520 



A HIM NDIX 



I. INTRODUCTION: STATISTICS 

 AND PARAMETERS 



There are numerous occasions when one 

 may wish i<> arrive a1 some genetic con- 

 clusion i»n the basis of experimental data. 

 Whenever these data arc subject to chance 

 variation, it is necessary to make use ol 

 biometrical ideas and techniques in order 

 to draw the mosl precise conclusions. 

 Let us consider, therefore, some of the basic 

 principles and methods which are likely to 

 be valuable in a study of genetics. (The 

 Table of Contents at the beginning of this 

 chapter will make it easier to find the sec- 

 tion that describes a particular biometrical 

 technique.) 



A statistic is a measurement obtained 

 from a sample. A sample can be consid- 

 ered as having been drawn from an ideal 

 population composed of an infinite number 

 of measurements. Whereas the measure- 

 ments of a sample are statistics, the meas- 

 urements of the ideal, infinitely large, popu- 

 lation are expressed in terms of parameters. 

 The difference between a statistic and a 

 parameter can be illustrated with a penny. 

 Let the ideal population be composed of the 

 results of an infinite number of tosses. In 

 this ideal population one would expect the 

 coin to fall heads up 50% of the time, and 

 tails up 50% of the time. The population 

 can be characterized in terms of a para- 

 meter, the probability of heads up, ex- 

 pressed as p = 0.5. If one actually takes 

 a sample of this infinite population by toss- 

 ing a penny a finite number of times, one 

 obtains the statistic, the frequency of heads 

 up relative to the total number of tosses. 



(liven a parameter, one may want to 

 predict the range of statistics expected to 

 comprise a sample (Figure A-1A). Alter- 

 natively, one might like to be able to deter- 

 mine from a statistic the range of para- 

 meters from which this statistic could have 

 been obtained by sampling (Figure A-1B). 



PARAMETERS 



12 3 4 5 6 7 



9 10 11 12 12 



1 2 



3 4 5 6 7 

 STATISTICS 



9 10 11 12 13 

 o ol o2 



figure A— I. Biometrical procedures to be 

 discussed with respect to discrete variables (see 

 text for explanation), o = observed, e = ex- 

 pected. Arrows show direction of prediction. 



One may want to determine the probabili- 

 ties (i.e., parameters) that different alterna- 

 tives will occur in samples drawn from an 

 ideal population (Figure A-1C). One may 

 wish to compare the statistics expected (e) 

 in a sample with those actually obtained 

 (o) (Figure A-1D). And finally, using a 

 parameter, one may wish to compare two 

 groups of statistics (ol and o2) (Figure 

 A- IE). Methods for making these and 

 other comparisons are presented here. 



Heads vs. tails, black vs. white, smooth 

 vs. rough, and tall vs. short all involve 

 discrete variables which are measured by 

 enumeration, since the outcomes or alter- 

 natives fall into discontinuous, easily dis- 

 tinguished and separable, classes. On the 

 other hand, the statistics of w r eight, height, 

 and intelligence are all quantitative, con- 

 tinuous, or indiscrete variables. The differ- 

 ence between the two lies in the number of 

 alternatives possible in each case; there is 

 an infinite variety of alternatives possible 

 in the indiscrete case, but only a limited 

 number of outcomes in the discrete one. 

 This difference disappears, however, once 

 the outcomes are tallied. For example, al- 



