Eight sliders, two groups. One option can lead in both groups and still trail in the total.
In the autumn of 1973 the graduate school at Berkeley counted who got in. Of 8,442 men who applied, about 44% were admitted. Of 4,321 women, about 35% were. Nine points across 12,763 applications is not a rounding error.
Three Berkeley researchers counted the same applications again, one department at a time. Their table ran in Science in 1975. Inside a department the gap could shrink or lean the other way.
Both counts came out of one file, and neither is a mistake. The sliders below build two groups from nothing. Push them until one side leads in group one and in group two. Then watch it lose the total.
Eight sliders run this page. Each option gets two sliders in each group. One sets how many trials it ran there. The other sets the share of those trials that succeeded. Nothing else goes in.
Press Berkeley 1973 (departments A and F). In Department A the model puts men at 62.1% admitted and women at 82.4%. In Department F it puts men at 5.9% and women at 7.0%. Women lead on both lines. Added together, the model puts men at 44.6% and women at 25.2%. The banner turns to Reversed. No rate slider moved.
The strip under the bars is the mixture. Its heading begins Where each option ran its trials, and that is the whole of it. The model puts 68.9% of the men's applications in Department A. On the women's side it puts 24.1%. Department A admitted most people who applied there. Department F admitted almost nobody. So the men's total leans on the generous line. The women's leans on the harsh one.
The second preset has the same shape. The model puts open surgery ahead on small stones, 93.1% against 86.7%. It is ahead on large stones too, 73.0% against 68.8%. Pooled, it puts the other procedure ahead, 82.6% against 78.0%. The two mixtures there are 24.9% and 77.1%.
In this model the quantity that moves the answer is where each option ran its trials, not how well it did in either group. The bars above draw all of it. Here is how to read them.
The top two bar pairs are the groups. Each pair shows one option against the other. Above the rule, those two pairs are the only like-for-like comparison the model makes. Same group, same conditions, two options.
The pair below the rule is titled Both groups added together. It is not a third measurement. It is the four figures above, re-added after the group labels are thrown away.
Each bar carries a count and a percentage, unless that cell has no trials at all. The count is there because a percentage on its own hides the mixture. The mixture is what moves. Every bar runs from zero to 100%, so the axis never flatters a gap. When two bars sit too close to tell apart, the note under the strips says so. It sends you to the figures instead.
The strip below the bars is the mixture. It shows one number per option. That is the share of its trials that sat in the first group. When the two first-group segments are the same length, a lead in both groups is a lead in the total. As those segments separate, the pooled pair drifts. Past some point it can cross — if the two groups differ in difficulty.
The banner marks that crossing. It reads No reversal while the pooled winner matches the group winners. It reads On the edge at the crossing itself, and Reversed on the far side. It reads No group winner when the two groups pick different sides, or when a group has no trials for one of the options. Where that crossing sits is what the next section works out.
Two options, A and B, are each tried in two groups. Eight numbers describe the whole thing. Each option gets a size n and a success rate p in each group. An option's pooled rate is its total successes over its total trials.
So a pooled rate is a weighted average of two group rates. The weight says where the option was tried, not how it did. Subtract the two pooled rates. Then add and remove wA · pB1 + (1 − wA) · pB2. That gives an exact identity.
The first bracket is a weighted average of the two group gaps. If A leads in both groups, both d's are positive, so the bracket is positive. The second term carries no d at all. It is the difference in mixing times the difference between the groups. Two things follow. If wA = wB, the second term is zero. Leading in both groups then forces leading in the total. The same holds if pB1 = pB2, the two groups being equally easy. In this model a reversal needs unequal mixing and unequal groups. The second term is not bounded by the first.
A small case makes that easier to trust. Here is one, counted by hand.
Load the Berkeley preset and follow the men's column.
Department A admitted 512 of 825, which is 0.6206. Department F admitted 22 of 373, which is 0.0590. Where were the applications? 825 of the 1,198 sat in Department A, so w = 0.6886. The weighted average is 0.6886 × 0.6206 + 0.3114 × 0.0590. Worked from the unrounded figures those two terms are 0.42738 and 0.01836, and they add to 0.44574.
Now count the other way. 512 + 22 = 534 admissions out of 1,198 applications. That is 0.4457 again. Same number, two routes.
The women's column runs the same way with a different weight. Department A admitted 89 of 108 (0.8241) and Department F 24 of 341 (0.0704). Only 108 of the 449 applications sat in Department A, so w = 0.2405. That gives 0.1982 + 0.0535 = 0.2517.
The second column has the higher rate on both lines and the lower total. The whole of the gap is the weight, 0.6886 against 0.2405. It applies to two rates that are ten times apart. Nothing in the second column was cut. Most of the first column sat on the line where most people got in. Most of the second sat on the line where almost nobody did. That is a fact about the mixture. It arrives inside a bar labelled as a rate. All of it rests on four rules.
No simulation. Every bar is a rate the sliders set, or a weighted average of two of them. The model's core arithmetic is a division and a weighted sum. The banner compares three signs.
Rounding. The rate sliders carry rates. A count on screen is n·p rounded, and a pooled count is the two group counts added after rounding. So a group count can sit half a case from its rate, and a pooled count up to a whole one. The bars and the banner use the unrounded rates. The presets go in as counts, so the rate sliders carry no step — a step would snap 512/825 to a neighbour the model does not hold.
No uncertainty. The model treats every rate as exact. There are no confidence intervals and no tests. Sample size never enters as uncertainty. A gap of 0.1 point weighs the same in four trials as in four thousand.
One split, two groups. The model divides the data one way. Real tables divide more ways. A table with no reversal on one split can show one on another.
Where the presets came from. The Berkeley preset takes departments A and F from a six-department table. That table is Bickel, Hammel and O'Connell (1975). The aggregate in the lead covers every department. The two sets of figures are not interchangeable. The stone figures are Charig and colleagues' 1986 series, as tabulated by Julious and Mullee (1994). The split is at a stone width of 2 cm. They are quoted as a published table in a statistics argument, not as a comparison of treatments.
Not one of the four rules lets the model say why.
The model cannot say why. It makes a sign flip out of eight numbers. It says nothing about why the mixtures differ. It cannot tell a group that explains the gap from a group the gap created. Nothing here says which bar is the number to use. That question is settled outside this model, by knowing what made the groups.
No statistics. No confidence interval and no test. The model reports a sign, and a sign carries no strength.
Two groups, one split, differences only. The model does not handle three or more strata. It does not handle a sliding scale in place of a split. Nor does it handle a grouping the option itself changed. It compares differences of proportions. Ratios and odds ratios behave their own way under pooling. None of that is on this page.
The 1973 figures are usually retold as a court case. Berkeley was sued for sex bias, the story goes. A statistician then won the case with Simpson's paradox. No suit was ever filed, as far as anyone has been able to trace: the lawsuit appears only in the retelling. An associate dean of the graduate school thought the university might be sued. Officials who feared bias asked Bickel to analyse the data.
So the two numbers in the lead never met in a courtroom. They met in a table. The sliders above build the same object in miniature: two options, two groups, two mixtures.
The finding in the paper was narrower than the retelling. It put most of the aggregate gap on where the applications went, department by department. What sent them there is not in the table. It is not in this model either.
Every model figure quoted above is worked out when the page is built, from the same model file the sliders run on, so the text and the widget cannot disagree about a value. The 1973 totals in the lead are the exception: they are quoted at second hand, from the secondary source listed below. A separate test compares the weighted-average route against the whole-count route and checks the identity on a sweep of settings. How I check things.
Sources.
UCBAdmissions table shipped with R, whose stated reference is the 1975 paper. The paper's own table was not consulted, so the two were not compared line by line.Chance
Move the door count and the number of doors the host opens. The model puts switching ahead of staying by the doors you did not pick, divided by the doors you could still swap to — two to one at three doors. Opening more doors raises that edge; adding doors without opening more lowers it.
0.667 the model's switch win rate at three doors
Chance
A single queue against one queue per counter, on average wait and on a bad visit (the 95th percentile).
5.4× the model's cut in mean wait from one line, at four counters 80% busy
Scale
Doubling thickness, on a linear axis and a log axis. The same numbers, two different stories.
42 folds until the model's stack passes the Moon, at 0.1 mm paper