ON BIOLOGICAL MEASUREMENTS. 293 



fraction of the information available from the original data, and for this reason, if 

 they be recorded alone, needless inaccuracy is introduced. 



For an important class of cases of homogeneous samples, an adequate summary 

 may be given by stating two statistics only ; namely, conventional estimates of the 

 mean and the variance of the population sampled. 



(a) The arithmetic mean of the measurements, defined as the sum of the measure- 

 ments divided by their number. 



(6) An estimate of the variance calculated from the sum of the squares of the 

 deviations from the arithmetic mean by dividing it either by (i) the number of indi- 

 viduals in the sample, n, or by (ii) one less than this number, n— 1. With large samples 

 it is seldom of importance which divisor is adopted, and the former method has been 

 the more widely employed in biology. Attention is called to the latter method for 

 no other reasons than that : (A) it is somewhat the more accurate in using the Normal 

 probability function of tests 5 (a) and 6 (a) (Section D), if the variance there employed 

 has been estimated from the sample ; (B) it is upon this convention that the table 

 of / for tests 5 (b) and 6 (b) has been calculated ; (C) it alone should be used when it 

 is desired to average the variance as estimated from several independent samples. 



From the variance two other quantities may be calculated : (i) the Standard 

 Deviation is the square root of the variance; (ii) the sampling variance of the mean 

 may be estimated by dividing the variance as estimated from the sample by the number 

 in the sample ; this measures the amount of variability to be expected among means 

 of different samples of the same size drawn fairly from the same population ; its square 

 root provides an estimate of the Standard Error of the mean. Large samples, if equally 

 homogeneous, will consequently enable finer distinctions to be drawn than can be 

 detected with confidence in smaller samples. 



For samples of the normal form (Section D, 4) the two quantities (a) and (b) 

 provide a complete statistical summary. Such a summary, though often valuable, 

 cannot be regarded as complete when the distribution of the sample is unsymmetrical, 

 or in other ways differs clearly from the normal form. 



4. BrvAEiATE Data. 



If two measurements are taken on each individual the sample may be completely 

 specified in a two-way or correlation table. The arrangement of such a table may 

 be illustrated by the example on page 292. ' In the table are recorded the measure- 

 ments of the Length and Breadth in cm. for each egg in a sample of 956 eggs of 

 the common Tern (Sterna Fluvialilis). The class interval is -05 cm. for each 

 measurement. The table shows, for example, that 18 eggs were found with a 

 breadth between 2-95 and 3-00 cm. and a length between 4-05 and 4-10. The figures 

 in the two margins give the total distributions for length and breadth respectively. 

 The table supplies in a compact and readily available form the whole of the infor- 

 mation supplied by this collection respecting (i) the variation in length, (ii) the 

 variation of breadth, and (iii) the co-variation of length and breadth. 



It is not necessary, however, that both or either of the variates should be quantities 

 capable of numerical measurement. For instance, observations upon the colouring 

 of eggs in a nest in relation to the type of environment in which the nest has been 

 placed could equally well be recorded in a two-way table. In such a case the group- 

 ing in one direction would be based upon a graded colour-scale and that on the other 

 by a series of environmental classes, such as green plants, speckled shingle, brown 

 sand, &c. In the following table, taken from the same source, the two characters 

 considered are (a) ' Value ' of ground-colour of one egg from a nest, and (b) ' Value ' 

 of ground-colour of a second egg from the same nest. 



The -75's, '5's and *25's among the frequencies arise because in cases of un- 

 certainty in classification a half-frequency was assigned to both, or a quarter to al 

 four of the possible groups. The colour-value classes w 2 — w K were described with 

 the aid of a coloured plate in the Memoir. (The table has been made symmetrical by 

 entering each pair of eggs twice, first 'with one egg as ' first egg ' and the other as 

 ' second egg ' and then in reversed order.) In this table there is seen to be a marked 

 clustering of frequencies along a diagonal ; for instance, when the ' first egg ' falls in 



1 This and the table on page 294 are taken from a paper in Biometrika, vol. xv, 

 pp. 294-345. 



