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The best possible methods are needed to evaluate the scientific excellence of individuals and research groups in order to award positions and distribute research grants with higher efficiency. It is shown here that for the symmetrical distribution of citations of an individual the currently used h-index is approximately half of the square root of the total number of citations, according to the rule of Hirsch. It is also shown that deviations from this “ideal” h-index are common and they are due to deviations in the citation distributions of different individuals. However, those deviations are not characteristic for the scientific excellence of an individual and therefore they lead only to confusion in scientific evaluation. Therefore the h-index is suggested here to be replaced by the k-index. The k-index of an individual is calculated from his/her all independent citations as self-citations cannot be considered as an indication of the excellence of any paper or its authors (the citation is independent if there is no overlap in the lists of authors of the citing and the cited paper). The k-index takes into account only partial citations for each author of multi-authored papers. In ideal case the shares of the authors in a paper are published in the same paper similarly as shares of the inventors are published in patents. If not, the share of each co-author is taken equal to the inverse of the number of authors of the given paper. The k-index of an individual is defined as the square root from the sum of his/her independent partial citations. The value of the k-index is dependent on the databank used for the citations and on the time of the measurement. If scientists of similar age working in similar fields are compared using the same databank, their personal scientific excellence will be proportional to their k-index. When the k-index is divided by the number of active scientific years, a correction can be made for different ages of different applicants. In average, the k-index has similar values, but a wider range compared to the h-index. More importantly the k-index is not biased by this or that type of citation distribution of an individual, not biased by the self-citations and not biased by the results of the co-authors. The squares of k-indexes of smaller units are additive, and so the k-index is extended to journals, publishing houses, departments, institutions, countries, continents and to the mankind.