396 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



US assume a null hypothesis that the variable in question is not a param- 

 eter or assignal)le cause, and select a critical value, F*, such that the 

 probability of ol)serving an F ratio j^reater than F* (when the null 

 hypothesis is true) is small (say 0.01). Then if the F ratio is computed 

 from our experimental observations and the null hypothesis rejected 

 when this ratio is larger than F*, on the average incorrect decisions 

 will be made not more than 1 per cent of the time. This method of 

 evaluation is not trivial and in complex situations reference should be 

 made to the literature or to experts. 



II. COMPONENTS OF VARIANCE 



A basic difference between the estimation of the Components of 

 Variance and the Analysis of Variance above is the concept of the 

 underlying model or law. The Analysis of Variance tests the hypothesis 

 that the treatment variable is not a parameter. In the estimation of 

 Components of Variance we assume that the observed effect of the 

 several values of a given variable is a random sample from a normal 

 population of effects from these values. If w;,, is the true effect of the i'' 

 value of the v variable, then the component of variance due to the 

 y' variable, a„ , is found from 



2 



Z2 _ 



k - 1 



2 



If Uiv = Uov = ■ • ■ = Ha,, , then ct„ = 0, and the mean square for variable 

 V contains only residual variation. If the variable y is a parameter, 

 o-p > 0; and the mean sciuare for variable v contains al -\- Mai (M meas- 

 urements being made at each of the k levels of the variable). It is desir- 

 able to estimate the component of variability of each variable, in order 

 to be able to estimate the variability of a measurement which is affected 

 by these variables. That is, if there are p variables whose components 

 of variance are a} , i = I , ■ • • , p respectively and if the measurement, x, is 

 influenced by all of these variables, then 



(T; = Z ^I + (tI , and <^x = /l/ ^<r • + afrrcr • 



Referring to Table XII and using the column of mean scjuares, we 

 make the following inferences: 



Since the ratio of mean square for variables to mean square for ex- 

 perimental error is "significantly" large for all the main variables, ex- 

 cluding Looks, these main variables are considered to be assignable 

 causes of variation. It is also evident that the three digit cards are caus- 



i 



