Elementary Biometrical Inferences 



521 



though the number of different weights pos- 

 sible in the range of weights between fat 

 and skinny people is infinite, weights are 

 scored with a scale whose number of pos- 

 sible readings is limited. In other words, 

 an infinite variety of outcomes must always 

 be scored or measured in a finite number of 

 ways. So far as statistics are concerned, 

 the only difference between indiscrete and 

 discrete outcomes is the possible occurrence 

 of a much larger number of scored out- 

 comes in the former case. In either group 

 of outcomes, scoring a statistic requires the 

 use of some measuring device, be it the eye, 

 ear, finger, etc., very often in combination 

 with a ruler, photoelectric cell, and so forth. 

 We will study first statistics and para- 

 meters for discrete outcomes (small number 

 of classes) and then those for indiscrete 

 outcomes (large number of classes). 



It should be emphasized at this point 

 that the accuracy of the conclusions reached 

 from the use of biometrical procedures de- 

 pends upon four major factors: (1) imagina- 

 tion and flexibility, (2) proper sampling 

 methods, (3) accurate recording of statis- 

 tics, and (4) correct choice and use of bio- 

 metrical procedures. It is unreasonable to 

 expect that good biometrical technique can 

 overcome poor data; the biometrical analy- 

 sis becomes more efficient the closer one ad- 

 heres to the first three factors in carrying 

 out experiments. 



II. DISCRETE VARIABLES 



A. Range of Statistics Expected from a 

 Parameter Involving One Variable 

 (Figure A-1A) 



One often formulates a hypothesis in terms 

 of the probability that an event will occur. 

 It is also often desirable to know the kind 

 of result one would obtain were this hy- 

 pothesis tested. For example, common 

 sense suggests that "unbiased" pennies 



tossed in an unbiased manner have equal 

 likelihood of falling heads up or tails up. 

 Let heads up be considered a success. We 

 can state as a hypothesis (Ho) that the pa- 

 rameter p, the probability of success, is 50 r ; , 

 or 0.5, of all the times the coin falls flat. 

 Note that there are only two alternatives 

 involved — success and failure. Since 50% 

 of the time we would expect failure, the 

 probability of failure is 1 — p. One need 

 only use a single variable, probability of 

 success, to describe all the outcomes pos- 

 sible. (If one were to toss an unbiased die, 

 there would be 6 different and equally pos- 

 sible outcomes, and 5 variables. But if one 

 considered as a success only when the die 

 falls "one" up, then there would be only one 

 variable and we could state as an hypothesis 

 that p = %.) What kind of statistics 

 would one expect to obtain from actual 

 tosses of an unbiased penny? Clearly the 

 result will depend upon whether 1, 2, or 

 many trials, i.e., tosses, are made. 



Expected range of f values 

 Let us represent the number of successes 

 by X, the total number of trials or size of 

 sample by N, and the proportion of success 

 by f. 



Therefore, ^ = f , 



our statistic. Suppose one collected many 

 relatively large samples. What f values 

 would result? It has been shown that this 

 can be determined by using the expression 



P(l ~ P) 



which is called the standard deviation of p, 

 or s p . If the value of N (p) (1 — p) is 

 equal to or greater than 25, it is found that 

 95% of the f values obtained lie between 

 p — 1.96 s p and p + 1.96 s p . 



If one stated that f can have only the 

 values included in this 95% confidence inter- 

 val, he would be right 95% of the time and 



