ccxl Tables for Statisticians and Biometricians [XLIX 



We may now test De Moivre's formula (v) against Laplace's (vii) given in 

 the footnote below for lower values of n. For 7i=100, (v) gives i 495*02 

 and (vii 6is ) 496*75. For most statistical purposes the shorter formula would in 

 this case give an adequate answer. 



Now let us take * = 20, ra= 10. De Moivre's (v) gives us P(10, 1, 20) = *2736. 

 But Laplace's (vii) does not readily give us 1/&, and so P(10, 1, 20) for a given i. 

 We are obliged to invert the problem and calculate i for given values of k. Taking 

 k = 3, 4, 5, we find from Laplace's formula 



P (10, 1, i) = -3333, *2500, *2000, 

 i = 22*80, 20*92, 19*74. 



On the basis of the first difference between the values of i for *25 and *20, the 

 difference between *25 and *3333 should be 2*00 for i, or we should by extrapolation 

 have the value for i of 22*92, instead of 22*80. It seems therefore reasonable to 

 interpolate for i = 20*00 linearly. We find that P (10, 1, 20) = -211,017, while the 

 actual value found from our Table XLIX is '214,737. Thus for n as low as 10, 

 i = 20, Laplace's formula gives better results than does De Moivre's. For n = 40, 

 P(40, 1, i) = \j Laplace's formula gives 161*94 and De Moivre's 160*52. Again 

 this is not statistically a very great divergence. Thus we may conclude that when 

 i is large, even when n is relatively small, there is not much difference between 

 the two formulae, but that when i and n are both small Laplace's formula, if more 

 laborious to compute, is the better. 



It may be noted that for high values of n and i, the general formula (iii) has a 

 De Moivre form of approximation 



) 

 i 

 1 - 



Of course u must be of a lower order than n. This formula may sometimes be 

 applied successfully with low values of n and i\ for example, if we apply it to the 

 example under Illustration (i) we find P (4, 2, 6) = '93895 instead of the correct 

 value *93763. We hold, however, that for low values of n and i it is better to use 

 Table XLIX, or when outside that table's range of values, to check by Laplace's 

 formula as modified in the footnote, p. ccxxxix. 



Laplace neglects the term - (1 -2 log k), and says we may usually neglect the (log n - log log k) % log k 



term, thus reducing the formula to 



t = (log n- log log k) (n~i) + 4 log k. 



It is a misfortune that all the logarithms in Laplace's formula are hyperbolic. Converted into 

 logarithms to base 10, we have 



i = 2-302,5851 (Iog 10 n - '362,2157 - Iog 10 log 10 k) (n-'5 + 1-151,2925' Iog 10 ft) 



-L + 2-302,5851 (-S + -\log lo k (vii). 



2n \ fi J 



If fe=2, i.e. P (n, 1, i) = \, this becomes 



i = 2-302,5851 (Iog 10 n + -159,1745) (n - -153,4264) + -346,5736 -' - (vii 6 * 8 ), 



the last term being rarely of any importance. 



When n = 10,000, we get Laplace's result 95767 -4. 



