162 THE QUANTITATIVE METHOD IN BIOLOGY 



given example n = lo) . The mean represents therefore a very 

 complex something. In order to realize its significance in 

 general we are compelled to have recourse to a rather compli- 

 cated argument (expounded in § 115), starting from [a + h)'\^ 

 In each peculiar case, we must know the numerical value of 

 the data a, b and n. 



It is, however, possible to give a definition of the mean value 

 by saying that — 



The mean value is the arithmetical mean of all the observed 

 figures (this definition is, for the naturalist, a fictitious some- 

 thing), 



Or— 



The mean value is the most frequent resultant of the combined 

 causes ; in other words, the most frequently observed (the most 

 probable) value. 



Although the latter definition is more satisfactory than the 

 former, we don't find in it the notion of a directly measurable 

 simple thing (a constant) . 



Moreover, the arithmetical mean and the most frequent value 

 do not always coincide. This coincidence exists in the tj^ical 

 symmetrical variation curves ; for instance, in the example of the 

 prisms (§ 115, p. 160). In this case we start from an urn con- 

 taining simple prisms a and b in equal numbers. Since the 

 frequencies of the simple events are equal, the variation of the 

 compound prisms is expressed by a symmetrical curve, and 

 the mean (25 cm.) coincides with the central ordinate (the 

 most frequent value ; hump of the curve). 



If it is supposed that the first urn contains 100 prisms a 

 (2 cm.) and 200 prisms b (3 cm.), the variation curve of the 

 compound prisms would be obtained by expanding (a + 6)^^,^ 

 the frequencies being a; = iw = i and & = lw = f (compare the 

 balls in § 99) — viz. (59,049 compound prisms being extracted) : 



This curve is asymmetrical. The most frequent value 



1 It is impossible to distinguish in each of the eleven observed values a 

 constant ± an error. 



2 See the eleven terms, p. 159. The numerical values are : a^*^ =(^)io = 5^? ; 

 ioa% = io(if (f ) = 5^Sf9 ; etc. 



^ The denominator of the eleven terms obtained by expanding (i+f)^*' is 



