702 APPENDIX 



differ but slightly from each other, but if the measurements are sufficiently 

 accurate there will be some differences. If we should find the mean of 

 these means (we may call it the second mean) we could plot a frequency 

 curve of the distribution of means just as we have for the deviations 

 of the original variates. Of course this curve would be much crowded 

 together, like A of Fig. 7. To make this general, with a very large popu- 

 lation, and with n in each group instead of 1000, the following result is 

 obtained : 



If E s is the probable error of a single variate, that of the mean of ;/ 

 variates is 



F 2- 



EM V' 



that is, to find the probable error of the mean, divide the probable error 

 of a single variate by the square root of the number of variates. 



Probable error in standard deviation. Taking up again the million cases 

 of stature divided into a thousand groups as an illustration, supposing that 

 the standard deviation of each of these thousand groups be found, we 

 should see that they differ but slightly. However, if the computations and 

 measurements be very refined there will be deviations. These standard 

 deviations constitute a frequency distribution whose standard deviation 

 can be found, and the probable error of the standard deviation can be 

 obtained just as we have shown in the case of a single variate. 



Generalizing this so as to have a very large number of groups each con- 

 taining n variates taken as a sample to represent the population, the prob- 

 able error of the standard deviation is 



that is, to find the probable error in the standard deviation, divide the 

 probable error in the mean by V2. 



Formulas for probable error in some important statistical constants. Enough 

 has now been said to give the conception of the probable error in any 

 statistical determination and a general notion of the methods by which 

 formulas for the probable error are derived. 



It is scarcely necessary to remark that the probable error does not take 

 into account evident mistakes either of observation or computation. We 

 are assuming that these have been eliminated. It has to do with errors 

 (deviations) due to an indefinitely large number of unassignable causes 

 such that the errors are distributed according to the laws of probability. 



It seems unnecessary to continue the discussion of probable error in 

 other determinations, but it does seem well to collect together, for purposes 

 of reference, the formulas for the probable error in some of the most 

 important statistical constants. 



