136 
High Profhict- Moment Coefficients 
we have 
Vi23i = Mean of {x-^x^x^x^ 
= Mean of {,Xj (Mean of x.^^Xr^x^ for a given value of Xi)} 
= Mean of 
^1 jMean of (^12 ^'^^1 + ('"is^'a^i + ^3) (''i-i^^i + ^4)] 
Now for normal distributions (and if the original distribution is normal, so is that 
within the x-^ array), Mean X.^ = 0, Mean X^X^X^ = 0, while 
UO'2) (iC^s) 1»'23 
^■3 ('*23 "~ '"12 '"13) 
Mean ZgZg 
vr 
13 
= (r, 
Hence 
?7i234 = Mean of 
>'l2^'l3j <^2<73 
.(10). 
^1 - '■l2''l3»'l4 f^2(^3(^4 -1 + - (''34 " '^iz'^u) ^3^4 
+ '■l3 0'3 — (''24 - »'l2''l4) <^2<^<l 
>"l2'"l3) 0'20-3 
or dividing by a^o^o-iO^, 
<ll1U = ^12'"l3»'l4'?l^ + {''12 (''34 - rizl'u) + ^'l^ ('•24 " »'l2''l4)} + »'l4 (»"23 " »"l2''l3) 
= *"l2''34 + '''23*'l4 + ''■l4''23> 
since q^^ = 1 and 3. Thus o\xr formula is established for the case of four 
variables. 
4. We will establish the case for n variables by induction, and it will be con- 
venient to denote by i9'234.,.« the vahie of the reduced product-moment coefficient 
for the variables 2, 3, 4, ... n within the x-^ array so that 
Mean value of {X^X^ ... X„) 
^ M M-M • 
where X2, X^, ... X„ denote as before the deviations of the variables from their 
means within the x-^ array. Of course when w is even, 
i^'234.. r, is zero since n — 1 is now odd. 
Let n be even and assume that our formula has been proved true for all even 
values of n up to n — 2 inclusive, then 
Vi2d - n = Mean (x^^x^Xo ... x„) 
= Mean j^i (^I'loU^ ^- -f X^ ^J-jgag ~ + Zgj ... (j\„a„ ^ + Z„j 
)\„a.,a., ...a„ Mean {x^«)/ai^-^ 
+ S {{>\ar^,r,, ...) (a„a,a, ...) Mean (Z.Z^)} Mean {x,"-^)/a,"-' 
+ S {{r.aruric •••) {<^a<^i>cTe •••) Mean {X^Xf^X.X,)} Mean {x,«-'')/a,-' 
+ ... 
+ ,S {ri„CT„ Mean (Z^Z^ ... Z,)} Mean {x,^)/^^ (11), 
