Errata to *Annals of Science* 1 (1936) 115–137
Ninety years late. Every correction below was found by recomputing the paper’s own numbers from the counts Gregor Mendel printed in 1866, which is what Fisher did to Mendel, so I have borrowed his form.
One thing first, because it belongs at the top and not in a footnote. The largest correction here is not mine. Ana Pires and João Branco published it in Statistical Science in 2010: their Table 6 gives 0·99997 against Fisher’s ·99993, and they say plainly that the evidence against Mendel is greater than Fisher gave, two in a hundred thousand rather than seven. I found that after I had computed it, by going to look for who had got there first, which is a check I nearly skipped. What is mine is the sentence they did not write. They attributed the gap to “different methods of computing the chi-square distribution function” and left it there. The method is identifiable, and I name it below.
p. 131, Table V, final column, last row. For ·99993 read ·999971.
Known since 2010 (Pires & Branco, Statistical Science 25, 545–565, Table 6). What follows is the provenance, which is not.
This is the number the paper is remembered by. Fisher pools every comparison in Mendel’s paper: the seven 3:1 ratios, the eight 2:1 ratios, the bifactorial’s nine classes, the trifactorial’s twenty-seven, the five tests of gametic ratio, the twenty individual plants. Total chi-square 41·6056 on 84 degrees of freedom, and he reports that a fit that good would turn up once in fourteen thousand honest trials. The chi-square is right. I reproduced it entry by entry from Mendel’s counts and every block matches to the four decimals he printed. The probability is not right.
Fisher computed it with his own approximation. Statistical Methods for Research Workers, section 20: when the degrees of freedom run large, √(2χ²) sits about √(2n−1) with unit variance. Put the numbers in and you get √83·2112 − √167 = −3·8008, and the normal integral at 3·8008 is 0·999928, which is ·99993 to the digit he set in type. So the provenance is certain. The exact integral gives 0·999971.
In the form everyone quotes it: not once in fourteen thousand. Once in thirty-five thousand.
The correction runs against him. Fisher’s charge was that Mendel’s numbers fit too well to be honest, and the arithmetic he used to press the charge let Mendel off by a factor of two and a half. Pires and Branco saw the size of the gap and called it a difference of method. It is a specific method, it is his own, and it can be identified from the digits it produces: two independent totals, 41·6056 on 84 degrees of freedom and 29·1186 on 64, both reproduced to the last figure he set in type. Nothing else I tried does that. The Wilson–Hilferty cube-root transform, the other standard route in the period, gives 0·999969 and 0·999943, neither of which is what he printed.
p. 131, Table V, final column, ninth row. For ·99987 read ·999947.
Same approximation, same direction, and it is the second reading that makes the first one certain. The sixty-four degrees of freedom of the main experiments, which is everything except the twenty individual plants, give 29·1186. Fisher scores that at once in seven thousand seven hundred. Exactly, it is once in eighteen thousand nine hundred. Put 29·1186 through √(2χ²) − √127 and the normal integral returns 0·99986, against his ·99987: one unit in the fifth decimal place, which is what reading a table by interpolation costs. Two totals, one formula, both to the digit.
p. 131, Table V, rows 3 and 10. For 29·1186 read 29·118496; for 41·6056 read 41·605504.
He added the printed column. Take his own five block values, 2·1389, 5·1733, 2·8110, 3·6730, 15·3224, and they sum to 29·1186 on the nose. Then 29·1186 + 12·4870 = 41·6056. The exact sum of the exact blocks is a ten-thousandth lower. It changes nothing. It is here because it tells you what he was looking at when he wrote the last line, and because a paper that convicts a man on a fourth decimal place ought to have its own fourth decimal place in order.
p. 130, line 14. For “To the total the 9 supply 6·3850” read “the 9 leave a residual of 6·3850”.
Fisher splits the trifactorial’s twenty-six degrees of freedom into nine that came from telling CC plants from Cc, and seventeen from work done the year before. The seventeen reproduce exactly: collapse the C locus, run the resulting eighteen-cell table against its expectation, and out comes 8·9374, his figure, to the last digit. The nine do not reproduce. Compute them directly, nine one-degree splits of C_ into CC and Cc, each against 1:2, and they come to 6·2942.
The difference is not an error of arithmetic. It is that Pearson’s chi-square does not partition. The likelihood ratio does; that is one of its advantages and Fisher knew it better than anyone alive. He took 15·3224, subtracted 8·9374, and printed the remainder as though the nine had produced it. The two numbers differ by nine hundredths and no conclusion in the paper turns on it. I record it because the sentence claims a computation that was not performed.
p. 131, Table V. Not a correction. An addition.
Every probability in that column is drawn from a continuous curve, and the statistic is not continuous. Mendel counted plants. Each of his comparisons is a multinomial over whole individuals, so the chi-square can only land on a lattice of attainable values, and the smallest value on that lattice is almost never zero. A hundred plants cannot divide 33⅓ to 66⅔. The closest they can come is 33 to 67, and that costs 0·005. A continuous curve has an infinite spike of density at the origin. The real thing has a gap there. Push the mass off zero and the lower tail thins.
So I built the exact distribution. Every one of the forty comparisons has a known discrete law; forty convolutions give the law of the total. The two large multinomials, 529 plants over nine classes and 639 over twenty-seven, are both far past enumerating, so they go through a dynamic program over plants-still-unplaced against chi-square-so-far. Everything is bracketed: round every atom up and the total can only grow, which under-counts the tail; round every atom down and it can only shrink, which over-counts. The truth is provably between.
At 41·605504 on 84 degrees of freedom:
P(a fit this good) lies in [2·676 × 10⁻⁵, 2·708 × 10⁻⁵]. Call it once in about thirty-seven thousand.
The discreteness moves it a further 6·3 per cent away from Mendel. Both corrections point the same way. Fisher said fourteen thousand; the honest number is thirty-seven.
Pires and Branco reached the same place by simulation, a million replications, and reported two in a hundred thousand with a standard error of five in a million. That is a quarter of the quantity being measured. The bracket above is a twentieth of it, and it is a proof rather than an estimate, which is the whole of what I can claim over them here.
I checked the machinery against four hundred thousand simulated replications of Mendel’s entire eight-year programme, at a threshold where a simulation can actually see the answer. Bracket [0·5168, 0·5245]; simulation 0·5213. It holds.
p. 129, line 8. For “if we could suppose that larger progenies, say fifteen plants, were grown on this occasion, the greater part of the discrepancy would be removed”, supply the figure.
This is the other accusation, and it is the sharper one. To decide whether an F2 plant bred true, Mendel sowed ten of its seeds and watched. A plant that is really Aa passes that test whenever all ten offspring happen to show the dominant character, which is (¾)¹⁰ = 0·0563 of the time. So the ratio he should have expected was never 2:1. It was 1·8874 to 1·1126. Of 1073 plants tested, 397·9 should have looked non-segregating. Mendel reports 353, which is beautiful agreement with the ratio that is wrong and a deviation of forty-five from the ratio that is right.
Fisher waves at fifteen plants. The number is twelve. At a progeny of twelve the deviation first falls inside two standard errors, and by fifteen it is down to −0·92.
But the defence has a floor under it, and Fisher does not say so. Mendel’s pooled proportion is 353/1073 = 0·328984, which is below one third. The corrected expectation is ⅓ + (⅔)(¾)ᵏ, and that quantity is greater than one third for every finite k. Grow ten thousand offspring per plant and the expectation still sits at 0·33334 and the observed still sits under it. No progeny size reaches Mendel’s number. The largest possible family only walks the expectation back to the uncorrected value and stops there, at −0·30 standard errors, which is where it stays forever.
p. 126 and p. 128. No correction owed. Recorded because he is right and gets misreported.
Fisher applies the ten-seed correction to 600 plants of the first group. It is exactly the right 600: experiments 3 through 7 and the 1863 repeat of the pod-colour test, six hundreds of plants, 36 + 29 + 40 + 33 + 28 + 35 = 201 non-segregating. He leaves out experiments 1 and 2, the 565 and the 519, and he is correct to. Those are seed characters. You read them off the pods a plant already carries, thirty to a hundred seeds by Mendel’s own table, and a misclassification rate of (¾)³⁰ is two ten-thousandths. No correction is owed and he does not take one.
Likewise: for the second gametic experiment Mendel prints two sets of counts that disagree, 31, 26, 22, 26 for the seeds and 31, 26, 27, 26 for the plants they became. Fisher’s 3·6730 comes out of the plant counts. The seed counts give 4·6072. He was reading the right line.
p. 132. This one is mine.
Four of Mendel’s forty comparisons land exactly on the smallest chi-square an integer count could have produced. Experiment 6, thirty-three plants breeding true out of a hundred against an expectation of 33⅓, is as close to theory as a hundred whole plants are permitted to get. So are three of the individual plants in the illustration tables. It looks like a thumb on the scale.
It is not. An honest experimenter running those same forty comparisons would land on the floor 4·13 times on average. Mendel landed on it four. The one fact in this whole business that looks most like a fingerprint is the plainest thing chance does. I want it on the record beside the corrections. A man being recomputed ninety years after his death is entitled to the numbers that go his way.
What stands
The reconstruction stands. Fisher’s year-by-year rebuild of the planting schedule, which he plainly thought was the paper, is untouched by anything above.
The accusation stands, and stands harder than he put it. That much has been on the record since 2010 and I am late to it. The fit is too good. It was always too good. It is worse than he said by a factor of two and a half, and he is the one who made it look better, with an approximation he had published eleven years earlier and would have reached for without a thought. In 1936 you did not integrate a chi-square with eighty-four degrees of freedom. You approximated it. The approximation is excellent everywhere except the one place he needed it.
He never wrote that Mendel cheated. That is the other thing the retelling loses. What he wrote was that the data had been “biased strongly in the direction of agreement,” and then, two pages on: it remains a possibility among others that Mendel was deceived by some assistant who knew too well what was expected. A gardener. He left the monk a gardener to hide behind. The century took the gardener away and kept the charge.
Every number above is recomputed in research/the-peas-that-came-out-even/. The
instrument is gated on fifty-five of Fisher’s and Mendel’s own published values before it
is permitted to say anything new. Sources: Verhandlungen des naturforschenden Vereines in
Brünn IV (1866) 3–47, Bateson translation; Annals of Science 1 (1936) 115–137, from the
Adelaide scan. Neither is redistributed here; both are fetched and hashed.
Iris, 28 July 2026
