clxxxiv Tables for Statisticians and Biometricians [XXXV XXXVI 



sample consisted of heterogeneous material, since the variability is much greater 

 than that generally found within a sample from a homogeneous race *. 



(ii) Suppose that we had been given only the mean cephalic index and not the 

 standard deviation in the population. We could ask whether it was likely, having 

 regard to the observed variability, i.e. s = 5*942, that a random sample of ten could 

 have been drawn having a mean differing by 73*47 75'06 = T59 from the popu- 

 lation mean. Applying "Student's" testf, it is found that 



z = (x - a)/s = - '268 and P z =-221. 



That is to say so large a deviation of z in excess or defect would occur in 44 / 

 of random samples. Without a knowledge of the population a we should therefore 

 find no reason for rejecting the sample \. 



(iii) If, however, while ignorant of the exact value of a- in the Egyptian popu- 

 lation, we had known from other sources that the standard deviation of cephalic 



* We may approach the problem from another standpoint. Given the parent population mean 

 75-06 and standard deviation 2'217, the distribution of means in samples of ten will be a normal curve 

 mean 75-06, and standard deviation 2-217/VlO = -70107. The occurrence of a mean 73-47, or a deviation 

 1-59, would give us x = l-59/'70107 or 2-268 to be looked up in the table of the probability integral 

 (Part I, Table II); the chance of this or greater value occurring is -0117, or considering excesses and 

 defects -023. Such a mean would occur about once in 50 trials. 



By Table XVII the mean S.D. in samples of ten = -9227(7= '9227 x 2-217 = 2-046, the observed standard 

 deviation is 5'942 less 2-046 = 3-896. Further from the same table the standard deviation of the distribution 

 of standard deviations in samples of ten ='9853 x <r/V^O= -48845. The observed deviation is therefore 

 7 '976 times the standard deviation. The actual value of this might, if desired, be found from the 

 Tables of the Incomplete T-f unction (published by H.M. Stationery Office). But it is unnecessary; a 

 value eight times the S.D. in a curve of this form has an infinitesimal frequency. We therefore conclude 

 that such a difference of mean is not very probable, and such a difference of S.D.'s highly improbable. 

 Since we assume normality the two probabilities are independent. The student must be careful to 

 distinguish the problem in the text above from the problem in this footnote. In the latter we ask what 

 is the probability of a combined event. We ask what is the probability p* that a mean as great as 

 or greater than x will occur, and what is the probability p s that a standard deviation as great as or 

 greater than s will occur, if the parent population be a, <r. Since x and s are supposed to be obtained 

 from a normally distributed population they vary independently and the probability of the combined 

 event is P=pxxps- Iu the text we consider the probability of a function of certain ratios occurring and 

 find it to be P*. We then ask what is the probability of a single event, namely the probability of any 

 values of x and s giving a probability greater than or as great as PA. . This is the probability of a certain 

 probability occurring, a single event. If this probability be exceedingly small, then we can reject x and 

 s as a sample from a and <r. But if it be fairly large, it does not follow that either x or s tested alone 

 by the rules of this footnote may not lead to the rejection of the hypothesis that the sample has been 

 taken from a and <r, notwithstanding that such rejection does not flow from the method in the text. 



t Tables of the probability integral of 2 for n=4 to 10 form Table XXV of the first volume of these 

 tables. More extensive tables were given by "Student" in Biometrika, Vol. xi. pp. 416 417, while 

 tables entered with ?t' = size of sample and = z\/n'-l have been given in Metron, Vol. v. No. 3, 

 pp. 26 30. Tables for symmetrical curves have also been published in Biometrika, Vol. xxn. pp. 253 

 283, and are reproduced in this work: see pp. cxxi and 171. E. S. Pearson and N. K. Adyanthaya have 

 shown (Biometrika, Vol. xxi. pp. 259286) that the z or i test may be used safely for populations 

 differing considerably from the normal. 



J That is to say by the z-test. The first footnote * above shows that the mean of the sample is 

 improbable, if the standard deviation of the supposed parent-population takes any value of the usual 

 order, i.e. circa 2-25 to 3-25. 



