Clarifying the Ambiguities in Bradford’s Law

Bradford’s law is one of the foundational principles of bibliometrics, yet it carries a quiet contradiction at its heart. When Samuel C. Bradford described how articles on a subject scatter across journals, he explained his finding in two ways: he stated it in words, and he drew it as a graph. For decades, students and researchers assumed these two versions said exactly the same thing. They do not. The two formulations are not mathematically equivalent, and this mismatch is what scholars call the ambiguity in Bradford’s law. Understanding this ambiguity is essential for anyone who wants to apply the law correctly to identify core journals or manage a library collection.

What Bradford actually proposed 🔗

Bradford first reported his observation in 1934 in the journal Engineering and later expanded it in his 1948 book Documentation. His central idea is simple: if you arrange journals in decreasing order of how many articles they publish on a given subject, you can divide them into groups that each contain roughly the same number of articles. The first group is a small nucleus of highly productive journals, and each successive group needs many more journals to match the same article count.

The key point that creates the later confusion is this: Bradford gave only a verbal explanation and a graphical illustration of his law, while the mathematical formulation was added later by other researchers. Bradford himself never wrote down an equation. He described a pattern in words and then drew a picture of it. This left two separate ways of representing the same phenomenon, and the gap between them is where the trouble begins.

The verbal formulation 🔗

The verbal formulation comes directly from Bradford’s written statement. It says that journals can be split into three zones, each producing an equal share of articles on the subject. The number of journals in these successive zones follows a multiplicative pattern.

If the nucleus (Zone 1) contains a certain number of journals, then Zone 2 contains that number multiplied by a constant, and Zone 3 contains it multiplied again. This gives the famous ratio of 1 : n : n², where n is called the Bradford multiplier. For example, if the core has 5 journals and the multiplier is 5, then Zone 2 would have 25 journals and Zone 3 would have 125, with each zone still containing the same total number of articles.

This verbal version is intuitive and easy to apply. You group your data into zones, count the journals, and check whether the counts grow by a roughly constant factor. Most practical library studies in India and elsewhere test the law this way because it produces clear, interpretable numbers.

The graphical formulation 🔗

The graphical formulation is built from the picture Bradford drew. It plots the cumulative number of articles on the vertical axis against the logarithm of the cumulative number of journals on the horizontal axis. When the data fits the law, this plot produces a characteristic shape with a rapid rise for the first few points, followed by a long straight line through the middle portion.

The straight central section is the heart of the graphical version. Its slope and shape are described by an equation. The most widely used mathematical expression was developed by B.C. Brookes, who refined Bradford’s idea into a clean logarithmic relationship. In this form, the cumulative output is a logarithmic function of journal rank. Leimkuhler also developed a closely related model. These equations turned Bradford’s hand-drawn curve into a precise mathematical statement.

Verbal vs. graphical form: where they diverge 🔗

Here is the core of the problem. Bradford stated his law in words and then illustrated it with a graph, without noticing that the graphical expression was not mathematically identical to the verbal formulation. The two descriptions look like the same thing but produce different predictions when you work through the mathematics.

This discrepancy was first observed by B.C. Vickery in 1948, shortly after Bradford’s book appeared. The most thorough investigation came later from E.A. Wilkinson, whose 1972 paper in the Journal of Documentation is titled, fittingly, “The Ambiguity of Bradford’s Law.” Wilkinson pointed out that discussion of the law had been founded on two formulations that are not mathematically equivalent. He developed a method to compare the two against real data.

Why the mismatch matters in practice 🔗

The disagreement is not just theoretical. When you apply the verbal zone-based version to a dataset, you may identify a certain number of core journals. When you fit the graphical equation to the same dataset, the predicted core size and the total number of journals can come out different. The two methods can hand you different answers about which journals belong in the nucleus.

This is a real concern for collection development. A librarian deciding which journals to subscribe to within a tight budget needs to know the core list precisely. If the verbal and graphical methods disagree, the librarian must understand why and choose deliberately rather than assuming both give identical results.

Wilkinson tested the two formulations against four existing datasets and found that one particular formulation was more consistent with the practical situation. This was an important step, because it showed the ambiguity could be examined empirically rather than just debated in the abstract.

The Groos droop: another wrinkle in the curve 🔗

A second complication appears when researchers plot real data. The ideal Bradford graph should rise, then run straight to the end. In practice, the line often bends downward at the far right, falling below the expected straight path. This deviation is named the Groos droop, after O.V. Groos, who described it in 1967 while studying the Keenan-Atherton data.

The droop appears in the zone of the least productive journals, the ones contributing only one or two articles each. A typical real-world plot shows an initial curve, a central linear portion, and a culminating deviation from linearity known as the Groos droop. The droop is a visible reminder that the graphical formulation is an idealisation that real data does not always obey.

Competing explanations for the droop 🔗

For a long time the droop was blamed on incomplete data: if your search misses some of the low-yield journals, the tail of the curve sags. That is one valid cause. But later analysis showed it is not the only one. Researchers demonstrated that combining several pure Bradfordian bibliographies can itself produce a droop, meaning the effect can always be expected in interdisciplinary bibliographies. In other words, the droop does not necessarily contradict Bradford’s law; it can arise naturally from it.

The shape of the curve also changes with time. Studies of the historical development of scientific fields have shown that the character of a Bradford distribution, including both the core zone and the Groos droop, depends on the stage of development of a field and on the time span examined. A young, fast-growing field produces a different curve from a mature one. This adds yet another layer to the ambiguity: the “correct” shape is a moving target.

Improving the understanding 🔗

Modern bibliometrics has not discarded Bradford’s law because of these ambiguities. Instead, professionals address them with sharper mathematics, careful method choice, and clearer reporting. Several approaches help resolve the confusion.

Treating the two formulations as a single family 🔗

Later theoretical work tied the verbal and graphical versions together. A relationship between the two forms was worked out so that they could be seen as two faces of one underlying distribution rather than two rival laws. Brookes’ logarithmic equation and Leimkuhler’s model gave the graphical curve a firm mathematical basis, and subsequent research linked these models to the zone-based verbal statement. Understanding that the formulations belong to one family lets researchers move between them knowingly.

Reporting both tests, not one 🔗

Good contemporary studies apply both the verbal and graphical tests to the same dataset and report both results. Indian doctoral-thesis studies in fields such as production engineering, for example, have tested the verbal formulation by dividing journals into zones and separately examined the graphical representation using the same data. Presenting both makes any discrepancy visible to the reader instead of hiding it behind a single number.

Fitting better models to the data 🔗

When standard formulations fail to fit, researchers turn to alternative models. Studies frequently find that the Leimkuhler model produces a very small error percentage, and that other distributions such as the log-normal can fit certain datasets better than the simple log-linear model. Recent work has extended Leimkuhler’s function specifically to account for highly productive core journals and to explain the reasons behind the Groos droop. The goal is no longer to force data into Bradford’s original shape but to model what the data actually shows.

Keeping the practical goal in focus 🔗

Perhaps the most important shift is in attitude. Even when a distribution does not perfectly conform to the expected ratios, the central practical insight still holds: a small number of core journals carries a large share of the literature. Many studies report that a significant majority of articles concentrate in a small number of core journals even when the distribution does not strictly follow Bradford’s law. For a librarian or researcher, identifying that core is the real objective, and the law remains a powerful tool for doing so. This is why Bradford’s law continues to be used to identify core journals across fields ranging from medicine to engineering.

Why this matters for students of bibliometrics 🔗

The ambiguity in Bradford’s law teaches a lesson that goes beyond the law itself. An empirical regularity can be described in more than one way, and those descriptions may not agree. A good analyst does not treat a law as a fixed truth but understands its assumptions, tests it carefully, and reports honestly when reality deviates. Bradford’s verbal statement and his graph were both attempts to capture the same scattering pattern, and the gap between them has driven decades of productive research. Far from weakening the law, the ambiguity has refined it into a more precise and useful instrument.

What do you think? If the verbal and graphical formulations of Bradford’s law give you different core-journal lists for the same dataset, which one would you trust when deciding a library’s subscription budget, and why? And do you think the Groos droop should be treated as a flaw in the data or as a genuine feature of how knowledge scatters?

References

1. https://arxiv.org/pdf/1305.0357
2. https://ebooks.inflibnet.ac.in/liscp10/chapter/bradford-distributions-an-overview/
3. https://www.academia.edu/108597387/Application_of_Bradford_s_Law_of_Scattering_to_the_Materials_Science_Literature_A_study_based_on_Web_of_Science_Database
4. https://www.researchgate.net/publication/280218558_Bradford's_Empirical_Law
5. https://www.emerald.com/jd/article-abstract/28/2/122/197553/THE-AMBIGUITY-OF-BRADFORD-S-LAW
6. https://pmc.ncbi.nlm.nih.gov/articles/PMC1250321/
7. https://link.springer.com/article/10.1007/BF02017105
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9. http://www.ijrls.in/wp-content/uploads/2020/08/Application-of-Bradford%E2%80%99s-Law-of-Scattering-in-the-Field-of-Production-Engineering-Literature-A-Bibliometric-Analysis-of-Ph.D.Theses.pdf
10. https://www.mdpi.com/2304-6775/12/4/36
11. https://files.eric.ed.gov/fulltext/EJ1115017.pdf