12 THE MATHEMATICAL TREATMENT OF DATA 



uncertainties of the factors making up M. As before, the most likely 

 value is the root-mean-square one: 



S M = Vs 2 A + S% 



Now, if one has a more complicated relationship, such as 

 the standard deviation may be written as 



SM //^i\ 2 /^B\ 2 / Sc+D \2 



m~ \ika) + u; + \C + Dj ' 



As we have just shown, 



S C+ D = s c + s D . 



Thus, finally, 



SB\2 , (Sc + Sd) 



SM / /SA\ 2 , /Sb\ 2 {S C j- Sp) 



M SKA) ^ KB) f (C+2)) 2 



APPLICATIONS OF STATISTICAL ANALYSIS 



The general idea in these analyses is to compute the probability that a 

 particular situation could have arisen by chance. If it could have oc- 

 curred by chance once in a million times, everyone would agree that 

 chance is not the explanation; there must be an underlying mechanism 

 for the deviation. Suppose, however, that by chance alone the situation 

 could have arisen once in 10 times — what this means is that a deviation 

 this great or greater will occur that often. Do we then say that a 1/10 

 chance is too unlikely, or that it is so likely to happen that the deviation 

 is not to be regarded as significant? 



Statisticians have found through experience that an event that can 

 occur once in 10 times by chance alone is not sufficiently unlikely to 

 warrant undue interest. They have arrived at the following statements: 



A probability of 0.05 is significant. 



A probability of 0.01 is highly significant. 



In the applications we shall present, these two levels of significance 

 will be utilized repeatedly. 



There are three main kinds of statistical computations that we shall 

 set forth. These involve: 



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



2. the ratio of one variance to another variance; 



3. the chi-squared test. 



