Volume XIII 
OCTOBER, 1920 
No. 1 
BIOMETRIKA 
THE FUNDAMENTAL PROBLEM OF PRACTICAL 
STATISTICS. 
By KARL PEARSON, F.R.S. 
(1) Some excuse must be given for once more returning to a pi'oblem which is 
as ancient as Baycs, has been apparently treated from most aspects by Laplace, and 
has been criticised and re-criticised by Boole, De Morgan, Venn and Edgeworth. 
The problem I refer to is that of " inverse probabilities " and in practical statistics 
it takes the following form : 
An "event" has occurred 'p times out of p + q = n trials, where we have no^y^'^^^^^^^'^^^^'^mSif^ 
a priori knowledge of the frequency of the event in the total population of J^^q n o -jq 
occurrences. What is the probability of its occurring ?• times in a furthi^r '"^U 
r + s = m trials ? "^v^rv,. ^ 
In statistical language a first sample of n shows p marked individuals, and we 
require a measure of the accordance which future samples are likely to give with 
this result. For example, a medical treatment is found to be successful in j) out of 
11 cases, we require some measure of the probable stability of this ratio. It is on 
this stability and its limits according to the size of the first sample that the whole 
practical doctrine of statistics, which is the theory of sampling, actually depends. 
We usually state the "probable errors" of results without visualising the strength 
or weakness of the logic behind them, and without generally realising that if the 
views of some authors be correct our superstructure is built, if not on a quicksand, 
at least in uncomfortable nearness to an abyss. 
As stated above, the problem had been considered in 1774 by Laplace* whose 
approximation by aid of Stirling's Theorem leads us directly to the normal curve. 
I shall later on repeat and to some extent modify on a broader basis Laplace's 
investigation. But Laplace was really only following Bayesf — and for our purposes 
* Memoires de mathematique et de physique presentes h V Academie par dwers savans, T. vi. p. 6. 
Paris, 1774. 
t Bayes' work was commuLiicated after Lis death to the Royal Society by Price : see Pliil. Trans. 
Vol. Liii. pp. 269 — 271, 370—375. Coudorcet also gave the main result in 1783, Histoire de V Academie, 
1786, p. 539, Paris, 1788 and also in the " Essai sur I'application de I'analyse a la probabilitc...," p. 188, 
Paris, 1785. Thus Condorcet wrote between the publication of Laplace's memoir of 1774 and of the TMorie 
analytique des Probabilites, Paris, 1812. Condorcet, however, supplied the combination factor omitted by 
Laplace. At the same time (Essai, p. Ixxxiii) he admitted Bayes and Price's priority, while remarking that 
Laplace had been the first to treat it analytically. He recognised the existence of the hypothesis of " the 
stability of the statistical ratios," it is not equally clear that he recognised the need for "the equal 
distribution of ignorance." I do not think it is correct to say that Laplace was the first to treat the 
problem analytically. It all turns on the evaluation of the incomplete /J-function. The methods of . 
quadrature of Bayes and Price may be somewhat primitive, but I cannot see that they are much rougher 
than those used on this occasion by Laplace. There is no special merit in reducing any integral to 
terms in exponentials, unless these give an adequate approximation to the sought value. And Laplace 
does not really measure the closeness of his approximation nor indicate where it fails. 
Biometrika xiii 1 
