clxx Tables for Statisticians and Biometricians [XXXP h , LI 



(iv) On the Determination of the Mode, r. 

 The mode is given by the following formula 



0-6 



, "i\rs . "* \rt "ay^v , v *(p) , /'vYY^ 



-P + ^l + (n-l)* + (n-l)* + (n^l)* + 



Table LI a provides the values of the first four v's for intervals of '05 of p. The 

 result will be correct to the 6th figure if n be 100 or greater, and correct to about 

 the 4th figure if n = 25. Below this value of ??, we cannot rety on formula (xxx) 

 with only four v's. If p be negative r has the opposite sign, but the same numerical 

 value as for p positive. 



A second convenient method is as follows: Let p 2 = pr, and z = p. Then the 

 following equation will give an approximation to z, and therefore to r : 



......... (xxxi). 



Illustration (vii). Find the mode for n = 16, p = '6. Our quadratic becomes 



62 2 - 169-922 + 27-216=0, 

 which gives, for the only possible root of z, the root less than unity, 



z = 1610,8575, 



whence P <? = *Jz = -4013,5489, 



and V=pjp = -668,925. 



The actual value of r correct to four figures is '6709. 



We see that equation (xxxi) gives us a reasonable first approximation to the 

 mode, even for a sample as small as 16. 



If we had applied formula (xxx) we should have from Table LI", 

 960 , T18272 -019,6608 S'57140,4288 



f 'nO -I- __ I- _ _ 



15 2-25 3375 50625 



or r = '669,180, 



which value is still more nearly correct. 



If for n small we wish to get a better approximation to /o 2 we suppose its true 

 value, p 2 , to be p 2 + e, where e is a small correction on p z . Then 



where E' and E" are the positive roots of the two quadratic equations 

 (-!)(!- po) E'* - (2n - 3) /a 2 & - (n - 2) = 0) 



r ( XXX111 ) 



n(l-p *)E" 2 -(2n-l)po z E"-(n-l) = ()) 



