1890.] occurring in certain Decapod Crustacea. 447 



lowing way, which is that adopted by Mr. Galton : — The values ob- 

 tained are sorted and arranged in order of magnitude ; then, at equal 

 distances along a given base, ordinates are erected equal in number to 

 the observations, one ordinate being proportional to each observed 

 value of the organ. By joining the tops of these ordinates, a curve is 

 obtained such as that drawn in fig. 2. 



If the base-line of such a curve be divided into one hundred parts, 

 then the proportion of individuals measured, which possess the 

 organ from which the curve is constructed, of a size greater or less 

 than any given magnitude, can be readily ascertained. For example, 

 in fig. 2, which shows the distribution of lengths of the carapace in 

 400 female shrimps from Plymouth, the ordinate, whose length is 

 256, stands at grade 20°, showing that 20 per cent, of the indi- 

 viduals examined had the carapace longer than 256 (the body length 

 being 1000), while in the remaining 80 per cent, the carapace was 

 shorter than this. 



A curve constructed in the manner directed is nearly always sym- 

 metrical about its middle point: and this point therefore closely 

 approximates to the average of the whole number of observations 

 from which the curve was constructed. The value of the middle 

 ordinate will always be taken, in what follows, as the average value : 

 it will, in accordance with Mr. Galton' s notation, be spoken of as the 

 Median, and denoted by the symbol M. Each curve, therefore, gives 

 by simple inspection the average value of the organ to which it 

 refers. 



In estimating the deviations from the average which occur in each 

 case, the magnitude of the average itself is evidently of no im- 

 portance : and the ordinates of the curve may therefore be considered 

 with reference to an axis passing through the point M, so that the 

 ordinate of M becomes zero. When measured from this axis, half 

 the ordinates of the curve are of course positive, the other half being 

 negative. 



If the frequency, with which the observed deviations from the 

 average occur, obeys the law of error, then the curve just described 

 should be a " curve of error," whose " probable error " is represented 

 by the ordinates at the 25th and 75th grades. These grades are the 

 boundaries of the first and third quarters of the base: they will, 

 therefore, be spoken of (again in accordance with Mr. Galton's nota- 

 tion) as Quartiles, and will be denoted by the symbols Q 1 and Q 3 respec- 

 tively. In a perfectly normal curve, Q l and Q 3 are of course equal in 

 magnitude and opposite in sign. In the observed curves there was 

 generally a slight difference between the two : and the mean value of 

 the two is therefore adopted as the " probable error," which will be 

 denoted by the symbol Q. 



In order to determine the correspondence between the observed 



