ON BIOLOGICAL MEASUREMENTS. 297 



D. Notes on Methods and References. 



The greater part of the procedure and methods of analysis indicated below Mas 

 first given in memoirs which appeared in journals such as The Philosophical Transac- 

 tions of the Royal Society, The Philosophical Magazine, Biometrika, The Journal of 

 the Royal Statistical Society, Metron, tfce. The fullest collected information is 

 probably contained in the following books, for which detailed page references are 

 given below : — 



(A) A. L. Bowley. ' The Elements of Statistics.' King & Son, London. 1920. 



(B) R. A. Fisher. ' Statistical Methods for Research Workers.' Oliver & Boyd, 

 Edinburgh. 1925. 



(C) T. L. Kelley. ' Statistical Method.' Macmillan, New York. 1923. 



(D) G. U. Yule. 'An Introduction to the Theory of Statistics.' Griffin & Co., 

 London. 1927. 



(1) Computation of the Arithmetic Mean, Variance and Standard Deviation. 



(A) pp. 251-255. (B) pp. 48-54. (C) pp. 45-48 ; 77-82. 

 (D)pp. 108-113; 134-141. 



(2) Definition and Computation of the Product Moment, Coefficient of 

 Correlation, and Regression Coefficients. 



(A) pp. 350-355 ; 380-383. (B) pp. 114-125 for regression. 



pp. 146-150 for correlation. 

 (C) pp. 101-164. (D) pp. 157-188. 



(3) The Standard Error and the Probable Error. 



The standard error of a descriptive constant or statistic measures the amount of 

 variability to be expected between the values of that quantity found in different 

 samples of the same size drawn at random from the same material. If a 2 represent 

 the variance in the population, then a/ -t/N is the standard error of the arithmetic mean 

 for samples of N individuals. Since in the study of natural variation the variance of 

 the population is unknown, we must use instead the variance as estimated from a 

 sample or group of samples, taking care to employ methods which made due allowance 

 for the sampling errors so introduced. 



Owing to the fact that if the distribution of a variate be Normal, 50 per cent, of 

 tlic observations will lie within the range taken from -6745 x standard deviation 

 below to a corresponding distance above the mean ; this multiple of the standard 

 deviation has been termed the Probable Error. Thus + -6745 a/ -%/N is termed the 

 probable error of a mean, by which it is implied that the means in samples of N will 

 as often fall inside as outside limits greater and less than the population mean by 



•(1745 .— As it is always necessary to find a or an estimate of a before the probable 



error can be calculated, it is always simpler, and more conformable to modern practice, 

 to measure variation by the standard error rather than the probable error. 



(4) Tests for Normality. 



The distributions given above of the length of Cuckoo's egg and the length and 

 breadth of Tern's egg may be taken as roughly representing the Normal form. This 

 is typified by a central concentration about the mean and a symmetrical tailing off 

 of the frequency for positive or negative deviations, in accordance with a definite 

 mathematical law. 



In dealing with samples containing only a few observations, it is only possible 

 to detect wide deviations from the normal form. The test, which is sensitiv only 

 for large samples, involves the calculation of the third and higher moments followed 

 by a comparison of the values obtained from the sample with those to be expected 

 from a normal distribution having the same variance. 



(B) pp. 54-56 or Phil. Trans, vol. 198 A, 1902, p. 278. 



