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tions 2 history 3 properties toggle properties subsection 3 1 relationship to arithmetic mean and variance 3 2 other relationships 4 uses in statistics 5 see also 6 references 7 external links toggle the table of contents contraharmonic mean 1 language français edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia in mathematics a contraharmonic mean or antiharmonic mean 1 is a function complementary to the harmonic mean the contraharmonic mean is a special case of the lehmer mean l p displaystyle l_ p where p 2 definition edit the contraharmonic mean of a set of positive real numbers 2 is defined as the arithmetic mean of the squares of the numbers divided by the arithmetic mean of the numbers c x 1 x 2 x n 1 n x 1 2 x 2 2 x n 2 1 n x 1 x 2 x n x 1 2 x 2 2 x n 2 x 1 x 2 x n displaystyle begin aligned operatorname c left x_ 1 x_ 2 dots x_ n right 1 over n left x_ 1 2 x_ 2 2 cdots x_ n 2 right over 1 over n left x_ 1 x_ 2 cdots x_ n right 3pt x_ 1 2 x_ 2 2 cdots x_ n 2 over x_ 1 x_ 2 cdots x_ n end aligned two variable formulae edit from the formulas for the arithmetic mean and harmonic mean of two variables we have a a b a b 2 h a b 1 1 2 1 a 1 b 2 a b a b c a b 2 a a b h a b a b 2 a b a b a b 2 2 a b a b a 2 b 2 a b displaystyle begin aligned operatorname a a b a b over 2 operatorname h a b 1 over 1 over 2 cdot left 1 over a 1 over b right 2ab over a b operatorname c a b 2 cdot a a b h a b a b 2ab over a b a b 2 2ab over a b a 2 b 2 over a b end aligned notice that for two variables the average of the harmonic and contraharmonic means is exactly equal to the arithmetic mean a h a b c a b a a b as a gets closer to 0 then h a b also gets closer to 0 the harmonic mean is very sensitive to low values on the other hand the contraharmonic mean is sensitive to larger values so as a approaches 0 then c a b approaches b so their average remains a a b there are two other notable relationships between 2 variable means first the geometric mean of the arithmetic and harmonic means is equal to the geometric mean of the two values g a a b h a b g a b 2 2 a b a b a b 2 2 a b a b a b g a b displaystyle operatorname g operatorname a a b operatorname h a b operatorname g left a b over 2 2ab over a b right sqrt a b over 2 cdot 2ab over a b sqrt ab operatorname g a b the second relationship is that the geometric mean of the arithmetic and contraharmonic means is the root mean square g a a b c a b g a b 2 a 2 b 2 a b a b 2 a 2 b 2 a b a 2 b 2 2 r a b displaystyle begin aligned operatorname g left operatorname a a b operatorname c a b right operatorname g left a b over 2 a 2 b 2 over a b right sqrt a b over 2 cdot a 2 b 2 over a b sqrt a 2 b 2 over 2 2pt operatorname r a b end aligned the contraharmonic mean of two variables can be constructed geometrically using a trapezoid 3 additional constructions edit the contraharmonic mean can be constructed on a circle similar to the way the pythagorean means of two variables are constructed 4 the contraharmonic mean is the remainder of the diameter on which the harmonic mean lies 5 history edit the contraharmonic mean was discovered by the greek mathematician eudoxus in the 4th century bce 6 properties edit the contraharmonic mean satisfies characteristic properties of a mean of some list of positive values x textstyle mathbf x min x c x max x displaystyle min left mathbf x right leq operatorname c left mathbf x right leq max left mathbf x right c t x 1 t x 2 t x n t c x 1 x 2 x n for t 0 displaystyle operatorname c left t cdot mathbf x _ 1 t cdot mathbf x _ 2 dots t cdot mathbf x _ n right t cdot c left mathbf x _ 1 mathbf x _ 2 dots mathbf x _ n right text for t 0 the first property implies the fixed point property that for all k 0 c k k k k it is not monotonic increasing a value of x displaystyle mathbf x can decrease the value of the contraharmonic mean for instance c 1 4 c 2 4 the contraharmonic mean is higher in value than the arithmetic mean and also higher than the root mean square min x h x g x l x a x r x c x max x displaystyle min mathbf x leq operatorname h mathbf x leq operatorname g mathbf x leq operatorname l mathbf x leq operatorname a mathbf x leq operatorname r mathbf x leq operatorname c mathbf x leq max mathbf x where x is a list of values h is the harmonic mean g is geometric mean l is the logarithmic mean a is the arithmetic mean r is the root mean square and c is the contraharmonic mean unless all values of x are the same the signs above can be replaced by the name contraharmonic may be due to the fact that when taking the mean of only two variables the contraharmonic mean is as high above the arithmetic mean as the arithmetic mean is above the harmonic mean i e the arithmetic mean of the two variables is equal to the arithmetic mean of their harmonic and contraharmonic means relationship to arithmetic mean and variance edit the contraharmonic mean of a random variable is equal to the sum of the arithmetic mean and the variance divided by the arithmetic mean 7 c x a x var x a x displaystyle operatorname c mathbf x operatorname a mathbf x frac operatorname var mathbf x operatorname a mathbf x the ratio of the variance and the arithmetic mean was proposed as a test statistic by clapham 8 since the variance is always 0 the contraharmonic mean is always greater than or equal to the arithmetic mean other relationships edit any integer contraharmonic mean of two different positive integers is the hypotenuse of a pythagorean triple while any hypotenuse of a pythagorean triple is a contraharmonic mean of two different positive integers 1 it is also related to katz s statistic 9 j n n 2 s 2 m m displaystyle j_ n sqrt frac n 2 frac s 2 m m where m is the mean s 2 the variance and n is the sample size j n is asymptotically normally distributed with a mean of zero and variance of 1 uses in statistics edit the problem of a size biased sample was discussed by cox in 1969 on a problem of sampling fibres the expectation of size biased sample is equal to its contraharmonic mean 10 and the contraharmonic mean is also used to estimate bias fields in multiplicative models rather than the arithmetic mean as used in additive models 11 the contraharmonic mean can be used to average the intensity value of neighbouring pixels in graphing so as to reduce noise in images and make them clearer to the eye 12 the probability of a fibre being sampled is proportional to its length because of this the usual sample mean arithmetic mean is a biased estimator of the true mean to see this consider g x x f x m displaystyle g x frac xf x m where f x is the true population distribution g x is the length weighted distribution and m is the sample mean taking the usual expectation of the mean here gives the contraharmonic mean rather than the usual arithmetic mean of the sample 13 this problem can be overcome by taking instead the expectation of the harmonic mean 1 x the expectation and variance of 1 x are e 1 x 1 m displaystyle operatorname e left frac 1 x right frac 1 m and has variance var 1 x m e 1 x 1 n m 2 displaystyle operatorname var left frac 1 x right frac me left frac 1 x 1 right nm 2 where e is the expectation operator asymptotically e 1 x is distributed normally the asymptotic efficiency of length biased sampling depends compared to random sampling on the underlying distribution if f x is log normal the efficiency is 1 while if the population is gamma distributed with index b the efficiency is b b 1 this distribution has been used in modelling consumer behaviour 14 as well as quality sampling it has been used longside the exponential distribution in transport planning in the form of its inverse 15 see also edit harmonic mean lehmer mean pythagorean means references edit 1 2 pahikkala jussi 2010 on contraharmonic mean and pythagorean triples elemente der mathematik 65 2 62 67 doi 10 4171 em 141 see means of complex numbers pdf texas college mathematics journal 1 1 january 1 2005 archived from the original pdf on september 9 2006 umberger shannon construction of the contraharmonic mean in a trapezoid university of georgia nelsen roger b 7 august 1997 proofs without words exercises in visual thinking mathematical association of america p 56 isbn 0 88385 700 6 slaev valery a chunovkina anna g mironovsky leonid a 2019 metrology and theory of measurement de gruyter p 217 isbn 9783110652505 antoine c 1998 les moyennes paris presses universitaires de france kingley michael c s 1989 the distribution of hauled out ringed seals an interpretation of taylor s law oecologia 79 79 106 110 bibcode 1989oecol 79 106k doi 10 1007 bf00378246 pmid 28312819 clapham arthur roy 1936 overdispersion in grassland communities and the use of statistical methods in plant ecology the journal of ecology 24 14 232 251 bibcode 1936jecol 24 232c doi 10 2307 2256277 jstor 2256277 katz l 1965 united treatment of a broad class of discrete probability distributions proceedings of the international symposium on discrete distributions montreal zelen marvin 1972 length biased sampling and biomedical problems biometric society meeting dallas texas banerjee abhirup maji pradipta 2013 rough sets for bias field correction in mr images using contraharmonic mean and quantitative index ieee transactions on medical imaging mitra sabry october 2021 contraharmonic mean filter kajian ilmiah informatika dan komputer 2 2 75 79 sudman seymour 1980 quota sampling techniques and weighting procedures to correct for frequency bias keillor bruce d d amico michael horton veronica 2001 global consumer tendencies psychology and marketing 18 1 1 19 doi 10 1002 1520 6793 200101 18 1 1 aid mar1 3 0 co 2 u amreen mohammed venkateswarlu bandi 2024 a new way for solving transportation issues based on the exponential distribution and the contraharmonic mean journal of applied mathematics and informatics 42 3 647 661 external links edit contraharmonic proportion at planetmath v t e statistics outline index descriptive statistics continuous data center mean arithmetic arithmetic geometric contraharmonic cubic generalized power geometric harmonic heronian heinz lehmer median mode dispersion average absolute deviation coefficient of variation interquartile range percentile range standard deviation variance shape central limit theorem moments kurtosis l moments skewness count data index of dispersion summary tables contingency table frequency distribution grouped data dependence partial correlation pearson product moment correlation rank correlation kendall s τ spearman s ρ scatter plot graphics bar chart biplot box plot control chart correlogram fan chart forest plot histogram pie chart q q plot radar chart run chart scatter plot stem and leaf display violin plot heatmap scatter plot matrix ecdf plot line chart statistical data processing transformations data transformation log transformation power transform box cox transformation yeo johnson transformation variance stabilizing transformation anscombe transform fisher transformation scaling and normalization feature scaling normalization standardization z score min max normalization unit vector normalization data cleaning data cleaning outlier winsorizing truncation missing data data reduction dimensionality reduction principal component analysis factor analysis time series preprocessing differencing detrending seasonal adjustment stationarity transformation data collection study design effect size missing data optimal design population replication sample size determination statistic statistical power survey methodology sampling cluster stratified opinion poll questionnaire standard error controlled experiments blocking factorial experiment interaction random assignment randomized controlled trial randomized experiment scientific control adaptive designs adaptive clinical trial stochastic approximation up and down designs observational studies cohort study cross sectional 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 l p 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 sensitivity analysis 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 g test score lagrange multiplier wald z test normal specific tests parametric student s t test f test goodness of fit chi squared kolmogorov smirnov anderson darling lilliefors jarque bera normality shapiro wilk 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 van der waerden test 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 template least squares and regression analysis linear regression simple linear regression ordinary least squares general linear model bayesian regression non standard predictors nonlinear regression nonparametric semiparametric isotonic robust homoscedasticity and heteroscedasticity generalized linear model exponential families logistic bernoulli binomial poisson regressions partition of variance analysis of variance anova analysis of variance ancova manova degrees of freedom categorical multivariate time series survival analysis categorical cohen s kappa contingency table graphical model log linear model mcnemar s test cochran mantel haenszel statistics multivariate regression manova principal components canonical 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