H. E. SoPER, A. W. Young, B. M. Cave, A. Lee, K. Pearson 371 
(iv) General Case of small Samples, n > 4. 
Equations (xxviii), (xxxi), (xxxii), and (xxxiv) enable us to express the moment- 
coefficients of a sample of w + 2 in terms of those of a sample of n. But we have 
found algebraic expressions for the moment-coefficients for « = 3 and n = 4 in 
terms of (a) the complete elliptic integrals and logarithmic functions, (6) trigono- 
metrical functions of a and cot a. Hence all even samples can have their moment- 
coefficients expressed in terms of a and cot a, and all odd samples can have their 
odd moment-coefficients expressed in terms of the complete elliptic integrals and 
their even moments in terms of logarithmic functions. The former result has been 
already noticed by Fisher*. The arithmetical calculation of the successive moment- 
coefficients after w = 4 by the difference formulae is, however, shorter than obtaining 
the algebraical expressions and then substituting arithmetical values, and has 
been followed in our calculations. 
(10) Approach of the Distribution as n increases to a Normal Character. 
It is well known that for the "probable error" to have meaning the distribution 
must approach the Gaussian for which /3j = 0, = 3. It is clear that these con- 
ditions are by no means fulfilled for samples of 25 or 50, whatever be the value 
of p. There is nearer approach in the loiv values of p in samples of 100, but there 
is considerable deviation for p = 5 and upwards. 
TABLE IV. Values of the Frequency Constants for the Correlation in 
Samples of 25. 
p 
)• 
mean 
Actual 
mode 
r from 
Pearson's 
f srmulaf 
Actual <r 
1 - p2 
Vn - 1 
Pi 
/3o 
0-0 
0 
0 
0 
•2041,241 
■2041,241 
0 
2-769,2305 
01 
■0979,577 
•11173 
•11127 
■2022,954 
■2020,829 
■012,3106 
2-791,6002 
0-2 
•1960,288 
•22258 
■22177 
■1967,883 
■1959,592 
■049,8655 
2-860,0511 
0-3 
•2943,287 
•33172 
•33090 
•1875,386 
■1857,530 
•114,6242 
2-978,8302 
0-4 
•3929,765 
•43840 
■43758 
•1744,356 
■1714,643 
■210,1771 
3-15.5,8537 
0-5 
•4920,974 
•54197 
■54149 
•1573,152 
■1530,931 
■342,3386 
3-404,2283 
0-6 
•5918,251 
•64194 
■64190 
•1359,499 
■1.306,395 
•520,2635 
3-745,3432 
0-7 
•6923,054 
•73792 
•73826 
•1100,322 
■1041,033 
■7.58,5549 
4-214,8982 
0-8 
•7937,001 
•82966 
■83025 
•0791,481 
•0734,847 
1-081,1286 
4-869,2635 
0-9 
•8961,933 
•91703 
•91736 
•0427,345 
■0387,836 
1-533,4124 
5-858,3872 
1-0 
1 
I 
1 
0 
0 
00 
00 
It will be reahsed that while the ordinary value for the standard deviation of 
r and the distribution of /• by a normal curve is fairly close for samples of 400, there 
is still a quite sensible deviation from normahty in the case of p = -8 or over. In 
fact it may be said that for the size of ordinary samples, there is always a sensible 
* Fisbei, Biometrihi, Vol. x. p. 516. 
t See Sections (4) and (5) above. 
t r= r+ M3(ft+ 3)/|mo(10/3, - 12(3,- ]8)|. 
