170 THE QUANTITATIVE METHOD IN BIOLOGY 



by the number of measured specimens developed at each 

 temperature : — 



Temperature 

 Frequency ^ 

 Groups . 



Temperature 

 Frequency ^ 

 Groups . 



10" 



I 

 a 



i6° 



210 

 g 



II" 

 10 



h 



120 



h 



12" 



45 



c 



i8° 

 45 



13" 



120 

 d 



19° 



10 



i 



14" 



210 



e 



20° 



I 

 k 



15^ 

 252 



/ 



Suppose, on the other hand, that the relation between the 

 length A and temperature (between 10° and 20° C.) is expressed 

 by the following table ^ : — 



Length A.g 



(for instance, 10 cm.) coincides with 10° and 20° C. 



^8 ( 



^9 ( 



II 

 12 

 13 

 14 

 15 



11^ 

 12^ 

 13^ 

 14^ 



19° C. 

 18° c. 

 17° c. 

 16° c. 



15° (=n 



From the above figures the following variation curve (curve 

 of frequency) of A is deduced : — 



Length Groups Frequency 



Ag (10 cm.) a + ^=I+I= 2 



^7 (11 „ ) 6 + /= 10+ 10= 20 



Ag (12 „ ) c + z = 45 + 45 = 90 



'^9 (13 » ) ^ + ^ =120 + 120= 240 



Xjo (14 „) ^+g = 2I0 + 2I0= 420 



11 (15 ., ) /= 252 



Total 



,10 



(A;. = ) A 



"" ' ' 1024 = 2^" Specimens 



This curve is asymmetrical : the most frequent value A^^ 

 (14 cm.) is the length of the spikes which have been developed 

 either at 14° or at 16° C. The arithmetical mean of A is 1377 

 cm. It may be possible to express (in a rather complicated 

 way) the relations between the mean value and the data. The 

 latter are : (i) the variation curve of temperature (absolute 

 values and frequencies) ; (2) the values A^, A^ . . . and the 

 curve which expresses the relation between these values and 

 the temperatures between 10° and 20° C. 



In the present state of biological science, however, the 

 problem is insoluble, because we have no exact information 



^Number of specimens (total 1024). 



2 Proceeding in Fig. 25 from both limits f and T° towards the highest ordi- 

 nate, we meet on each side successively the values Xj, Xg . . . Xn. The 

 length Xi (practically o) coincides with the critical temperatures f and T°. 

 The length Xg coincides with two temperatures (see Fig. 25) which are nearer 

 the optimum (^°), etc. The length X^ (=X/i) coincides with ^°. 



