Sec. 2.8 PROBLEMS CREATED BY DRAWING SAMPLES 43 



2.8 SOME OF THE PROBLEMS CREATED WHEN ONLY 



A SAMPLE OF A POPULATION IS AVAILABLE 



FOR STATISTICAL STUDY 



Suppose that we wished to study ACE scores of college students 

 but could not afford the time or the expense required to analyze all 

 their scores, and hence took only a portion of them, say 50. Although 

 an economy of time and money will be obtained, several new prob- 

 lems will be created. 



First, how should the 50 students be chosen for the sample? 

 Ideally, they should be representative, in all important respects, of 

 the whole group which is being sampled. But this cannot be ascer- 

 tained without studying the ACE scores of the whole group — and 

 then no sampling would be needed. If the first 50 on an alphabetical 

 list were to be taken, the Macintoshes, McTaverishes, Swensons, and 

 Swansons never would be chosen ; and they might differ fundamentally 

 from those who would be chosen. If the first 50 who came into the 

 counseling bureau were taken as a sample, they might differ as re- 

 gards ACE scores from those who came in later, or who never came 

 into the bureau at all. In view of these and similar dangers of 

 acquiring a biased sample from such procedures, it is necessary to 

 devise a sampling method such that every eligible student has an 

 equal and independent opportunity to be chosen in the sample. The 

 net result of these requirements is to make it true that every pos- 

 sible sample of the chosen size (50 in the example above) will be 

 equally likely to be drawn. This is the fundamental requirement of 

 random sam-pling. 



There are various ways to draw a random sample of 50 from among 

 1290 members of a population. One would be to assign each person 

 who took the ACE test a different number, place these numbers on 

 pieces of cardboard, and draw 50 of them at random (in the popular 

 sense) from a bowl containing all of the pieces of cardboard. If the 

 scores in the population are recorded in rows and columns, as in 

 Table 2.01, we can assign numbers to the rows and to the columns, 

 and then draw a row number and a column number at random as 

 before. These two numbers together will uniquely designate a score 

 for the sample. If this is done 50 times — ignoring any repeats of 

 exactly the same row-column combination — this sample also will be 

 a random sample because every possible set of 50 scores among the 

 1290 in the population will have had an equal opportunity to have 

 been drawn. 



