1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

obtaining a deviation from the theory assumed to be valid, scientistshave been attempting to calculate, if possible, the probability of adeviation not less than that. If that probability was high, 1/2, say,everything was in order; otherwise, supposing that its order was1/1000, it was advisable to look for the cause of the deviation. If,finally, it was moderate, its order being 1/10, say, the case wasdoubtful and a final decision impossible. Can we object to such kind ofapplying the confidence level?I do not understand Alimov’s concept of independence either. On p.80 he thinks that secondary trials, that is, data on the assortment ofindications in different families, unconnected with each other, can bestatistically dependent. But how could that occur with the outcomes ofdifferent trials unconnected with each other? If as a result of one trialevents A and B can either happen or fail, they can be statisticallydependent and, when treating this dependence according to Mises, weshould use a single record. But in case of two absolutely different trialswe should apparently introduce something like a direct product of tworecords.Finally, concerning my treatment of Enin’s data, Alimov remarksfirst of all that his number of families is so small (11 + 14 = 25), thattheir treatment did not warrant the waste of either time or paper with aspecial non-linear scale. I will answer that by stating that, on thecontrary, I aimed at showing that the image of a distribution functionunlike that of a histogram allows to obtain sensible results even whenhaving such a small sample size.Then, Alimov states that it was possible to arrive at my conclusionsby compiling an extended sample 1 . To some extent this is correct, butto some extent wrong. After taking samples of about the same size, thefrequencies in Enin’s second sample will be closer to the theoreticalmagnitudes than Ermolaeva’s similar frequencies. This is seen inAlimov’s table (1978, p. 78). It can be therefore concluded, ifErmolaeva’s data are considered as a standard, that there is sometrouble with Enin’s materials.However, after calculating the chi-squared statistic (Tutubalin [iii]),a standard is not needed. Actually, Alimov (1978, pp. 80 – 81)believes that Enin’s data should be treated not by means of the normaldistribution of the normed frequencies, but by a more subtle model. Inprinciple, I completely agree, only that model should not be a mixtureof binomial distributions (Alimov, p. 80, formula (21)), but it shoulddirectly consider the actual numerical strength of the families. A seriesof binomial trials would be obtained having a known number of trialsand a known probability of success. Understandably, such a model isbarely convenient and therefore the stupidest Monte Carlo method 2will apparently be most effective for calculating the various pertinentprobabilities. Thus, for example, the true distribution of theKolmogorov statistic or some other statistic measuring the deviationfrom the Mendelian law can be determined. Since such statistics arerather diverse, we conclude that not only the electron or the atom butalso the certainly carelessly constructed Ermolaeva’s tables areinexhaustible 3 .137