30 THE QUANTITATIVE METHOD IN BIOLOGY 



idea, the variation of a large number of properties of numerous 

 species has already been investigated. According to QUETE- 

 LET, the mean value may be taken as a measure of the property 

 under consideration. 



By means of a variation curve we may calculate not only the 

 mean value, but also the probable error. 



This may be represented in the following way (according to 

 GALTON) : — Suppose that a given property has been measured 

 in n specimens ^ (for instance, the height of n soldiers taken at 

 random in a certain district, or the length of the spike of the 

 main stem of n specimens of rye taken at random from a field). ^ 

 When all the specimens are placed side by side, in the order of 

 their individual lengths, the smallest one being N"! and the 

 largest one being Wn, the specimen M, which is in the middle 



(at the place -), represents very approximately the mean 



value. ^ The difference D between the figure of M and that of 



the specimen Q^ which is at the place - is the negative mean 



deviation. In the same way the difference D^ between M and 



'\n 

 the figure of the specimen ^2 which is at the place — is the 



-positive mean deviation. In a perfectly normal (symmetrical) 

 curve Z) and D^ are equal in value, but as no experimental 

 result is perfect, they usually differ slightly in amount. A 

 mean between the two is therefore taken : this mean is the 



frooame error = . 



EXAMPLE : I put down n = 1000. The mean between the 

 specimens 500 and 501 (these specimens differ very little or not 

 at all in amount) is the measure of M. In the same way the 



mean between the specimens 250 and 251 (place -) is Q^ and the 



4 



mean between the specimens 750 and 751 (place — ) is Q^. 



The probable error is a measure of the degree of spread of the 

 curve ; it is an index of the variabihty of the property under 

 consideration.* 



1 n being a large number ; for instance, looo. 



^ The latter example (rye) was studied in the Botanic Garden at Ghent by 

 Dr C. DE BRUYKER. 



' That is to say that 50 per cent, of all the measurements fall below it in 

 magnitude and 50 per cent, above it. 



* The quotient probable error : mean value is the relative probable error. By 

 means of the latter value the degree of variabihty of two or several series of 

 observations (differing by their mean values) may be compared. 



Another method of expressing the variability of a property is to calculate 

 the arithmetical mean error. 



