200 STATISTICAL TREATMENTS 



enough measurements to assure meaningful results, without taking a 

 wastefully large number of measurements. More important, perhaps, we 

 can be sure to take the right kind of measurements. 



Probability 



Statistics, a field based upon the laws of probability, treats events that 

 occur at random. Most people have an intuitive notion about simple 

 probability and agree readily that an honest coin comes up heads half 

 the time and that seven is most likely to appear in a roll of a pair of dice. 

 People are gamblers, not because they understand probability, but be- 

 cause they do not. The professional gambler stays in business because 

 he knows that any individual event has a certain probability of occurring, 

 regardless of what has happened before. In the long run, the distribution 

 of events follows the individual probabilities. If a million coins are tossed, 

 very nearly half a million come up heads. 



Truly random events occur according to probabilities that are inherent 

 in the events themselves and are not influenced by outside conditions 

 or time sequences. Any individual atom of a radioactive isotope has a 

 certain probability of decaying within the next second. This likelihood 

 is not influenced by the presence of other atoms of the same kind. Ran- 

 dom events are completely unpredictable on an individual basis. If meas- 

 urement is influenced only by random variations, it is just as likely that 

 the measurement will be a little too large as a little too small. 



In theory, events occur at random because they are not influenced by 

 outside conditions. In practice, it is too easy for events to be aff^ected by 

 bias of some sort, and therefore not to occur at random. In the analysis 

 of experimental results, procedures are used which assume that the errors 

 occur at random. The error or variability of measurement can be treated 

 statistically only if the variability is random. For this reason, special steps 

 must be taken in planning the experiment to assure the randomness of 

 the errors. Laboratory biologists do not commonly carry out formal ran- 

 domization steps but, instead, hope to obtain experimental results that 

 answer the hypothesis even without statistical treatment of the data. 

 Unfortunately, too many of them perform statistical analyses, even with- 

 out prior planning. If the assumption of true randomness is ignored, the 

 statistical analysis is not only meaningless but deceptive as well. 



Statistics ideally treats populations of things or events. By a population 

 we mean all the possible things in a particular class, just as the human 



