636 TRA'NSACTIONS OF SECTION F. 



difficulty of appreciating the nature of mathematical probability is overcome, a 

 difficulty which rather grows than diminishes as one works at it, there still 

 remains the greater task of deciding in what cases it can properly be applied and 

 on the method of calculation. It has, in my opinion, often been used where 

 it is not appropriate, where the chances of deviation are not those indicated 

 by a normal curve, where it is a mere numerical value without involving the 

 superstructure that makes the measurement of precision real. Thus it has some- 

 times been argued that if pn cases of a particular kind are found in n instances, 

 then (without further analysis of the relation of the cases to the whole group) the 



' statistical co-efficient ' for the class is j?± \/ ^^ ^^', a deduction not 



based on sound theory ; if, in fact (here I follow Lexis), the deviation found from this 

 formula is compared with that actually found from several observed values of ^, the 

 two do not in general coincide. In general, two lines of analysis are possible : we 

 may find an empirical formula (with Professor Karl Pearson) which fits this class of 

 observations, and by evaluating the constants determine an appropriate curve of fre- 

 quency, and hence allot the chances of possible differences between our observation 

 and the unknown true value ; or we may accept Professor Edgeworth's analysis of 

 the causes which would produce his generalised law of great numbers, and determine 

 (i priori or by experiment whether this universal law may be expected or is to be 

 found in the case in question. It is to the latter method that my next remarks 

 apply and on which the example I give depends. It can be shown' that if quan- 

 tities are distributed according to almost any curve of frequency satisfying simple 

 and common conditions, the average of successive groups of, say ,10, 20, 100 . . ?? of 

 these conform to a normal curve (the more and more closely as n is increased) 

 whose standard deviation diminishes in inverse ratio to Ihe number in each 

 sample. My own practice is to take, first, a number of small samples (say of 4 or 

 ot 10 in each) and observe the curve of frequency for these ; if there is a reason- 

 able indication of the shape of the normal curve appearing, I calculate the 

 ' standard deviation ' for this grouping, say a-, and proceed with confidence to 

 deduce that the average of a much larger sample, say of «, will have a normal 



curve of frequency, with deviation nearly o- a/ — , where 10 was the number 



in the first group of samples. If we can apply this method — and for clearness I 

 give an example immediately — we are able to give not only a numerical average, 

 but a reasoned estimate for the real physical quantity of which the average is a 

 local or temporary instance. It is the main weakness of statistical estimates, 

 whether of those on a great scale supplied by English or foreign public departments 

 or of more intensive inquiries by private investigation, that no measure of precision 

 is given, and consequently that no determination can be made as to whether 

 observed differences (in wages, in death-rates, in diet, in prices) are the accidents 

 of observation or are really significant. 



The example I have taken for illustrating the use of samples is worked rather 

 roughly, but when we are calculating the chances of unknown deviations it seems 

 unnecessary to go beyond the first decimal place. I took a copy of the ' Investor's 

 Eecord ' in which is given the yield per cent, to an investor at current prices on 

 the basis of last year's dividends for 3,878 companies, and set to work to find the 

 average of these percentages and the numbers giving various rates per cent, 

 by sampling. Having numbered the list consecutively, I took the ' Nautical 

 Almanac ' and read down the last digits of one of the tables, in groups of four : 

 if the number so read was over 3,878 I ignored it, if under (including such a 

 number as 0063) I wrote down the corresponding interest from the table I was 

 sampling. I thus secured equal chances for each of the 3,878 entries, and took 

 one at a venture 400 times. It is necessary to make certain, in some such wi^y as 



' See Professor Edgeworth's paper in the Jubilee nnpiber of the Statistical 

 Society, and subsequent papers thpre and elsewhere till that of Jupe 1906, 



