cxx Tables for Statisticians and Biometriciam [XXIII XXIV 



with /3 2 , increasing steadily as the population becomes more leptokurtic. In the 

 case of the skew Type III population the mean range is in close agreement with 

 "normal theory" but the standard error is for all sizes of sample somewhat larger. 



These results show that while Tippett's Table XXII may be used in a fairly wide 

 area of populations for estimating the population standard deviation from the 

 sample range when the samples are small, the reliability of this method of esti- 

 mation depends very much on the population form. In particular the method is 

 of little value if the population be leptokurtic. 



For some practical purposes it is necessary to know the chance of drawing a 

 sample with a range greater than certain multiples of the population standard 

 deviation. The results given in Table XXIV give some idea of the position; here 

 the permilles for the experimental sampling groups have been found by rough 

 smoothing of the data. The very considerable deviations of the fti and /3 2 of the range 

 curves for leptokurtic parent systems (/3 2 >3) are obvious in Table XX III, and should 

 warn the student to be cautious in assuming that results deduced theoretically from 

 a normal curve will in every case apply to small samples where we do not know the 

 parent population. This is still more strikingly illustrated in Table XXIV, which 

 is a rough probability integral of the range distribution curves. " The great length 

 of the tails of the range curves obtained in sampling from leptokurtic populations 

 will be seen at a glance. This is of considerable importance. ' Student ' has found 

 for example that leptokurtic error systems are common in routine analysis* and a 

 value of fi z = 7'0 is probably not unduly exceptional. The analyst must decide 

 therefore whether he should reject extreme observations as excessively improbable 

 deviations on 'normal theory' or accept them as perhaps rare but perfectly genuine 

 variants in a leptokurtic system f." 



For example, from a symmetrical population of /3 2 = 5*6, we should anticipate 

 that 8 out of 1000 samples of 10 would have a range greater than seven times the 

 parent population's standard deviation, or about 1%. In the case of a normal 

 parent population, only one individual range in 1000 samples may be expected 

 to exceed six times the parental standard deviation and less than 0'5 per 1000 

 seven times that quantity. Skewness appears (see Table XXIV, lower half) to cause 

 greater deviations from normality for lesser values of fa. Thus, for fti = 0*50 and 

 /3 2 = 373, we find only 5 - 5 / of samples of 20 give a range greater than 5a in a 

 normal parent population, but 7'7/ in the skew population. Such results indicate 

 the amount of caution needful in applying results deduced from the normal curve to 

 small samples. 



(ii) The Median. The median in the population is that value of the variable 

 which divides the frequency curve into two equal portions. If the observations in a 

 sample of n be arranged in order of magnitude the median may be defined as the 

 | (n + l)th observation if n be odd, and as the mid-point between the |nth and the 



* Biometrika, Vol. xix. pp. 151 164. 

 t Ibid. Vol. XX A . p. 358. 



