Log In Sign Up

The h-index and the Harmonic Mean

The Harmonic Mean between the number of papers and the citation number per paper is proposed as a simple single-value index to quantify an individual's research output. Two simple comparisons with the Hirsch h-index are performed.


page 1

page 2

page 3



I propose the P-Index that genuinely constitutes a well-defined, compact...

Mean and dispersion of harmonic measure

In this note, we provide and prove exact formulas for the mean and the t...

A two-dimensional index to quantify both scientific research impact and scope

Modern management of research is increasingly based on quantitative bibl...

A modification to Hirsch index allowing comparisons across different scientific fields

The aim of this paper is to propose a simple modification to the origina...

Machine learning assisted Bayesian model comparison: learnt harmonic mean estimator

We resurrect the infamous harmonic mean estimator for computing the marg...

Does the h_α index reinforce the Matthew effect in science? Agent-based simulations using Stata and R

Recently, Hirsch (2019a) proposed a new variant of the h index called th...

Towards a Theory of Domains for Harmonic Functions and its Symbolic Counterpart

In this paper, we begin by reviewing the calculus induced by the framewo...

The single-number criteria commonly used to evaluate scientific output of a researcher are the number of papers , a rough measure of the productivity, and the total number of citations , a rough measure of the impact. A third simple single-number index is the ratio between and , the citations per paper . Advantages and disadvantages of these simple single-numbers indices are discussed in [1], where the celebrated h-index is proposed: A scientist has index h if h of his/her papers have at least h citations each, and the other ( - h) papers have no more than h citations each. One basic motivation was to avoid unjustified reward or penalization of low productivity and low impact. The merits of the Hirsch h-index are provided by an academic example. Let us consider the lives of five researchers characterized scientifically by the following parameters

Researcher 1 1 10000 10000
Researcher 2 10 10000 1000
Researcher 3 100 10000 100
Researcher 4 1000 10000 10
Researcher 5 10000 10000 1

We remark that they have the same number of citations but very different productivity and impact, and their h-indices are given by

Name h
Researcher 1 1 10000 10000 1
Researcher 2 10 10000 1000 10
Researcher 3 100 10000 100 100
Researcher 4 1000 10000 10 10
Researcher 5 10000 10000 1 1

Strangely enough the merits of another simple index have not been explored in detail, at least to the knowledge of the author. Let us consider the half Harmonic Mean between the productivity, quantified by the number of papers , and the impact, quantified by the citations per paper


If we compare in our academic example the HM-index with the Hirsch h-index we have

Name h HM
Researcher 1 1 10000 10000 1 1
Researcher 2 10 10000 1000 10 10
Researcher 3 100 10000 100 100 50
Researcher 4 1000 10000 10 10 10
Researcher 5 10000 10000 1 1 1

and we see that the -index agrees with the -index to penalize outrageous productivity and impact, but is a little less disposed to reward the middle.

Another example is more realistic. Thirty years ago three researchers gathered at the Center of Turbulent Research in Stanford and elaborated with the Professor Parviz Moin the so called dynamic model, a subgrid procedure useful in the Large Eddy Simulation of turbulent flows [2]. They were Massimo Germano, Ugo Piomelli and William H. Cabot. Since then they dispersed in the world, the first is now at Duke University, the second at Queen’s University and the last at the Lawrence Livermore National Laboratory. The Professor Moin is always at the CTR, and during this long time everyone developed his own life of researcher. They are very interesting from the point of view of the quantification of the research. From Scopus, we derive on September , the following data

Name h
Germano 37 6235 9
Piomelli 150 11467 39
Moin 288 38042 86
Cabot 39 9128 21

We see that two of them have a large number of papers and citations, the other two a relatively large number of citations but a lower productivity. The -index penalizes them severely. The Harmonic Mean -index is more generous. We have

Name h HM
Germano 37 6235 168 9 30
Piomelli 150 11467 76 39 51
Moin 288 38042 132 86 91
Cabot 39 9128 234 21 33


  • [1] J. E. Hirsch, An index to quantify an individual s scientific research output. Proceedings of the National Academy of Sciences of the United States of America, 102(46), 16569-16572 (2005).
  • [2] M. Germano, U. Piomelli, P. Moin and W. H. Cabot, A dynamic subgrid-scale eddy viscosity model, Phys.Fluids A 3, 1760-1765 (1991).