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f genetic variation in the population this is a result of the fact that following a bottleneck very few individuals contribute to the gene pool limiting the genetic variation present in the population for many generations to come when considering fuel economy in automobiles two measures are commonly used miles per gallon mpg and litres per 100 km as the dimensions of these quantities are the inverse of each other one is distance per volume the other volume per distance when taking the mean value of the fuel economy of a range of cars one measure will produce the harmonic mean of the other i e converting the mean value of fuel economy expressed in litres per 100 km to miles per gallon will produce the harmonic mean of the fuel economy expressed in miles per gallon for calculating the average fuel consumption of a fleet of vehicles from the individual fuel consumptions the harmonic mean should be used if the fleet uses miles per gallon whereas the arithmetic mean should be used if the fleet uses litres per 100 km in the usa the cafe standards the federal automobile fuel consumption standards make use of the harmonic mean in chemistry and nuclear physics the average mass per particle of a mixture consisting of different species e g molecules or isotopes is given by the harmonic mean of the individual species masses weighted by their respective mass fraction beta distribution edit this section does not cite any sources please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed december 2019 learn how and when to remove this message harmonic mean for beta distribution for 0 α 5 and 0 β 5 mean harmonicmean for beta distribution versus alpha and beta from 0 to 2 harmonic means for beta distribution purple h x yellow h 1 x smaller values alpha and beta in front harmonic means for beta distribution purple h x yellow h 1 x larger values alpha and beta in front the harmonic mean of a beta distribution with shape parameters α and β is h α 1 α β 1 conditional on α 1 β 0 displaystyle h frac alpha 1 alpha beta 1 text conditional on alpha 1 beta 0 the harmonic mean with α 1 is undefined because its defining expression is not bounded in 0 1 letting α β h α 1 2 α 1 displaystyle h frac alpha 1 2 alpha 1 showing that for α β the harmonic mean ranges from 0 for α β 1 to 1 2 for α β the following are the limits with one parameter finite non zero and the other parameter approaching these limits lim α 0 h undefined lim α 1 h lim β h 0 lim β 0 h lim α h 1 displaystyle begin aligned lim _ alpha to 0 h text undefined lim _ alpha to 1 h lim _ beta to infty h 0 lim _ beta to 0 h lim _ alpha to infty h 1 end aligned with the geometric mean the harmonic mean may be useful in maximum likelihood estimation in the four parameter case a second harmonic mean h 1 x also exists for this distribution h 1 x β 1 α β 1 conditional on β 1 α 0 displaystyle h_ 1 x frac beta 1 alpha beta 1 text conditional on beta 1 alpha 0 this harmonic mean with β 1 is undefined because its defining expression is not bounded in 0 1 letting α β in the above expression h 1 x β 1 2 β 1 displaystyle h_ 1 x frac beta 1 2 beta 1 showing that for α β the harmonic mean ranges from 0 for α β 1 to 1 2 for α β the following are the limits with one parameter finite non zero and the other approaching these limits lim β 0 h 1 x undefined lim β 1 h 1 x lim α h 1 x 0 lim α 0 h 1 x lim β h 1 x 1 displaystyle begin aligned lim _ beta to 0 h_ 1 x text undefined lim _ beta to 1 h_ 1 x lim _ alpha to infty h_ 1 x 0 lim _ alpha to 0 h_ 1 x lim _ beta to infty h_ 1 x 1 end aligned although both harmonic means are asymmetric when α β the two means are equal lognormal distribution edit the harmonic mean h of the lognormal distribution of a random variable x is 22 h exp μ 1 2 σ 2 displaystyle h exp left mu frac 1 2 sigma 2 right where μ and σ 2 are the parameters of the distribution i e the mean and variance of the distribution of the natural logarithm of x the harmonic and arithmetic means of the distribution are related by μ h 1 c v 2 displaystyle frac mu h 1 c_ v 2 where c v and μ are the coefficient of variation and the mean of the distribution respectively the geometric g arithmetic and harmonic means of the distribution are related by 23 h μ g 2 displaystyle h mu g 2 pareto distribution edit the harmonic mean of type 1 pareto distribution is 24 h k 1 1 α displaystyle h k left 1 frac 1 alpha right where k is the scale parameter and α is the shape parameter statistics edit for a random sample the harmonic mean is calculated as above both the mean and the variance may be infinite if it includes at least one term of the form 1 0 sample distributions of mean and variance edit the mean of the sample m is asymptotically distributed normally with variance s 2 s 2 m e 1 x 1 m 2 n displaystyle s 2 frac m left operatorname e left frac 1 x 1 right right m 2 n the variance of the mean itself is 25 var 1 x m e 1 x 1 n m 2 displaystyle operatorname var left frac 1 x right frac m left operatorname e left frac 1 x 1 right right nm 2 where m is the arithmetic mean of the reciprocals x are the variates n is the population size and e is the expectation operator delta method edit this section does not cite any sources please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed december 2019 learn how and when to remove this message assuming that the variance is not infinite and that the central limit theorem applies to the sample then using the delta method the variance is var h 1 n s 2 m 4 displaystyle operatorname var h frac 1 n frac s 2 m 4 where h is the harmonic mean m is the arithmetic mean of the reciprocals m 1 n 1 x displaystyle m frac 1 n sum frac 1 x s 2 is the variance of the reciprocals of the data s 2 var 1 x displaystyle s 2 operatorname var left frac 1 x right and n is the number of data points in the sample jackknife method edit a jackknife method of estimating the variance is possible if the mean is known 26 this method is the usual delete 1 rather than the delete m version this method first requires the computation of the mean of the sample m m n 1 x displaystyle m frac n sum frac 1 x where x are the sample values a series of value w i is then computed where w i n 1 j i 1 x displaystyle w_ i frac n 1 sum _ j neq i frac 1 x the mean h of the w i is then taken h 1 n w i displaystyle h frac 1 n sum w_ i the variance of the mean is n 1 n m w i 2 displaystyle frac n 1 n sum m w_ i 2 significance testing and confidence intervals for the mean can then be estimated with the t test size biased sampling edit assume a random variate has a distribution f x assume also that the likelihood of a variate being chosen is proportional to its value this is known as length based or size biased sampling let μ be the mean of the population then the probability density function f x of the size biased population is f x x f x μ displaystyle f x frac xf x mu the expectation of this length biased distribution e x is 25 e x μ 1 σ 2 μ 2 displaystyle operatorname e x mu left 1 frac sigma 2 mu 2 right where σ 2 is the variance the expectation of the harmonic mean is the same as the non length biased version e x e x 1 e x 1 displaystyle e x 1 e x 1 the problem of length biased sampling arises in a number of areas including textile manufacture 27 pedigree analysis 28 and survival analysis 29 akman et al have developed a test for the detection of length based bias in samples 30 shifted variables edit if x is a positive random variable and q 0 then for all ε 0 31 var 1 x ϵ q var 1 x q displaystyle operatorname var left frac 1 x epsilon q right operatorname var left frac 1 x q right moments edit assuming that x and e x are 0 then 31 e 1 x 1 e x displaystyle operatorname e left frac 1 x right geq frac 1 operatorname e x this follows from jensen s inequality gurland has shown that 32 for a distribution that takes only positive values for any n 0 e x 1 e x n 1 e x n displaystyle operatorname e left x 1 right geq frac operatorname e left x n 1 right operatorname e left x n right under some conditions 33 e a x n e a x n displaystyle operatorname e a x n sim operatorname e left a x n right where means approximately equal to sampling properties edit assuming that the variates x are drawn from a lognormal distribution there are several possible estimators for h h 1 n 1 x h 2 exp 1 n log e x 2 1 n x h 3 exp m 1 2 s 2 displaystyle begin aligned h_ 1 frac n sum left frac 1 x right h_ 2 frac left exp left frac 1 n sum log _ e x right right 2 frac 1 n sum x h_ 3 exp left m frac 1 2 s 2 right end aligned where m 1 n log e x displaystyle m frac 1 n sum log _ e x s 2 1 n log e x m 2 displaystyle s 2 frac 1 n sum left log _ e x m right 2 of these h 3 is probably the best estimator for samples of 25 or more 34 bias and variance estimators edit a first order approximation to the bias and variance of h 1 are 35 bias h 1 h c v n var h 1 h 2 c v n displaystyle begin aligned operatorname bias left h_ 1 right frac hc_ v n operatorname var left h_ 1 right frac h 2 c_ v n end aligned where c v is the coefficient of variation similarly a first order approximation to the bias and variance of h 3 are 35 h log e 1 c v 2 n 1 1 c v 2 2 h log e 1 c v n 1 1 c v 2 4 displaystyle begin aligned frac h log _ e left 1 c_ v right 2n left 1 frac 1 c_ v 2 2 right frac h log _ e left 1 c_ v right n left 1 frac 1 c_ v 2 4 right end aligned in numerical experiments h 3 is generally a superior estimator of the harmonic mean than h 1 35 h 2 produces estimates that are largely similar to h 1 notes edit the environmental protection agency recommends the use of the harmonic mean in setting maximum toxin levels in water 36 in geophysical reservoir engineering studies the harmonic mean is widely used 37 see also edit mathematics portal rate mathematics contraharmonic mean generalized mean weighted mean weighted geometric mean hm gm am qm inequalities harmonic mean p value harmonic number parallel operator whose result is half of the harmonic mean mediant mathematics a fraction which lies between two fractions notes edit if nm a and pm b am am of a and b and radius r aq ag using pythagoras theorem qm ² aq ² am ² qm aq ² am ² qm using pythagoras theorem am ² ag ² gm ² gm am ² ag ² gm using similar triangles hm gm gm am hm gm ² am hm references edit course archived 2022 07 11 at the wayback machine srivastava u k shenoy g v sharma s c 1989 quantitative techniques for managerial decisions new age international p 63 isbn 978 81 224 0189 9 jones alan 2018 10 09 probability statistics and other frightening stuff routledge p 42 isbn 978 1 351 66138 6 1 2 3 4 weisstein eric w harmonic mean mathworld wolfram com retrieved 2023 05 31 1 2 boyd stephen vandenberghe lieven 2004 convex optimization pdf cambridge university press exercise 3 17 p 116 doi 10 1017 cbo9780511804441 isbn 978 0 521 83378 3 marshall albert w olkin 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technical support document for water...