406 H. S. JENNINGS AND K. S. LASHLEY 



differs from it; particularly in the fact that its upper limit is 

 much farther from the mode than is the lower limit. This is not 

 characteristic of all polygons obtained in this way, but apparently 

 so only of cases where w and n are rather large. In many cases 

 we have, in place of such a polygon, merely a point, a straight 

 line, or a broken line, which may be of various forms. Some of 

 these are illustrated in figure 2. 



In view of the form of the polygons obtained, it would appear 

 to be difficult to obtain any simple formula to express the proba- 

 bility of a given deviation: one must use such a method as that 

 set forth above (see rule (5), Appendix). 



The procedure which we have just described of course makes 

 it possible to determine with absolute accuracy what is the most 

 probable number of pairs when an odd number of units is drawn; 

 a point which we left undecided in our previous account. When 

 this is the only question to be answered, it is done as follows: 

 Find by formula (1) the average number of pairs to be obtained 

 when the given odd number is drawn. Then find by formulae 

 (3) and (4) the probabilities for the two numbers nearest this 

 average; this will of course show which of the two is the most 

 probable. 



Formula (3) gives unmanageable numbers when the operations 

 indicated are directly performed; thus 186! gives a number con- 

 sisting of 343 integers. In practical work therefore logarithms 

 must be employed. The logarithm for any factorial number 

 n! is of course the sum of the logarithms of all integral numbers 

 from 1 up to and including n (since n ! is the product of all numbers 

 from 1 to n). With a table of such sums of logarithms the com- 



Fig. 1 Polygon showing the relative probabilities for obtaining the different 

 possible numbers of pairs, when 51 specimens are drawn from 186 (93 pairs). The 

 ordinates give the probabilities in per cent, for each of the numbers of pairs 

 indicated on the base line. (These probabilities are the percentage of all draw- 

 ings in which the given number of pairs would be obtained, if the drawing of 51 

 from 186 were repeated a great number of times.) They are plotted from the data 

 given in table 36, page 412. 



The polygon extends at the left to 0, at the right to 25 pairs, but the probabilities 

 for and for all numbers from 15 to 25 pairs are so minute that the ordinates do not 

 appear at all in a polygon drawn to this scale, so that in these regions the outline 

 of the polygon as drawn coincides with the base line. 



