ON BIOLOGICAL MEASUREMENTS. 295 



entered in corresponding positions on the different cards. A key card should always 

 be prepared giving the significance and units of the several entries on the individual 

 cards. 



An incomplete but valuable record of a large number of individuals measured in 

 more than two characters is provided by the preparation of every possible two-way 

 table. Thus with seven variates, twenty-one tables will specify the simultaneous 

 distribution of the samples for every pair of variates. Such a record, though 

 incomplete (because it does not specify which values of all seven characters were 

 associated together in an individual, but oidy considers them in pairs), will yet 

 provide a basis for all calculations ordinarily conducted. 



7. Gkaf-hic Methods. 



Diagrams should be freely used in exploring the character of the relationship 

 between two closely related variates. In plotting two sets of values against each other, 

 we may take absolute values, or the reciprocals of the absolute values of one or both, 

 or the logarithms of one or both, and so forth. If a straight graph is obtained b%' any 

 one of these methods, it suggests a particular type of mathematical relationship, the 

 recognition of which may facilitate the detection of the biological process or mechanism 

 involved. 



Diagrams provide no adequate substitute for the tabular presentation of data, 

 or for the critical tests necessary to examine their conformity with the hypotheses 

 they suggest. In the publication of results their purpose is to illustrate and make 

 plain particular facts selected for emphasis by the author, and not to establish such 

 facts. It is not necessary to publish every diagram which has proved useful in 

 studying the data. 



C. The Interpretation of Results and Tests of Significance. 



1. In carrying out any statistical analysis it is necessary to bear in mind the 

 distinction between the following : — 



(1) The population which has been sampled. 



(2) The true measurements of the sample available. 



(3) The measurements of these individuals as recorded. 



Provided that the specification is adequate and that the errors of measurement are 

 small compared with the real biological variation among the individuals of the sample, 

 it may be assumed that (3) provides no adequate description of (2). The problem 

 that remains is to consider what may be inferred legitimately from the measurements ( 3 ) 

 regarding the population (1). It needs little experience to realise that the average 

 measure of some character found in a sample, or the percentage of individuals falling 

 into certain groups, may often differ considerably from the values in the population 

 sampled, and further that two samples will themselves often differ considerably 

 from one another. The problem is therefore to obtain criteria which will enable 

 a judgment to be formed as to whether the variation in a sample is of statistical 

 significance (see Section D, 3-8); or is not more than might be expected to arise from 

 the chance fluctuations of random sampling. 



By mathematical analysis it has been found v possible to determine the variation 

 due to random sampling of some of the most important descriptive measures or 

 statistics, such as the mean or the standard deviation of a series of observations. 

 A definite measure of probability can therefore be assigned to the occurrence of a 

 particular value of the statistic in a random sample. In general, the procedure is to 

 calculate the ratio of (a) ; the difference between the statistic and the quantitative 

 character of the population of which it is the estimate or between the corresponding 

 statistics in two samples, to (b), the Standard Error or an estimate of the Standard 

 Error of that difference, and then to obtain the probability from the appropriate 

 table. 



The nature of the problem can be indicated most readily by considering two typical 

 examples. 



(1) Suppose there to be a population of individuals whose frequency distribution 

 for measurements of a single character is Normal (Section D, No. 4) ; and that the 

 mean measurement is known to be 22-56 cm. while the standard deviation is 1-54 cm. 

 Then it is possible to state by reference to the appropriate table that only in about 

 two cases out of one thousand should we expect to find a mean of 23-56 cm. or more 

 in a random sample of twenty individuals. Or supposing that the only available 



