114 
Miscellanea 
Type II. A curve from some unpublished material in the hands of the writer. 
Ti/pe III. Curve of frequency of barometric heights, Churchstoke station*. This curve is 
figured as Fig. V, of Plate II, of the memoir cited. 
Ti/pe IV. Curve of variation in length of the carpopodite of leg III in the crayfish Camharus 
propinqims Girard t. 
Type V. Curve of variation in the number of lips in the medusa, P.pentata%. The curve 
is figured on p. 456 of the memoir cited. 
Type VI. Curve of variation in leaf number in main stem whorls of Ceratophyllum%. The 
curve is figured on p. 34 of the memoir cited. 
Certain of the constants of these curves are given in Table I. From the constants it will be 
seen that all these curves differ widely from the normal or Gaussian curve. Taken as a whole 
they are fairly representative of the conditions which one finds in frequency curves in practice 
when large numbers of individuals are dealt with. It will be noted that in some of these curves 
as tabled, Sheppard's corrections of the moments have been used, while in other cases they 
have not. 
TABLE I. Constants of Curves. 
Constant 
Curve of 
Type I 
Curve of 
Type 11 
Curve of 
Type 111 
Curve of 
Type IV 
Curve of 
Type V 
Curve of 
Type VI 
N 
1954 
275 
4018 
283 
996 
374 
Unit of grou^jing 
1 leaf 
15 
1/10 in. 
3/10 mm. 
1 
1 leaf 
1-5350 
6-3761 
12-6642 
7-2794 
-3090 
•8798 
5-8052 
111-2316 
511-4453 
246-7453 
1-1817 
4-0691 
-2112 
-0008 
-1258 
-7276 
4-1683 
1^1928 
2-4637 
2-7360 
3-1889 
4-6565 
12-3760 
5 •2568 
Kl 
- 1 -7060 
-5303 
+ -0005 
+ 1-1302 
+ 6-2469 
+ ^9354 
- -1002 
•0011 
+ 198-5780 
+ -5738 
+ 1-0659 
+ 1-2455 
o-ll 
1 -2390 
37-8765 
-3559 
-809 
-5559 
-9380 
Skewness 
- -6119 
-0171 
+ -1773 
+ -3293 
- -5294 
- -4270 
Sheppard's corrections 
Not used 
Used 
IT 
Used 
Not used 
Not used 
We may turn now to the consideration of our first problem. Taking the values of the 
moments given in Table I for the six curves I have calculated the probable error of the standard 
deviation for each curve, first according to equation (i) above, and then according to equation (ii). 
It will be noted that this procedure takes no account of whether Sheppard's corrections have 
been applied to the moments of the several distributions. It is here assumed that (i) is the 
absolutely true formula in every case. The results are shown in Table II. This table further 
gives the absolute differences for each pair of probable errors calculated in the two ways. 
Finally an expression of the relative magnitude of the error made by the use of the approximate 
formula was obtained by finding in each case what percentage of the true probable error the 
* Pearson, K., and Lee, A. Phil. Trans. Vol. 190, A, pp. 4'23— 469, 1897. 
t Pearl, R., and Clawson, A. B. "Variation and Correlation iu the Crayfish." Carnegie In- 
stitution of Washington, Publication No. 64, p. 7, Table I. 
X Pearson, K., Phil. Trans. Vol. 197, A, p. 455, 1901. 
§ Pearl, Pepper, and Hagle. Loc. cit. p. 32. 
II Given throughout in concrete units, not in units of grouping. 
T The moments in this case were corrected by considering the frequency areas as trapezia. Cf. 
Pearson, Phil. Trans. Vol. 186, A, p. 350, 
