative category assignments. Geometric mean the nth root of the product of the data values, where there are n of these. This measure is valid only for data that are measured absolutely on a strictly positive scale. Harmonic mean the reciprocal of the arithmetic mean of the reciprocals of the data values. This measure too is valid only for data that are measured absolutely on a strictly positive scale. Weighted arithmetic mean an arithmetic mean that incorporates weighting to certain data elements. Truncated mean or trimmed mean the arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded. Interquartile mean a truncated mean based on data within the interquartile range. Midrange the arithmetic mean of the maximum and minimum values of a data set. Midhinge the arithmetic mean of the two quartiles. Trimean the weighted arithmetic mean of the median and two quartiles. Winsorized mean an arithmetic mean in which extreme values are replaced by values closer to the median. Any of the above may be applied to each dimension of multi-dimensional data, but the results may not be invariant to rotations of the multi-dimensional space. In addition, there are the Geometric median which minimizes the sum of distances to the data points. This is the same as the median when applied to one - dimensional data, but it is not the same as taking the median of each dimension independently. It is not invariant to different rescaling of the different dimensions. Quadratic mean ( often known as the root mean square ) useful in engineering, but not often used in statistics. This is because it is not a good indicator of the center of the distribution when the distribution includes negative values. Simplicial depth the probability that a randomly chosen simplex with vertices from the given distribution will contain the given center Tukey median a point with the property that every halfspace containing it also contains many sample points Solutions to variational problems Several measures of central tendency can be characterized as solving a variational problem, in the sense of the calculus of variations, namely minimizing variation from the center. That is, given a measure of statistical dispersion, one asks for a measure of central tendency that minimizes variation : such that variation from the center is minimal among all choices of center. In a quip, `` dispersion precedes location ''. This center may or may not be unique. In the sense of L spaces, the correspondence is : L dispersion central tendency variation ratio mode average absolute deviation median standard deviation mean maximum deviation midrange The associated functions are called p - norms : respectively 0 - `` norm '', 1 - norm, 2 - norm, and ∞ - norm. The function corresponding to the L space is not a norm, and is thus often referred to in quotes : 0 - `` norm ''. In equations, for a given ( finite ) data set X, thought of as a vector x = ( x 1,..., x n ) ( displaystyle mathbf ( x ) = ( x_ ( 1 ), ldots, x_ ( n ) ) ), the dispersion about a point c is the `` distance '' from x to the constant vector c = ( c,..., c ) ( displaystyle mathbf ( c ) = ( c, ldots, c ) ) in the p - norm ( normalized by the number of points n ) : f p ( c ) = ‖ x − c ‖ p : = ( 1 n ∑ i = 1 n x i − c p ) 1 / p. ( displaystyle f_ ( p ) ( c ) = left mathbf ( x ) - mathbf ( c ) \right _ ( p ) : = ( \bigg ( ) ( \frac ( 1 ) ( n ) ) sum _ ( i = 1 ) ^ ( n ) left x_ ( i ) - c \right ^ ( p ) ( \bigg ) ) ^ ( 1 / p ). ) Note that for p = 0 ( displaystyle p = 0 ) and p = ∞ ( displaystyle p = infty ) these functions are defined by taking limits, respectively as p → 0 ( displaystyle p \to 0 ) and p → ∞ ( displaystyle p \to infty ). For p = 0 ( displaystyle p = 0 ) the limiting values are 0 0 = 0 ( displaystyle 0 ^ ( 0 ) = 0 ) and a 0 = 1 ( displaystyle a ^ ( 0 ) = 1 ) for a ≠ 0 ( displaystyle a \neq 0 ), so the difference becomes simply equality, so the 0 - norm counts the number of unequal points. For p = ∞ ( displaystyle p = infty ) the largest number dominates, and thus the ∞ - norm is the maximum difference. Uniqueness The mean ( L center ) and midrange ( L center ) are unique ( when they exist ), while the median ( L center ) and mode ( L center ) are not in general unique. This can be understood in terms of convexity of the associated functions ( coercive functions ). The 2 - norm and ∞ - norm are strictly convex, and thus ( by convex optimization ) the minimizer is unique ( if it exists ), and exists for bounded distributions. Thus standard deviation about the mean is lower than standard deviation about any other point, and the maximum deviation about the midrange is lower than the maximum deviation about any other point. The 1 - norm is not strictly convex, whereas strict convexity is needed to ensure uniqueness of the minimizer. Correspondingly, the median ( in this sense of minimizing ) is not in general unique, and in fact any point between the two central points of a discrete distribution minimizes average absolute deviation. The 0 - `` norm '' is not convex ( hence not a norm ). Correspondingly, the mode is not unique -- for example, in a uniform distribution any point is the mode. Relationships between the mean, median and mode Main article : Nonparametric skew § Relationships between the mean, median and mode For unimodal distributions the following bounds are known and are sharp : θ − μ σ ≤ 3, ( displaystyle ( \frac ( \theta - mu ) ( sigma ) ) leq ( sqrt ( 3 ) ), ) ν − μ σ ≤ 0.6, ( displaystyle ( \frac ( \nu - mu ) ( sigma ) ) leq ( sqrt ( 0.6 ) ), ) θ − ν σ ≤ 3, ( displaystyle ( \frac ( \theta - \nu ) ( sigma ) ) leq ( sqrt ( 3 ) ), ) where μ is the mean, ν is the median, θ is the mode, and σ is the standard deviation. For every distribution, ν − μ σ ≤ 1. ( displaystyle ( \frac ( \nu - mu ) ( sigma ) ) leq 1. ) See also Expected value Location parameter References Jump up ^ Weisberg H.F ( 1992 ) Central Tendency and Variability, Sage University Paper Series on Quantitative Applications in the Social Sciences, ISBN 0 - 8039 - 4007 - 6 p. 2 ^ Jump up to : Upton, G. ; Cook, I. ( 2008 ) Oxford Dictionary of Statistics, OUP ISBN 978 - 0 - 19 - 954145 - 4 ( entry for `` central tendency '' ) Jump up ^ Dodge, Y. ( 2003 ) The Oxford Dictionary of Statistical Terms, OUP for International Statistical Institute. ISBN 0 - 19 - 920613 - 9 ( entry for `` central tendency '' ) Jump up ^ Johnson NL, Rogers CA ( 1951 ) `` The moment problem for unimodal distributions ''. Annals of Mathematical Statistics, 22 ( 3 ) 433 -- 439 Jump up ^ Hotelling H, Solomons LM ( 1932 ) The limits of a measure of skewness. Annals Math Stat 3, 141 -- 114 Jump up ^ Garver ( 1932 ) Concerning the limits of a mesuare of skewness. Ann Math Stats 3 ( 4 ) 141 -- 142 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 - 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 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=Central_tendency&oldid=826381045 '' Categories : Summary statistics Probability theory Talk Contents Interaction About Wikipedia বাংলা Català Deutsch Español Euskara فارسی Français Bahasa Indonesia Italiano עברית Bahasa Melayu Nederlands Polski Русский Simple English Basa Sunda Suomi తెలుగు Українська 中文 Edit links This page was last edited on 18 February 2018, at 20 : 09. 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Question: mean mode and median are measures of what
Answer: central tendency