The formula to find the harmonic mean is given by: For Ungrouped Data: New York: Wiley Interscience, Davidov O, Zelen M (2001) Referent sampling, family history and relative risk: the role of length‐biased sampling. ,xn are n individual values and f1, f2, f3, …..,fn are the frequencies, then, H.M = f1+f2+f3+…+fnf1x1+f2x2+f3x3+…+fnxn\frac{f_{1}+f_{2}+f_{3}+…+f_{n}}{\frac{f_{1}}{x_{1}}+\frac{f_{2}}{x_{2}}+\frac{f_{3}}{x_{3}}+…+\frac{f_{n}}{x_{n}}}x1f1+x2f2+x3f3+…+xnfnf1+f2+f3+…+fn = ∑f∑(fx)\frac{\sum f}{\sum (\frac{f}x{})}∑(xf)∑f. Again, if three terms are in HP, then the middle term is called the Harmonic Mean between the other two, so if a, b, c are in HP, then b is the HM of a and c. Let n positive numbers be a1, a2, a3, …, an and H be the HM of these numbers, then, 1. The harmonic mean takes into account the fact that events such as population bottleneck increase the rate genetic drift and reduce the amount of genetic variation in the population. showing that for α = β the harmonic mean ranges from 0 for α = β = 1, to 1/2 for α = β → ∞. [29], In geophysical reservoir engineering studies, the harmonic mean is widely used. 21 (2) 24, Sung SH (2010) On inverse moments for a class of nonnegative random variables. Significance testing and confidence intervals for the mean can then be estimated with the t test. Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions Vol 1. Hx−xy=−+Hy+xy⇒H(x+y)=2xyHx-xy=-+Hy+xy\Rightarrow H(x+y)=2xyHx−xy=−+Hy+xy⇒H(x+y)=2xy, i.e. The American Statistician. Harmonic Mean of two numbers is an average of two numbers. That is the appropriate average for the two types of pump is the harmonic mean, and with one pair of pumps (two pumps), it takes half this harmonic mean time, while with two pairs of pumps (four pumps) it would take a quarter of this harmonic mean time. Both the mean and the variance may be infinite (if it includes at least one term of the form 1/0). Find the Harmonics mean of the given numbers. H=2xy(x+y)H=\frac{2xy}{(x+y)}H=(x+y)2xy. 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. Hx=2yx+y and Hy=2xx+y\frac{H}{x}=\frac{2y}{x+y}\ and\ \frac{H}{y}=\frac{2x}{x+y}xH=x+y2y and yH=x+y2x, By componendo and dividendo, we have H+xH−x=2y+x+y2y−x−y=x+3yy−x\frac{H+x}{H-x}=\frac{2y+x+y}{2y-x-y}=\frac{x+3y}{y-x}H−xH+x=2y−x−y2y+x+y=y−xx+3y and H+yH−y=2x+x+y2x−x−y=3x+yx−y\frac{H+y}{H-y}=\frac{2x+x+y}{2x-x-y}=\frac{3x+y}{x-y}H−yH+y=2x−x−y2x+x+y=x−y3x+y Assume a random variate has a distribution f( x ). Example 2: Find the harmonic mean for integers from 15 to 24. For example, Terms t1, t2, t3 is HP if and only if 1t1,1t2,1t3,…\frac{1}{{{t}_{1}}},\frac{1}{{{t}_{2}}},\frac{1}{{{t}_{3}}},…t11,t21,t31,… is an AP. U.L. Harmonic mean = 2/(160 + 120) = 30 km/h Check: the 10 km at 60 km/h takes 10 minutes, the 10 km at 20 km/h takes 30 minutes, so the total 20 km takes 40 minutes, which is 30 km per hour The harmonic mean is also good at handling large outliers . This apparent difference in averaging is explained by the fact that hydrology uses conductivity, which is the inverse of resistivity. where Cv is the coefficient of variation. Let a and b be two given numbers and H1, H2, H3,….., Hn are n HM’s between them. HM gives less weightage to large values and more weightage to small values and thus does the balancing act properly. 2. Example 1: If H be the harmonic mean between x and y, then show that H+xH−x+H+yH−y=2\frac{H+x}{H-x}+\frac{H+y}{H-y}=2H−xH+x+H−yH+y=2, Solution: We have, H=2xyx+yH=\frac{2xy}{x+y}H=x+y2xy J Biostats 1 (2) 189-195, Chuen-Teck See, Chen J (2008) Convex functions of random variables. and n is the number of data points in the sample. A Harmonic Progression is a sequence if the reciprocals of its terms are in Arithmetic Progression, and harmonic mean (or shortly written as HM) can be calculated by dividing the number of terms by reciprocals of its terms. This is known as length based or size biased sampling. The harmonic mean is one of the three Pythagorean means. McGraw-Hill, New York, Learn how and when to remove this template message, inequality of arithmetic and geometric means, http://ajmaa.org/RGMIA/papers/v2n1/v2n1-10.pdf, "Average: How to calculate Average, Formula, Weighted average", Multivariate adaptive regression splines (MARS), Autoregressive conditional heteroskedasticity (ARCH), https://en.wikipedia.org/w/index.php?title=Harmonic_mean&oldid=986244527, Articles needing additional references from December 2019, All articles needing additional references, Creative Commons Attribution-ShareAlike License, This page was last edited on 30 October 2020, at 17:34. Water Resour Res 16(3) 481–490, Limbrunner JF, Vogel RM, Brown LC (2000) Estimation of harmonic mean of a lognormal variable. Where Arithmetic mean is denoted as A, Geometric Mean as G and Harmonic Mean as H. If x1,x2,….,xn are the n individual items, the Harmonic mean is given by, Harmonic Mean = n1x1+1x2+1x3+….1xn\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+….\frac{1}{x_n}}x11+x21+x31+….xn1n. In population genetics, the harmonic mean is used when calculating the effects of fluctuations in the census population size on the effective population size. 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, where H is the harmonic mean, m is the arithmetic mean of the reciprocals, s2 is the variance of the reciprocals of the data. Step 2: Set up the harmonic mean formula (Given above). Harmonic Mean in statistics is the reciprocal of the arithmetic mean of the values. A second harmonic mean (H1 − X) also exists for this distribution. where μ is the arithmetic mean and σ2 is the variance of the distribution. Information on what the harmonic mean is, how to calculate it for two or three numbers, formulas for harmonic mean, and example applications in physics, geometry, finance (P/E ratios), and other sciences. Assuming that the variates (x) are drawn from a lognormal distribution there are several possible estimators for H: Of these H3 is probably the best estimator for samples of 25 or more. The Harmonic mean has an application in many fields like physics, finance, geometry, trigonometry etc. 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. Then, a, H1, H2, H3, …., Hn, b will be in HP, if D be the common difference of the corresponding AP. Johnson, H Smith eds. 1H1=1a+D,1H2=1a+2D,…,1Hn=1a+nD\frac{1}{{{H}_{1}}}=\frac{1}{a}+D,\frac{1}{{{H}_{2}}}=\frac{1}{a}+2D,…,\frac{1}{{{H}_{n}}}=\frac{1}{a}+nDH11=a1+D,H21=a1+2D,…,Hn1=a1+nD This harmonic mean with β < 1 is undefined because its defining expression is not bounded in [ 0, 1 ]. It is calculated by dividing the number of observations by the sum of reciprocal of the observation. where k is the scale parameter and α is the shape parameter. A jackknife method of estimating the variance is possible if the mean is known. J Pharm Sci 74(2) 229-231, Cox DR (1969) Some sampling problems in technology. [19] This method is the usual 'delete 1' rather than the 'delete m' version. 4/6 + 4 = 4.8. Da-Feng Xia, Sen-Lin Xu, and Feng Qi, "A proof of the arithmetic mean-geometric mean-harmonic mean inequalities", RGMIA Research Report Collection, vol. Harmonic Mean of Two Numbers. n = 10, Sum of reciprocal of all the terms = (1/15)+(1/16)+(1/17)+(1/18)+(1/19)+(1/20)+(1/21)+(1/22)+(1/23)+(1/24)+(1/25) = 1/1.906, HM = (number of terms) / (Sum of reciprocal of all the terms), Relationship between Arithmetic mean, Geometric Mean and Harmonic Mean, CBSE Previous Year Question Papers Class 10, CBSE Previous Year Question Papers Class 12, NCERT Solutions Class 11 Business Studies, NCERT Solutions Class 12 Business Studies, NCERT Solutions Class 12 Accountancy Part 1, NCERT Solutions Class 12 Accountancy Part 2, NCERT Solutions For Class 6 Social Science, NCERT Solutions for Class 7 Social Science, NCERT Solutions for Class 8 Social Science, NCERT Solutions For Class 9 Social Science, NCERT Solutions For Class 9 Maths Chapter 1, NCERT Solutions For Class 9 Maths Chapter 2, NCERT Solutions For Class 9 Maths Chapter 3, NCERT Solutions For Class 9 Maths Chapter 4, NCERT Solutions For Class 9 Maths Chapter 5, NCERT Solutions For Class 9 Maths Chapter 6, NCERT Solutions For Class 9 Maths Chapter 7, NCERT Solutions For Class 9 Maths Chapter 8, NCERT Solutions For Class 9 Maths Chapter 9, NCERT Solutions For Class 9 Maths Chapter 10, NCERT Solutions For Class 9 Maths Chapter 11, NCERT Solutions For Class 9 Maths Chapter 12, NCERT Solutions For Class 9 Maths Chapter 13, NCERT Solutions For Class 9 Maths Chapter 14, NCERT Solutions For Class 9 Maths Chapter 15, NCERT Solutions for Class 9 Science Chapter 1, NCERT Solutions for Class 9 Science Chapter 2, NCERT Solutions for Class 9 Science Chapter 3, NCERT Solutions for Class 9 Science Chapter 4, NCERT Solutions for Class 9 Science Chapter 5, NCERT Solutions for Class 9 Science Chapter 6, NCERT Solutions for Class 9 Science Chapter 7, NCERT Solutions for Class 9 Science Chapter 8, NCERT Solutions for Class 9 Science Chapter 9, NCERT Solutions for Class 9 Science Chapter 10, NCERT Solutions for Class 9 Science Chapter 12, NCERT Solutions for Class 9 Science Chapter 11, NCERT Solutions for Class 9 Science Chapter 13, NCERT Solutions for Class 9 Science Chapter 14, NCERT Solutions for Class 9 Science Chapter 15, NCERT Solutions for Class 10 Social Science, NCERT Solutions for Class 10 Maths Chapter 1, NCERT Solutions for Class 10 Maths Chapter 2, NCERT Solutions for Class 10 Maths Chapter 3, NCERT Solutions for Class 10 Maths Chapter 4, NCERT Solutions for Class 10 Maths Chapter 5, NCERT Solutions for Class 10 Maths Chapter 6, NCERT Solutions for Class 10 Maths Chapter 7, NCERT Solutions for Class 10 Maths Chapter 8, NCERT Solutions for Class 10 Maths Chapter 9, NCERT Solutions for Class 10 Maths Chapter 10, NCERT Solutions for Class 10 Maths Chapter 11, NCERT Solutions for Class 10 Maths Chapter 12, NCERT Solutions for Class 10 Maths Chapter 13, NCERT Solutions for Class 10 Maths Chapter 14, NCERT Solutions for Class 10 Maths Chapter 15, NCERT Solutions for Class 10 Science Chapter 1, NCERT Solutions for Class 10 Science Chapter 2, NCERT Solutions for Class 10 Science Chapter 3, NCERT Solutions for Class 10 Science Chapter 4, NCERT Solutions for Class 10 Science Chapter 5, NCERT Solutions for Class 10 Science Chapter 6, NCERT Solutions for Class 10 Science Chapter 7, NCERT Solutions for Class 10 Science Chapter 8, NCERT Solutions for Class 10 Science Chapter 9, NCERT Solutions for Class 10 Science Chapter 10, NCERT Solutions for Class 10 Science Chapter 11, NCERT Solutions for Class 10 Science Chapter 12, NCERT Solutions for Class 10 Science Chapter 13, NCERT Solutions for Class 10 Science Chapter 14, NCERT Solutions for Class 10 Science Chapter 15, NCERT Solutions for Class 10 Science Chapter 16, JEE Main Chapter Wise Questions And Solutions.
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