cxxvi Tables for Statisticians and Biometriciam [XXV 



If r=2a = -458,8314, then # = r 2 = '210526. If r = '50, x ='25. 



We have accordingly to find from Table XXV, for n = 19, the value of the func- 

 tion tabled for x= '210526 and x = '25. 



The latter comes without interpolation at once as ^(1 + 03) = '987,6152, or 

 a 2 = '487,6152, hence doubling, we find the chance is '975,2304, or the odds are 

 about 975 to 25, or 39 to 1, that in taking a sample of 20 individuals from a 

 normal population two characters of zero correlation will not show a correlation 

 in the sample exceeding numerically + '50. 



In the first we have to interpolate between the values for x of '21 and '22, i.e. 

 U Q = -978,9245, MI = -981,5217, 



Fourth differences are here unnecessary. 



6 = '0526, </> = -9474, <9</> = '0083,0554, 

 u e = -9790,6111 + '0000,0827 = '979,0694. 



The chance therefore of r falling within the range + 2<r is "958,1388. Had we 

 assumed the distribution of r to be a normal curve, the chance of r falling within 

 the range + 2<r would be '954,4998. 



Illustration (ii). In a sample of 12, the correlation coefficient is found to be '3. 

 What is the chance that in the original population there was no correlation ? 



In this case p 12 and m z = (p 4) = 4 = (n 3), 

 or w = ll, x = r* = '09. 



Our table under w = ll gives for x ='09 the value '828,2807. The chance 

 accordingly, of r exceeding + '30, if the correlation were zero, would be 



2(1- -828,2807), 



or, if the population sampled had no correlation between the variants considered, 

 a correlation of numerical intensity '30 or more would occur in 343 out of 1000 

 samples, i.e. in more than one sample in three. We cannot therefore assert that the 

 correlation found in the sample marks a significant correlation in the parent 

 population. 



Even if the observed correlation in the sample were "50, there would still be 

 98 samples in 1000 with a correlation of + '50 or more if the parent population 

 had no correlation. Indeed correlation coefficients found from very small samples 

 are of small service in indicating significant correlation in the parent population 

 unless the correlation in the sample be very high. For example, if the correlation 

 in the sample of 12 were '80, samples from an uncorrelated population would only 

 give rise to such a value once in 500 trials. 



Illustration (iii). What is the chance in a sample of 31 that the regression 

 coefficient will not differ from that of the parent population, supposed normal, by 

 more than twice its standard deviation ? 



