260 
Miscellanea 
IV. On the Modal Value of an Organ or Character. 
The mode has been defined as that value of an organ or character for which the frequency 
per unit variation of the organ or character is a maximum. The mode is so important for many 
purposes that its accurate determination is essential. Unfortunately rough and often practi- 
cally worthless determinations of its value have been given somewhat frequently in recent papers. 
It has to be remembered that in nine cases out of ten the mode cannot be settled by 
inspection of the observations, and that its calculation involves processes far more elaborate than 
those necessary to find the mean, and unless such processes are used the probable error of the 
mode will be so large as to render its determination of no practical service*. 
The method frequently adopted is the following : The whole range of the organ measured is, 
say, 6 to 9 mm. This is divided into elementary ranges of -4 or -5 mm., and 500 to 1000 indivi- 
duals being measured they are grouped into these elementary ranges. One of these elements 
exhibiting a maximum fi'equency, the mid-point of this element is selected as the modal value of 
the organ in question. 
Now such a frequency grouping as that referred to is ample for a good determination of the 
mean value of the organ or, with the use of the proper corrective term, for that of the standard 
deviation. It is generally idle for determining the mode by inspection. The mode will 
extremely rarely be at the mid-point of the element of observed maximum frequency, and 
in very many cases will not lie in that element at all. To be reasonably certain that it does, 
there must be a very large preijonderance of frequency in that element. In order to recognise 
the truth of the first statement, i.e. that the mode does not bisect the element of maximum 
frequency, consider the three consecutive frequencies 52, 93 and 84 which actually occur in 
a certain system of 400 measurements t. The group 93 falls into a certain range of 8'15 to 
8'55 mm. and the modal value is supposed on inspection to fall into this range. If we accept 
this, it will surely fall nearer to the 84 than to the 52 side of the range, and therefore it is clearly 
erroneous to place it in the middle at 8-35. We might allow for this bias of the mode by 
interpolating a curve through the tops of ordiuates of 52, 93 and 84 and finding its maximum 
ordinate, or better still by taking a curve, the areas of which on the corresponding ranges had 
these values. 
This might give fairly good results if 52, 93 and 84 were the true values of the frequencies of 
the given ranges for the whole population. But they are not ; they are subject to really 
considerable errors of random sampling. Let us find these errors. If an individual falls 
93 times in 400 trials into a certain group, its chance of falling into this group is and the 
chance that it will not is Hence the S.D. of the random sampling of this group is 
>/400 X fg^ = 8-45. Similarly the S.D.'s of the 84 and 52 groups are 8-15 and 6-73 respec- 
tively. Multiplying by "67449 to obtain the probable errors our system of frequencies reads : 
52 ±4-54, 93 ±5-70, 84 ±5-49. 
Now it will be clear that three numbers subject to probable errors of this kind can hardly 
give a very accurate determination of the mode ! Indeed the difference between 93 and 84 is 9, 
and the probable error of this difference is 8'97 J. Thus the difference is sensibly equal to the 
* The whole subject is well discussed by G. U. Yule: "Notes on the History of Pauperism," 
Journal R. Stat. Soc. Vol. lix., pp. 318 — 357. See especially the " Supplementary Note on the 
Determination of the Blode," with illustration of various approximate determinations (p. 398), and 
also the account of the discussion on this point at the meeting (p. 356). 
+ Left marginal length of left-handed fiddler crabs. Diagram No. 44, of Robert M. Yerkes' paper: 
"A Study of Variation in the Fiddler Crab, Gelasimus Pugilator, Latr.," Proe. American Academy of 
Arts and Sciences, Vol. xxxvi. p. 429. 
J It must be remembered that the errors in 93 and 84 are negatively correlated, and that accordingly 
the probable error of the difference is not the square root of the sum of the squares of the probable 
errors of 93 and 84. 
