Brainerd-Robinson Coefficient: Dating Sites by Pottery

The Brainerd-Robinson coefficient measures how similar two archaeological assemblages are by comparing the percentage of each pottery type they contain. You sum the absolute differences between matching type percentages and subtract that total from 200, giving a score from 0 (nothing in common) to 200 (identical proportions). Archaeologists then order sites so the most similar ones sit adjacent in time.

It sounds almost too simple to be useful, yet this single subtraction underpinned one of the first rigorous ways to build a calendar-free chronology. Published in 1951 by W. S. Robinson and George W. Brainerd in back-to-back papers in American Antiquity (volume 16), the coefficient turned a vague intuition (sites that look alike are probably close in age) into a number you can rank. You can try the calculation yourself on the Brainerd-Robinson similarity coefficient calculator by entering two rows of type counts.

The formula, step by step

The coefficient works on proportions, not raw counts. First, each assemblage is converted to percentages so that all its type counts sum to 100. Then, for every pottery type that the two assemblages share as a category, you take the absolute difference between the two percentages. Sum those differences and subtract from 200.

BR = 200 − Σ |P₁ₖ − P₂ₖ|

Here P₁ₖ is the percentage of type k in the first assemblage and P₂ₖ is the percentage of type k in the second. The maximum possible sum of differences is 200: when two assemblages share no types at all, each is concentrated 100% in categories the other holds at 0%, and the two sets of differences add to 200. Subtracting from 200 flips the scale into a similarity measure. Statisticians call this a city-block, or Manhattan, metric, because you add up the gaps coordinate by coordinate rather than measuring a straight-line distance.

A quick example. Suppose Site A is 60% redware and 40% greyware, and Site B is 50% redware and 50% greyware. The differences are |60−50| = 10 and |40−50| = 10, summing to 20. So BR = 200 − 20 = 180, a high score indicating the two are very alike. Now compare Site A with Site C, which is 20% redware and 80% greyware: the differences are 40 and 40, summing to 80, so BR = 120, a much weaker match. Changing one proportion noticeably moves the score, which is exactly what makes the coefficient sensitive enough to order sites.

Why proportions beat raw counts

The coefficient's most prized virtue is that it is independent of sample size. One excavation might yield 5,000 potsherds and a neighboring trench only 300, but because everything is converted to percentages first, a small assemblage and a huge one can be compared on equal footing. A site that produced ten times more pottery does not automatically look more or less similar to its neighbors; only the shape of its type distribution matters.

This is why the same measure was reinvented in ecology, where it is called the percentage similarity or proportional-similarity index. Its first formulation there traces to Renkonen in 1938, more than a decade before Robinson and Brainerd, though the archaeologists arrived at it independently. Ecologists value the identical property: comparing the species composition of two habitats without one sample's sheer size skewing the result.

That said, sample size still matters in a subtler way. A tiny assemblage gives a noisier estimate of the true type proportions, so two small samples can score lower simply because of sampling chance. The coefficient itself is scale-free, but interpreting it responsibly means keeping an eye on how many sherds each percentage is built from, which is precisely why later workers added significance tests.

From a number to a chronology

A single coefficient tells you how alike two sites are. The real power comes from computing it for every pair of assemblages and arranging the results in a square similarity matrix, where each site is compared with every other. Brainerd and Robinson's insight was that this matrix can be reordered to reveal time.

The reasoning rests on a behavioral assumption: pottery styles rise and fall in popularity along a bell curve, often drawn as a "battleship" shape. A style appears, grows common, peaks, then fades as the next fashion takes over. If that holds, assemblages close in time share styles in similar proportions and score high, while assemblages far apart in time share little and score low. So you shuffle the rows and columns until the matrix is well behaved.

The matrix-monotonicity goal

A correctly ordered matrix has a distinctive look. The diagonal (each site compared with itself) holds the maximum value of 200. Moving away from the diagonal along any row or column, the coefficients should decrease steadily, never jumping back up. When the values fall off uniformly in this way, the matrix is said to be monotonic, and it is now called a Robinson matrix in the formal literature. Because the chronological order that produces this clean falloff is essentially unique, finding it gives you the temporal sequence of the sites directly, with no radiocarbon dates required.

Before this approach, James Ford physically arranged paper strips bearing bar graphs of type frequencies, sliding them up and down by hand until each type formed a battleship curve. Many archaeologists considered that manual, eye-balled process a critical weakness. The Brainerd-Robinson coefficient replaced subjective fiddling with a reproducible number, which is why M. J. Baxter's survey of statistical landmarks in archaeology lists Robinson's 1951 paper as its very first entry.

The gotcha: high scores can deceive

The coefficient has a well-documented trap. Two assemblages that share only one or two pottery categories can still post a deceptively high score. Imagine two sites that are each overwhelmingly dominated by a single ubiquitous type, say 90% of one common ware, with their remaining sherds scattered across types they do not share. The dominant category alone forces the absolute differences to stay small, so the formula can return a value around 170 (equivalent to 0.85 on a 0-to-1 scale) even though the two assemblages have almost no diversity in common.

In other words, a high BR value does not always mean a rich, meaningful overlap; it can simply mean both sites lean on the same one or two abundant types. Researchers address this with corrections that account for the number of non-shared categories, and with significance tests. Matt Peeples' widely used R script, following a method from DeBoer, Kintigh, and Rostoker, runs a Monte Carlo simulation, drawing about 1,000 random samples from a combined global pool, to estimate the probability that a given coefficient could have arisen by sampling chance alone. The brsim R package does something similar with permutation tests.

Where it still matters

Frequency seriation using the coefficient was foundational to archaeology as a discipline, precisely because it could build chronologies from artifacts alone. Today it is more often a starting point than a final answer; modern workers pair it with multivariate techniques like correspondence analysis and nonmetric multidimensional scaling, which can recover the same ordering while handling messier, real-world data. Still, almost no consumer software computes the coefficient, which is why the practical tools remain R scripts. If you want to feel how it behaves, enter two type-count rows into the Brainerd-Robinson calculator and watch how the score reacts when you change a single proportion. For the underlying popularity curves, the Zipf's law calculator offers a related look at how a few categories dominate a distribution.

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