Miscellmiea 
279 
With the preceding calculations we can now complete the Table on p. 529 of Biometrika, 
Vol. X, with the following 
TABLE I. 
Table of Values of the Constants of the Frequency Distributions of the Standard 
Deviations of Samples of Size 2 and 3 drawn at random from a Normal 
Population. 
Size of 
Mode 
Mean 
Standard Deviation 
Measures of Deviation from Normality 
sample 
Skewness 
ft 
2 
0-0 
■5642 
•4263 
■8525 
1^3236 
■9906 
3-8692 
3 
•5774 
7236 
■3782 
■9265 
■3867 
•3983 
3-2451 
Experimental Veriji cat/ion for the Case of Samjdes of Two. 
The frequency distribution of the number of stigmatic bands on the capsules of a growth of 
Shirley poppies is as follows * : 
TABLE II. 
No. of Stigmatic 
Bands 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
Total 
No. of Capsules 
1 
11 
32 
56 
148 
363 
628 
925 
954 
709 
397 
155 
51 
12 
1 
4443 
the standard deviation of the distribution, as calculated by the ordinary method, being 1-8977. 
We will examine the frequency distribution of the values of the standard deviation of this 
series which are given by samples of two capsules taken at random. Now the standard deviation 
of two measurements and X2 is easily shown to be ^, taken with the plus sign, so that 
in this case, the variate being measurable only in units, the possible standard deviations will 
be 0, -5, 1, 1-5, and we can find the theoretical frequencies (column (&) of Table III) of these 
values by taking the areas of the strips of the half-Gaussian 
^ 1-8977 \/7r 
whose bounding ordinates cut the axis of 2 at 0, -25; -25, -75; -75, 1-25; .... Thus the breadth 
of the first strip is only haK that of the others and it will be found that it is the value -5 which is 
the real mode of the probability. 
In the case of such a distribution we can find the frequencies of the standard deviations of 
samples of two by actual calculation. For if we denote the chance of a capsule oceiu'ring with 
5, 6, 7, 8, 9, ... stigmatic bands by a, b, c, d, e, ... respectively, it is clear that the chance of the 
value 0 of the standard deviation occurring is + + + + and the chance of the value -5 
occurring is the chance of a sample of two flowers whose numbers of stigmatic bands differ by 1, 
i.e. is 
ab + b {a + c) + c {b + d) + ... = 2 {ab + be + cd + ...), 
and similarly for 2 = -I the chance is 
ac + bd + c {a + e) + d (b + f) + ... = 2 {ac + bd + ce + ...), 
and so on. 
By this means the theoretical frequencies given in column (a) of Table III were calculated. A 
histogram of these values will be found to be well in accord with the half-Gaussian given above. 
* Pearson, Phil. Trans. Vol. 197 A, p. 314, Hampden Series. 
