METHODS OF STATISTICAL ANALYSIS. 21 



significantly from zero, which would be found if an infinitely large 

 series of observations were available. For example, in table 18 we 

 show that the correlation between stature and pulse-rate in 121 men 

 is +0.0916 0.0608, while for 90 women it is -0.0669 0.0708. These 

 constants differ from zero by 1.51 and 0.94 times their probable errors 

 and consequently would not be considered to prove the existence of a 

 real positive correlation between stature and pulse-rate in the case of 

 men as a class or of a real negative correlation in the case of women as 

 a class. In short, the probable error indicates that the series of deter- 

 minations available is too small to justify any generalization concerning 

 the numerical magnitude of the correlation between stature and mini- 

 mum or basal pulse-rate other than that it is exceedingly small if it- 

 exists at all. A comparison of the coefficients obtained in the sub- 

 samples shown in table 18 justifies this view, for in the several series 

 available for adult males the coefficients are sometimes positive and 

 sometimes negative in sign. 



If we turn from the relationship between stature and pulse-rate 

 to that between stature and total heat-production given in table 32, 

 Chapter IV, we note that the correlation for the total males is +0.6149 

 0.0360, while for the total females it is +0.23 18 0.0629. The first 

 of these two constants is 17.1 while the second is 3.7 times as large as 

 its probable error. Thus there can be no question whatever concerning 

 the statistical significance of the deviation of these correlation coeffici- 

 ents from the zero which would be the average value if there were no 

 correlation between stature and total heat-production. We may con- 

 clude, therefore, that as far as the relationship between stature and 

 total heat-production is concerned the series of determinations available 

 furnish a fair basis for generalization concerning the numerical rela- 

 tionship between stature and total heat-production in men and women 

 at large. 



This discussion of the probable error has been of the most general 

 nature, but it may be sufficient to dispel the confusion which seems to 

 exist in the minds of some between technical errors of measurement and 

 the probable errors of random sampling of statistical constants, and to 

 enable the reader unaccustomed to statistical reasoning to follow argu- 

 ments based on probable errors in the following pages. 



Finally a few words concerning the actual routine of calculation 

 are in order. The formulas for the determination of r used in explaining 

 this coefficient above are not the most useful for practical work. In 

 the calculation of the standard deviation it is quite unnecessary to ob- 

 tain the actual deviation in each case. If the deviations are not wanted 

 for other purposes the standard deviation is easily obtained from 7 



7 Harris, Am. Nat., 1910, 44, p. 693. 



