330 Prof. Karl Pearson. Contributions to [Nov. 16, 



Weldon, for the measurements of a certain organ in crabs, by Mr. 

 Thompson, for prawns, by Mr. Bateson, for earwigs. They occur, 

 however, in physics, e.g., Dr. Venn's barometric and thermometric 

 frequency curves ; in anthropology, e.g., Signer Perozzo's curves for 

 Italian recruits, and Dr. C. Roberts' curves for the eyesight of 

 Marlborough College boys and in fever mortality statistics ; in 

 economics, Mr. Edgeworth's curves of prices, and curves I have had 

 drawn for rates of interest. 



Frequency curves may, however, be abnormal and yet symmetrical. 

 These are much more likely to deceive even the trained statistician ; 

 such curves might arise in target practice, and would be due, for 

 example, to firing with equal precision, but with a change of sighting 

 at mid-firing. 



2. Abnormal frequency curves fall into three distinct classes : 



a. Asymmetrical curves best represented by a point-binomial, or 



by its limit a continuous curve. 



b. Asymmetrical curves which are the resultant of two or more 



normal curves, with different positions of axes, different areas, 

 and different standard deviations a term used in the memoir for 

 what corresponds in frequency curves to the error of mean 

 square. 



c. Symmetrical abnormal curves, which are compounded of two or 



more normal curves having coincident axes but different areas 

 and standard deviations, or of two normal curves with the 

 same areas and standard deviations but different axes. 



3. Let a. be the area of any frequency curve, let the vertical 

 through its centroid, or the line through its centroid perpendicular to 

 the axis of measurement be drawn, and let the second, third, 

 fourth, fifth, and sixth moments about this centroid- vertical, a/*2, 

 aft*, xfjL 5 , and /t 6 , be ascertained. This can be done by graphical or 

 arithmetical processes indicated in the memoir, tables being given 

 to assist the calculation in the latter case. Then we can treat the 

 three classes of abnormal curves in the following manner : 



4. Class a. Let the binomial corresponding to the curve be : 

 a (p_|_2) } where p = probability in favour of an isolated event, q = 

 probability against, and n = number of contributory " causes " in a 

 single trial. For example : the simultaneous spinning of n teetotums 

 with black and white sides proportional respectively to p and q, and 

 the total number of times the group of n is spun. Then it is easy 

 to fit this point-binomial to a frequency curve of which the centroid 

 vertical and p l9 /i 2 , [**, and ^ 4 are known. The solution for this case 

 is not discussed in the memoir, having been already dealt with by the 

 author. 



If it be desired to draw a continuous curve corresponding to the 



