h-Index Calculation: Formula & Examples

The h-index is calculated by sorting a researcher’s publications by citation count in descending order and finding the largest number h such that h papers have at least h citations each. You enter a list of citation counts (one per paper) and the calculator returns the h-index, along with the m-index and g-index as supplementary metrics.

Formula

The h-index is defined mathematically as:

h = max{ i ∈ ℕ : at least i papers have ≥ i citations }

In practice, you sort all citation counts in descending order: c₁ ≥ c₂ ≥ … ≥ c_n. Then h is the largest rank i where c_i ≥ i. If no paper meets this condition, h = 0.

Two related metrics are often computed alongside:

• g-index (Egghe, 2006): the largest g such that the top g papers together have at least g² total citations. It rewards sustained high productivity.
• m-index (Hirsch, 2005): h divided by the number of years since the researcher’s first publication. It normalizes for career length.

Worked Example

Using the default input 25, 18, 12, 8, 5, 3, 2, 1 (8 papers):

1. Sort descending: 25, 18, 12, 8, 5, 3, 2, 1.
2. Compare each rank i with the citation count c_i:

• i=1: 25 ≥ 1 ✓
• i=2: 18 ≥ 2 ✓
• i=3: 12 ≥ 3 ✓
• i=4: 8 ≥ 4 ✓
• i=5: 5 ≥ 5 ✓
• i=6: 3 ≥ 6 ✗

The largest i that works is 5, so h = 5.

For the g-index: sum the top g citation counts. The top 5 sum to 25+18+12+8+5 = 68. Check g=8: total sum is 74, and 8²=64, so 74 ≥ 64 works; g=9 would require 9²=81, but only 8 papers exist, so g=8. The m-index (with career length 10) is 5/10 = 0.5, displayed as 0.50.

How to Choose the Right Data

The h-index is only as good as the citation data you feed it. You need a complete list of an author’s publications and their citation counts from a single source (e.g., Google Scholar, Scopus, or Web of Science). Mixing sources double-counts or misses citations. For a fair comparison, use the same source for all authors.

Include all document types that the source counts (articles, reviews, conference papers) but be consistent. Exclude self-citations if you want a stricter measure – some databases let you toggle this. For early-career researchers, the h-index is small (often 1–3) and not very informative; the m-index or field-normalized metrics are better.

Common Mistakes and Misconceptions

A frequent error is sorting in ascending order or forgetting to sort at all. The h-index is not the number of papers with more than h citations – it’s the point where the rank crosses the count. Another mistake: using total citations instead of per-paper counts. The h-index is not the average citations per paper, nor the total citations divided by something.

People also assume a higher h-index always means better research. That’s not true across fields: biomedical papers accumulate citations faster than mathematics or engineering. A mathematician with h=20 may be as distinguished as a biologist with h=60. Also, the h-index can never decrease, so it rewards longevity over peak impact. A single blockbuster paper does not raise h much after the first few years.

Limitations of the h-Index

The h-index ignores the distribution of citations. Two authors can have the same h but very different total impact: one might have 5 papers with 100 citations each, another 5 papers with 5 citations each. It also penalizes authors with many low-cited papers and rewards those who publish moderately cited papers steadily.

Field normalization is missing – citation rates vary by discipline. The h-index does not account for the number of co-authors, so a paper with 100 authors counts the same as a solo paper. It also treats all citations equally, including negative citations (papers that point out flaws). For a quick snapshot it’s useful, but for hiring or funding decisions it should be supplemented with other metrics like the i10-index (number of papers with ≥10 citations) or field-weighted citation impact.

Alternatives to the h-Index

• i10-index (Google Scholar): number of papers with at least 10 citations. Simple and robust for small samples.
• g-index: gives more weight to highly cited papers. Better for recognizing top performers.
• e-index: the excess citations beyond the h-core, useful for distinguishing authors with the same h.
• Field-normalized metrics (e.g., CNCI, FWCI): compare citation counts to the global average in the same field and year.
• Altmetrics: measure online attention (downloads, mentions) but do not replace citations.

For a single-number index, h is the most widely used, but always pair it with total citations and a field-normalized score for a fair assessment.

Practical Tips for Using the Calculator

• Enter citation counts as a comma-separated list, e.g., 10, 7, 5, 3, 1.
• Use a consistent source – do not mix Scopus and Google Scholar numbers.
• If you have more than 100 papers, the h-index will still compute quickly; the list can be long.
• For the m-index, the calculator assumes a 10-year career. Adjust the divisor manually if needed – the formula is h / years.
• Round the m-index to two decimals for reporting.

Frequently Asked Questions

What is a good h-index?

It depends on the field and career stage. For a junior researcher, h=5–10 is typical; for a tenured professor in biomedicine, h=40–60 is common, while in mathematics h=20–30 is already strong. There is no universal threshold.

Can the h-index be calculated for a specific topic or institution?

Yes, you can compute the h-index for any set of papers, not just an author. For example, the h-index of a journal or a research group is calculated the same way: sort the papers by citations and find the h point.

Does the h-index include self-citations?

It depends on the database. Google Scholar includes self-citations by default; Scopus and Web of Science let you exclude them. For a conservative estimate, use the version without self-citations.

How does the h-index differ from the i10-index?

The i10-index simply counts papers with at least 10 citations, ignoring the exact number. The h-index requires a balance between number of papers and citation count, so it is more stringent for low-cited papers.

Can the h-index decrease over time?

No, once a paper has a certain citation count, it never loses it (unless a database corrects errors). New citations only increase h, so it is a non-decreasing metric.

Why is my h-index different on Google Scholar vs. Scopus?

Each database indexes different journals and conference proceedings. Google Scholar covers more grey literature and preprints, while Scopus is more selective. Also, citation counts may lag or differ due to indexing delays.

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in c | Citation counts (comma-separated) | - | 25, 18, 12, 8, 5, 3, 2, 1 | 0
out h | h-index | - | hindex(c) | 0
out g | g-index | - | gindex(c) | 0
out m | m-index (career length 10) | - | hindex(c) / 10 | 2