C o n s e q u e n t ly th e e q u a t io n o f th e c u r v e o f p ro b a b ilit y c o r r e sp o n d in g t o th is s e r ie s is
y = 2 5-13^ ^3-84) ’
a n d th e m e an lin e is a t 73*98, o r p r a c t ic a l ly a t 74.
T o d r aw th is c u r v e w e fin d th e v a lu e s o f th e o rd in a ls a t p o in ts d i s t a n t , - , c, — an d i c o n e ith e r 3 a
s id e o f th e m e an , i . e . a t p o in ts d is ta n t , 1-9 3, 3.84, 5 -7 5 , 7.68 o n e ith e r s id e o f i t in th e p re s en t ca se.
W e t a k e th e s e p o in ts b e c a u s e w e h a v e
w h en x — — y = 25*13*“ ^
x ~ c ƒ = 35-13^-1
3 C B
x ~ 2 y - W S ' -*
x = z c y — 2 5 * 13 * -* .
B u t a s w e k n ow e — 2*718 a p p ro x im a t e ly , w e c a n fin d
e\ = 1*284
an d w h en
e \ — 9*287
e * = 54*6 ;
23*13
* = i -9a ^ = i 4 4 = i9-®
— 3-84 , = ^ = 9-«
* = ’ 68 * 6-
W e th u s g e t fo u r p o in ts o n e a c h s id e o f th e m e an : a s w e k n o w th e g en e r a l s h a p e o f t h e C u r v e o f
P r o b a b i l i t y th e s e p o in ts o n i t e n a b le u s t o d r aw it . A f t e r b e in g d r aw n in c u r v e fo rm i t is r ed raw n in
b lo c k fo rm in th e s am e fig u r e a s th e C u r v e o f F r e q u e n c y .
T h e m is fi t is n ow c a lc u la ted b y a d d in g u p the, a r e a s o f th e d iffe r e n c e s b e tw e e n th e tw o cu r v e s .
In th e p re s en t c a s e w e h a v e d iffe r e n c e o f e x c e s s o f C u r v e o f F r e q u e n c y o v e r C u r v e o f P r o b a b i l i t y
*7 + i* 8 + 5 * 6 + 3 * 6 4 .1 * 8 + 2*8 + * 7 = 1 7 ,
a n d d iffe r en c e o f d e fe c t == *i 4- *i 4-* 3 4- 1 *44-4*6 4-1*6 4- 2*14-4*44- 2*2 4- 1 4 - * i = 1 7 * 9 .
T h e s e sh o u ld , o f c o u r s e , b e th e s am e , b u t th e a r e a o f th e C u r v e o f P r o b a b i l i t y in b lo c k fo rm is n o t
a c c u r a t e ly 1 7 1 , b u t is 7 1 1*9 , an e r ro r o f a b o u t i p e r cent.
T h e m is fit , th e n , is 34*9, an d th e p e r c e n ta g e m is fit is
34*9 x 10 0 3490
— — ------= ^ — = 20 a p p ro x im a t e ly .
In a l l th e c a se s in th is w o r k th e v a lu e s o f th e M a th em a t ic a l c o n s tan ts h a v e b e e n o b ta in ed in this
m an n e r , b u t p ra c t ic e e n a b le s s e v e ra l sh o r t cu t s to b e t a k e n w h ic h s im p li fy t h e w o r k t o a s lig h t e x t e n t .
I t h a s b e e n th o u g h t a d v is a b le to p re s en t th e w o r k in g h e r e in th e fu lle s t d e ta il.
TH E NAGADA SERIES. (See p. 77.)
(S p e c ia l N o t e by D . R a n d a ll-M a c lv e r ■ •)
I t is im p o s s ib le t o a l lo w D r . C i c e l y F aw c e t t ’s e s s a y o n th e N a g a d a c r an ia (B iom e tr ik a , v o l. i. no . 4)
t o p a s s w ith o u t c r itic ism . I t w o u ld b e o u t o f p la c e , in d eed , t o em p h a s iz e m in o r d e ta ils o n w h ic h w e
d is a g r e e w ith th e a u th o r an d h e r c o l la b o r a to r s ; b u t i t is n e c e s s a r y t o p o in t o u t th a t th e r e a re
fu n d am e n ta l e rro r s o f t re a tm e n t w h ic h c om p le t e ly d e s t r o y a n y v a lu e t h a t m ig h t b e a t ta ch ed t o th is
la b o r io u s s tu d y . T h e a u th o r s m a n ife s t th r o u g h o u t t h a t p e c u lia r p re ju d ic e , w h ic h s e em s in e r a d ic a b le
in th e w o r k e r s o f th e ir s ch o o l, fo r s u p p o s in g a n y s e r ie s o f o b s e r v a t io n s w h ic h com e u n d e r th e ir n o t ic e
to b e h om o g e n e o u s in ch a r a c te r . T h e r e is n o th in g w h a ts o e v e r t o ju s t i f y th is p r e ju d i c e ; in d e ed , in
a n th r o p o lo g y th e a p r i o r i p ro b a b ilit ie s a r e r a th e r o n th e s id e o f th e o p p o s i te h y p o th e s is . M o s t
a n th r o p o lo g is t s w o u ld fin d i t h a rd t o n am e a n y s in g le r a c e w h ic h is u n q u e s t io n a b ly o f p u re s to c k , a n d
c e r ta in ly th e o n u s p r o b a n d i m u s t a lw a y s l ie w ith th o s e w h o w o u ld m a in ta in i t t o b e so . B u t th e
B iom e t r ic ia n s ig n o r e a l l h is to r ic a l an d a ll a r c h a e o lo g ic a l e v id e n c e , an d h a b i tu a l ly a s sum e n o t o n l y th a t
s e r ie s o f a s in g le lo c a l i t y o r o f a s in g le d a t e a r e r ep r e s e n ta t iv e o f a n e n t ir e c o u n t r y , an d o f t h a t c o u n t r y
th r o u g h o u t it s w h o le h is to r y , b u t a ls o t h a t t h e y r ep r e s e n t in e v e r y c a s e a s in g le an d h om o g e n e o u s
s to c k . T h u s , o n p. 409 o f th e m em o ir in q u e s t io n , th e a u th o r s s ta te t h a t ‘ th e m a n n e r in w h ic h
v a r ia t io n is d e a lt w ith , e v e n in su ch a c la s s ic a l w o r k a s R u t im e y e r a n d H is ’s C r a n ia H e lv e tica , is
a s to n is h in g t o th e s ta t is t ic ia n w h o h a s r e a liz ed th e d is t r ib u t io n o f a n y c h a r a c te r in a h om o g e n e o u s
p o p u la tio n ,’— a s th o u g h h om o g e n e o u s p o p u la tio n s w e re r e a d i ly to b e fo u n d . A n d o n p. 4 1 4 t h e y
t r e a t a s ‘ a dm i t te d ly h om o g e n e o u s m a te r ia l ’ su ch s e r ie s a s B r o c a ’s m e a su r em en ts o f th e s k u lls in th e
P a r is c a ta c om b s a n d P r o f. T h a n e ’s m e a su r em e n ts o f W h i t e c h a p e l s k u lls . S u r e l y i t d o e s n o t n e ed an
e x p e r t to p ro v e t h a t th e F r e n c h an d th e E n g l i s h p e o p le s a r e com p o s e d o f m o re th a n o n e r a c e . W e r e
i t n o t fo r th e b lin d n e s s w h ic h is m a n ife s te d in su ch a c a s e a s th is , w e m ig h t b e a s to u n d e d a t th e
a s sum p tio n (p . 4 1 2 ) th a t o n l y o n e r a c e - t y p e h a s e x i s t e d in I t a ly , o r a t th e t em e r i t y w h ich a llow s th em
t o t r e a t a s p u re s e r ie s r e s p e c t iv e ly (p . 4 2 5 ) n o t o n l y T h e b a n m um m ie s an d m o d e rn E g y p t ia n s k u lls
from a c em e t e r y o f w h ic h th e h is to r y is u n k n ow n , b u t e v e n m o d em n e g ro e s from N o r th A f r i c a , a n d
w o r s t o f a l l a m is c e lla n y o f n e g ro e s from su ch d iffe r e n t lo c a lit ie s a s A lg i e r s , th e S o u d a n , an d th e W e s t
C o a s t o f A f r i c a . A f t e r th is i t s e em s a lm o s t v e n ia l t h a t t h e y s h o u ld su p p o s e m e r e g e o g r a p h ic a l
p ro p in q u ity to ju s t i f y th em in c la s s in g c e r ta in p o p u la t io n s a s a llie d a n d o th e r s a s n o t a llie d to t h a t o f
N a g a d a .
T h e a u th o r s , th e n , h a v e a p p a r e n t ly n o t a t t em p t e d t o t e s t th e m a te r ia l s e le c ted fo r c om p a r is o n b y
e v e n su ch c u r s o r y p ro c e d u r e a s t h e y h a v e a p p lie d t o th e N a g a d a s p e c im e n s ; t h e y h a v e s im p ly
a s sum e d t h a t i t w a s o f su ch a c h a r a c te r a s t h e y w o u ld h a v e w ish ed i t t o b e fo r th e p u rp o s e s o f th e ir
w o r k . A s t o th e a c tu a l s u b je c t o f th e ir s tu d y , v iz . th e s k u lls from N a g a d a , t h e y w e r e in d e e d a l i t t le
le s s h a s t y , an d s ta te d e x p l i c i t l y th a t i t w a s im p o r ta n t t o e x am in e w h e th e r i t m ig h t fa i r ly b e t r e a t e d a s
h om o g e n e o u s o r no t . B u t th e e x am in a t io n w h ic h w a s in s t itu te d w ith th is o b je c t w a s m o s t in a d e q u a te
an d su pe r fic ia l. W h e n i t is a p p re c ia te d t h a t th e h om o g e n e i t y o f th is s e r ie s w a s th e in d isp en s a b le
fo u n d a t io n o f th e ir w o r k , i t s e em s in c r ed ib le t h a t t h e y s h o u ld h a v e th o u g h t i t su ffic ien t t o t e s t i t b y
o n l y tw o o u t o f a l l th e n um e ro u s fe a tu re s w h ic h w e r e a t th e ir d isp o sa l. T h e s e tw o m o re o v e r , th e
h e a d - le n g th a n d h e a d -b r e a d th , w e r e a p e c u lia r ly u n fo r tu n a te s e le c t io n , th o u g h n o d o u b t th e a u th o r s
m a y b e p a r t ly e x c u s e d b y t h e fa c t t h a t an th r o p o lo g is t s h a v e lo n g a t t a c h e d an e x a g g e r a te d im p o r tan c e
t o th e c e p h a lic in d e x . E v e n h a d t h e y s u c c e e d ed in p ro v in g th e s e r ie s t o b e h om o g e n e o u s in r e g a rd
t o th e s e tw o fe a tu re s— a s in o u r o p in io n t h e y h a v e h o t— i t w o u ld s t i l l b e q u ite u n ju s t ifia b le to su p p o s e
w ith o u t fu r th e r e x am in a t io n t h a t a l l th e o th e r fe a tu re s d is p la y e d h om o g e n e ity .
A n d y e t i t w a s th e m o r e n e c e s s a r y t o fu l l y d em o n s t ra te th e h om o g e n e it y o f th e s e r ie s b e fo re
p ro c e e d in g to fo u n d co n c lu s io n s u p o n t h a t b a s is , in a sm u ch a s th e a r c h a e o lo g ic a l p ro b a b ilit ie s w e re
r a th e r a g a in s t th an in fa v o u r o f th e h y p o th e s is . T h e s p e c im e n s from N a g a d a r a n g e (a s w a s fu l ly
r e c o g n iz e d b y th e t im e D r . F a w c e t t ’s m em o ir w a s p u b lish e d ) o v e r th e w h o le o f th e p r e -d y n a s t ic an d
a t th e le a s t a g r e a t p a r t o f th e p ro to -d y n a s t ic t im e . T h e y o u g h t th e r e fo r e t o h a v e b e e n t r e a t e d a s
r ep r e s e n ta t iv e , n o t o f a s in g le p e r io d , b u t o f tw o o r t h r e e ; o r , i f i t w a s im p o s s ib le to a v o id com b in in g
th e p e r io d s , i t s h o u ld a t le a s t h a v e b e e n r em em b e re d t h a t th e fa c t o f th e s e r ie s b e in g th u s c om p o s i te
w a s a n a d d i t io n a l re a so n fo r s u sp e c t in g it s h om o g e n e it y . T o su ch c o n s id e r a t io n s , h ow e v e r , th e a u th o r s
f>aid n o h e e d , an d t h e y a t t a c h e d a p p a r e n t ly s o l i t t le im p o r ta n c e t o th e a r c h a e o lo g ic a l e v id e n c e o f
d a t in g w h ic h w a s su p p lie d to th em t h a t th e y h a v e n o t e v e n p u b lish e d i t fo r th e b en e fi t o f th e ir
re a d e r s . I f t h e B iom e t r ic ia n s w is h to c o n v in c e c r a n io lo g is t s t h a t th e ir w o r k is o f s c ie n t ific v a lu e t h e y
w ill n e e d t o t r e a t i t in a d iffe r e n t s p i r i t from t h a t w h ic h is e x h ib i t e d in th e m em o ir o n th e s k u lls from
N a g a d a .