# Is there any (comparative) statistics on h-indices of old scientists during their lives?

In my country (I live in Russia) the government puts in place some reforms in Education, and, in particular, the possibility for a person to have a position at a university now depends on his bibliometric rates. This causes numerous debates among academics because there is a difference between this bibliometry and the real value of a scientific research.

Since in the West this way to estimate the success of an academic is not new, I believe there are many investigations in this field, and, in particular, I think there must be some statistics on how the modern bibliometry evaluates the research of old scientists during their lives. Did anybody hear someting about this?

In other words,

Does anybody know if there exist statistical researches on bibliometric rates (like h-index) of old scientists (like Karl Weierstrass) during their lives?

(I stress this "during their lives" since this is important in the debates I am talking about.)

I am a mathematician, that is why I am intrested first of all in the statistics on mathematics, but this is not necessary, I would appreciate any references. Of course, the more data such a statistic would contain, the better.

P.S. Dear moderators, please, do not change the title. My question is not about a technical possibility to compute the bibliography, I wonder whether (and where) the statistics that I am talking about exists.

• I don't think finding such a resource will give you the information that you are looking for, since it is well-known that H-index cannot be meaningfully compared against different bodies of literature. That would include the fact that H-index long ago is necessarily much less because there were many less scientists working at that time. Dec 27, 2016 at 12:57
• The value in computing these numbers might be just to show how poorly they actually measure the value of a particular researcher's output! Dec 27, 2016 at 13:26
• David, of course, my aim is not to justify those bureaucratic games. I think this statistics could help to show what you say. Dec 27, 2016 at 13:29
• to the OP: an H-index will be tremendously different in a society with a total of 50 mathematicians vs a society with 50 thousand mathematicians. Dec 27, 2016 at 15:16
• David, I also have a feeling that people don't understand what I want. Nobody says here that these rates, the H-indices in 19 century, will be the same, or even comparable with what we have now. And I don't say this. But this doesn't mean that what I am talking about is totally senseless. Because, for example, apart from absolute values there are comparative values, and it would be interesting to compare the h-indices of "great scientists", like Weierstrass, with those that other scientists had at that time. And this is the first what comes to mind about such a statistic. Dec 27, 2016 at 16:20

As mentioned in comments, this is hard to estimate offhand, as it could depend on things like nationality, field of study, age, degree of "success", and so on. I suppose that's why a question is justified.

I'm less familiar with other indices, so I'll focus on the h-index.

### h-index compared to age?

Nevertheless, some have considered this. Here's a quick post by Colin Mathers: H-index and age: a new criterion for success?. Among some commentary, Mathers suggests that having a h-index greater than one's age is a good sign.

Also in that post:

JE Hirsch, the author of the h index estimated that after 20 years a “successful scientist” will have an h-index of 20, an “outstanding scientist” an h-index of 40, and a “truly unique” individual an h-index of 60.

### Estimating h-index

The h-index does not provide a significantly more accurate measure of impact than the total number of citations for a given scholar. In particular, by modeling the distribution of citations among papers as a random integer partition and the h-index as the Durfee square of the partition, Yong arrived at the formula h ~ 0.54*sqrt(N), where N is the total number of citations, which, for mathematics members of the National Academy of Sciences, turns out to provide an accurate (with errors typically within 10–20 percent) approximation of h-index in most cases.

Source: Alexander Yong, Critique of Hirsch’s Citation Index: A Combinatorial Fermi Problem, Notices of the American Mathematical Society, vol. 61 (2014), no. 11, pp. 1040–50.

We can then estimate N as a function of time. Assuming academics begin publishing at age 25 yrs and publish an average of two papers per year and two conference proceedings (see also) -- which is clearly a very rough estimate -- P ~ 4*(A-25 yrs), where P is the number of papers published and A is the academic's age in years and must be greater than 25 yrs. One could apply some weighting function to estimate citations per year per paper and integrate, but for the sake of a crude estimate, we could assume the average paper garners roughly 10 citations/year (source). The number of citations per year for just Career Year t (where t~A-25 yrs) is then 10 * P(t) == 40*t. The cumulative number of citations N(t) is then the integral of that over the person's career up to year t: 20 * t^2. Finally, we plug this into the formula above (h ~ 0.54 * sqrt(N(t)) and get 0.54*sqrt(20)*sqrt(t^2) ==> h ~ 2.4 * t.

After Hirsch's 20 years, this gives an estimate of a h-score of 48.

Again, this is an incredibly crude estimate. It likely overestimates the h-index for older researchers because most papers receive attention in the first decade or so after publication, but the average number of citations per year per paper would likely decay off in subsequent decades after publication. It also assumes that the researcher is continually publishing.