APPLICATIONS OF STATISTICAL ANALYSIS 13 



1 . The ratio of a given deviation to the standard deviation 



As an illustration, suppose we are studying the effect of adding various 

 biochemicals to a nutrient medium and find that, after a given time, there 

 is an average of 183 organisms per test tube, with a standard deviation 

 of 9 organisms. One test tube (that has one particular biochemical addi- 

 tive) contains 212 organisms. Is this increase significant? 



The deviation is 29 organisms, which is slightly more than 3s. From 

 our previous reference table we find that a deviation of this much or more 

 will occur by chance alone about one time in 800. Therefore we say 

 that the deviation is highly significant and it is therefore highly likely 

 that the particular biochemical had a real, positive effect. 



If there had been another tube with 198 organisms, the deviation would 

 be 15 organisms, and the ratio of this deviation to the standard deviation 

 is almost 2. Thus the probability of this happening by chance alone is 

 almost 0.025, and is therefore considered almost significant. Thus we 

 don't quite know whether to consider that the particular additive had a 

 real effect. We have to do more or better experiments to decide. 



The ratio of the observed deviation to the standard deviation should 

 really be evaluated by using tables which depend on the number of 

 independent measurements, as explained previously. This computation 

 was done many years ago by a man publishing under the pseudonym 

 Student. He called the ratio t, and you will therefore find this test in the 

 textbooks under the name Student's t test. 



The evaluation of n is not entirely trivial. When we choose a par- 

 ticular average a, the variability in the set of measurements is reduced 

 by the choice. Indeed, given n — 1 measurements and a, one can recon- 

 struct the nth measurement. Thus, knowing a, there are only n — 1 

 independent measurements. Similarly, every time we fix a parameter 

 like a or s or the total number of things counted, we impose a condition 

 which allows us to reconstruct the entire series of measurements with one 

 fewer measurement. 



In a way entirely similar to that for individual measurements, we can 

 compare deviations of averages with standard deviations of averages. 

 If two laboratories measure the concentration of viruses in a vaccine 

 preparation, the deviation between their average values should be com- 

 pared with the standard deviation of the averages (the standard error) 

 to see whether there is agreement or systematic deviation between their 

 results. 



2. The ratio of one variance to another variance 



It happens with surprising frequency that there is need to compare 

 two variances. The analysis of the problem ends up with an estimate 

 of whether the ratio of the two variances is sufficiently different from 



