XXIII XXV] Introduction cxxi 



(J?j + l)th if At be even. In dealing with samples from symmetrical populations the 

 median determined in this manner lr<.in the sample may be used as an estimate of 

 tin- population mean. If the population be nonnal the standard error of the median 

 tends to the value of 1'253 x standard error of the mean, or l*S580'/Vf^ where a is 

 the population .standard deviation. This value is not however exact for very small 

 samples, DI.I- is it applicable to non-normal populations. Table XXIII, which corn- 

 Itiin -s theoretical with experimental results, shows the value of this standard error 

 as a multiple of a/V?*, the standard error of the mean. It must be remembered 

 that as the definition of the median differs in the two cases, the values of the 

 ratio corresponding to odd and even samples will converge separately on the 

 limiting value. 



It will be seen from the table that as the $ 2 of the population increases the 

 median improves as an estimate of the population mean, and the results suggest 

 that for $j > V'O it is probably a better central estimate than the sample mean; but 

 of course the evidence is only for samples of 20 or less. 



In skew distributions the median and mean do not coincide, and the value of the 

 former in the sample would only be of use to estimate the population median. 



(iii) The Mid-point between Extremes. If u be the highest and v the lowest value 

 of the variates in a sample of n, then |(M + v) may be termed the mid-point between 

 extremes, or briefly the "centre" of the sample. In dealing with symmetrical 

 populations the centre calculated from the sample is another form of estimate of 

 the population mean. Table XXIII contains certain theoretical and experimental 

 values of the standard error of the centre expressed as a multiple of o-/Vn, the 

 standard error of the mean. It will be seen that this estimate increases in reliability 

 as the /3 2 of the population decreases, but it is not until the rectangular popu- 

 lation is approached (/3i = 0, # 2 = 1'8) that it becomes more reliable than the mean. 



In sampling from skew populations the sample centre is not likely to be of much 

 value, for its mean position will change with the sample size and not correspond 

 to any fixed value in the population. Its use can only therefore be recommended 

 when the samples are very small and the parent population is known to be sym- 

 metrical and with /3 2 $ 3. In such cases it will give a rapid and not too crude measure 

 of the position of the population mean, which may be of some practical value. 



TABLE XXV. 



Table of the Probability Integral for Symmetrical Curves, /8i = 0, 2 = 1 to 3 and 

 3 onwards. (K. Pearson and B. Stoessiger, Biometrika, Vol. xxn. pp. 253283.) 



1. The symmetrical curves to be considered are those for which & = and $ 2 

 takes any value from 1 to oo . The curves are supposed completely determined by 

 their /3 2 's and their standard deviations. 



B. II. 



