398 THE BELL SYSTEM TECHNICAL JOURNAL, MAKCH 1954 



the set contains only one measurement {n = 1), we cannot estimate 

 ffy, , and if- there is only one unit (^ = 1) we cainiot estimate al . 

 When both n, k > 1 we can estimate simultaneously both al and crl, . Let 

 Xij be the j measurement on the i' ' iniit, 



J = 1, • • • , n; I = 1, ... , k. 



and Xi be the mean of the t' unit, 

 X be the mean of all the units. 

 We can estimate o-^ directly by computing 



k n 



E Z (X, - xf 



k{n — 1) 

 (Xb can only be estimated indirectly by first estimating cri + H(rl from 



n E {Xi - Xy- 



k 



Then the estimate of ab is 



k n \ 



»-i j-i I 



(w - l)fc / ■ 



1 ^ 



Since o-^ = ^ (o-b -\ — -), it is clear that if al is large relative to al, , then 

 k n 



X, for fixed M=tik, will have greater precision if k is large and n is small. 

 The estimate of al, is called the sampling variance or sampling attriluit- 

 able to repeated measurements. The estimate of al is called the component 

 of variance due to experimental variation free of sampling error. For a 

 given experiment the estimate of al, -\- nal is called the experimental error 

 term and measures the precision of measurement of a unit. In the experi- 

 ment ay, = as, and ab = a e • 



BIBLIOGRAPHY 



1. L. N. Hampton and J. B. Newsom, The Card Translator for Nationwide Dial- 



ing, B.S.T.J., 32, pp. 1037-1098, Sept., 1953. 



2. W. G. Cochran and G. M. Cox, "Experimental Designs", J. Wiley and Sons, 



New Yor,, 1950. 



3. J. W. Tukev, "Comparing Individual Means in the Analysis of Variance", 



Biometrics, Vol. 5, No. 2, 1949. 



4. W. J. Dixon and F. J. Massev, "Introduction to Statistical Analysis" McGraw 



Hill, New York, 1951. 



5. R. A. Fisher, "Design of Experiments", Oliver and Boyd, Lontlon, 1950. 



6. S. Lee C'rump, "The Estimation of Variance Components in the Analysis of 



Variance", Biometrics, Vol. 2, No. 1, 1946. 



7. C. Eisenhart, "Assumptions Underlying the Analysis of Variance", Biometrics, 



Vol. 3, No. 1, 1947. 



