In sabermetrics, a player's Power–speed number is the harmonic mean of their home run and stolen base totals. Both the mean and the variance may be infinite (if it includes at least one term of the form 1/0). =11a+(n+2−1)D=\frac{1}{\frac{1}{a}+(n+2-1)D}=a1+(n+2−1)D1, Therefore, 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 It is calculated by dividing the number of observations by the sum of reciprocal of the observation. ,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. It is based on all observations and is rigidly defined. 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. 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. 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. Arithmetic mean = a1+a2+a3+….+ann\frac{a_{1}+a_{2}+a_{3}+….+a_{n}}{n}na1+a2+a3+….+an, Geometric mean = a1.a2.a3….ann\sqrt[n]{a_{1}.a_{2}.a_{3}….a_{n}}na1.a2.a3….an, Harmonic mean = n1a1+1a2+1a3+…+1an\frac{n}{\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+…+\frac{1}{a_{n}}}a11+a21+a31+…+an1n. For a random sample, the harmonic mean is calculated as above. U.L. H+xH−x+H+yH−y=x+3yy−x+3x+yx−y=x+3y−3x−yy−x=2(y−x)(y−x)=2\frac{H+x}{H-x}+\frac{H+y}{H-y}=\frac{x+3y}{y-x}+\frac{3x+y}{x-y}=\frac{x+3y-3x-y}{y-x}=\frac{2(y-x)}{(y-x)}=2H−xH+x+H−yH+y=y−xx+3y+x−y3x+y=y−xx+3y−3x−y=(y−x)2(y−x)=2, Now, H+xH−x+H+yH−y=2\frac{H+x}{H-x}+\frac{H+y}{H-y}=2H−xH+x+H−yH+y=2 Harmonic Mean is used when we need to give greater weights to smaller items. In numerical experiments H3 is generally a superior estimator of the harmonic mean than H1. 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. ⇒(H+xH−x−1)=(1−H+yH−y)⇒2xH−x=−2yH−y\Rightarrow \left( \frac{H+x}{H-x}-1 \right)=\left( 1-\frac{H+y}{H-y} \right)\Rightarrow \frac{2x}{H-x}=\frac{-2y}{H-y}⇒(H−xH+x−1)=(1−H−yH+y)⇒H−x2x=H−y−2y, i.e. 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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