The h-index attempts to measure both the scientific productivity and impact and has been extensively used in scientometrics. It has become so popular that it has now been used to assess individuals, institutions, countries and journals. Several modifications to the h-index like g-index, h-b index etc have been introduced.
One of the main problems of the h-index is that it does not account for the distribution. Consider two cases: A scientist publishes 20 papers each cited twenty times. Another scientist publishes 40 papers, with twenty papers cited twenty times each and the rest of the 19 papers cited 19 times each. Now, both the scientists will have an identical h-index of 20 but clearly scientist two is better.
How about h-index of batsmen? Bradman, Tendulkar, Gavaskar and Dravid have played 79, 233, 211 and 189 innings, respectively. Their h-indexes are 43, 63,63, 61, respectively. Bradman has scored more than 43 runs 43 times while Tendulkar has scored more than 63 runs 63 times. But the number of innings played by Bradman is much lesser than Tendulkar because, in those days, they did not play so much cricket.
To correct these ambiguites, Gangan Prathap, has introduced a new index, called the p-index or the mock h-index, which is given by (C*C/P)^(1/3), where C is the number of citations and P is the number of publications.
In case of batsmen, C and P will refer to the number of runs scored and the number of innings played. Then the p-indexes of the batsmen listed above become 85, 82, 78,78 respectively.
Because the p-index represents a combination of size and quality, it would be ideal to compare institutions and countries on this index. Going further, the ratio of p/h would yield interesting information of the actual distributions between citations and publications. For example, if the ratio of p to h is more than 1.5 (like Bradman), it means that there is a very small tail and not many papers below h. Similarly, if the ratio of p to h is around 0.5, it means a lot of papers that have not been cited (i.e., long tail).