o f co n v e n ien c e . A n o th e r w h ich is s om e tim e s u sed , a n d w h ic h h a s a d v a n ta g e s from th e a n a ly t ic a l p o in t
o f v iew , m a y b e a r r iv e d a t b y r e a so n in g a s f o l lo w s :—
C o n s id e r th e e x p a n s io n o f th e e x p r e s s io n ( 1 + x ) n w h e n n is v e r y l a r g e ; i f w e a s sum e a v a lu e h
fo r th e la r g e s t c o e ffic ie n t in th e e x p a n s io n , i. e. i f w e a s sum e a d e f in i te v a lu e fo r th e m a x im um o rd in a te
in th e cu r v e , th e v a lu e o f th e p™ t e rm o n e ith e r s id e o f th e m a x im um te rm c a n b e sh ow n t o b e
p i
k x e » , w h e r e e is th e b a s e o f th e N a p ie r ia n L o g a r i thm s a n d e q u a ls 3 -7 18 2 8 18 3 . . . .
I f w e e x p r e s s th is in t h e o r d in a r y la n g u a g e o f A n a l y t i c a l G e om e t r y w e s a y th a t th e e q u a t io n o f th e
C u r v e o f P r o b a b i l i t y is y — h e n , w h e r e x , y a r e th e a b s c is s a a n d o rd in a te o f a n y p o in t o n th e
cu r v e , i. e . it s d is ta n c e s f r om tw o f ix e d p e rp e n d ic u la r s t r a ig h t lin e s . I t is a w e ll-k n o w n th e o r em
in G e om e t r y t h a t a n y e llip s e c a n b e tra n s fo rm e d in to a n y o th e r e llip se , o r in to a c ir c le , b y su ita b le
c h a n g e s in th e u n i ts o f le n g th t a k e n in m e a su r in g th e a b s c is s a e an d o rd in a te s . I n e x a c t l y th e s am e
w a y a l l C u r v e s o f P r o b a b i l i t y c a n b e t ra n s fo rm e d in to o n e an o th e r, o r in to s om e S t a n d a rd C u r v e
o f P r o b a b ilit y . T h e m o s t o b v io u s fo rm t o t a k e fo r th e s ta n d a rd , from th e a n a ly t ic a l p o in t o f v iew , is
th e o n e in w h ic h th e e q u a t io n is red u c ed t o it s s im p le s t fo rm , n am e ly y — e~ a*. T h i s h a s o f te n b e e n
t a k e n a s th e s ta n d a rd fo rm , b u t t o a v o id th e e x t r em e sm a lln e s s o f th e c u r v e a s d e fin e d b y th is
e q u a tio n , a d if f i c u lt y w h ic h is u n a v o id a b le u n le s s th e u n i ts o f m e a su r em e n t a r e in c o n v e n ie n t ly la r g e ,
th e o rd in a te s a n d a b s c is s a e a r e e a c h m u lt ip lie d b y th e s am e n um b e r . T h i s p re s e r v e s th e fo rm o f th e
c u r v e , w h ile w e c a n d e te rm in e i t s s iz e a t w ill.
N O T E I I I .
T h e C a lcu la tion s by w h ich a C u r v e o f P r o b a b ility i s a d apted to a n y r e q u ir e d S e r ie s . (S e e p . Jo.)
A s s h ow n in A p p e n d ix , N o t e I I , b y a lt e r in g th e u n its o f m e a su r em e n t in a p a r t ic u la r m a n n e r a n y
C u r v e o f P r o b a b i l i t y c a n b e t ra n s fo rm e d in to a n y o th e r . T h e s ta n d a rd fo rm o f th e a n a ly t ic a l e q u a t io n
is y = e~**, a n d b y a s u it a b le t ra n s fo rm a t io n th is c a n
b e t ra n s fo rm e d in to y — b e d' , w h en b a n d d m a y
h a v e a n y v a lu e s w e w ish to a s s ig n t o th em . T h e
A
fo rm w h ic h w e u s e is y =■ e w h e r e A a n d c
h a v e m e an in g s w h ic h a r e e x p la in e d b e low . T h is fo rm
h a s m a n y p ro p e r t ie s o f w h ic h w e c a n t a k e a d v a n ta g e .
T w o q u an t it ie s , th e n , a r e r e q u i r e d to c om p le t e ly d e te r m
in e a C u r v e o f P r o b a b ilit y , n am e ly A . an d c. A s
s h ow n la te r , th e r e a r e fo u r q u a n t it ie s , c, tj, r , an d
e , a n y o n e o f w h ic h s e r v e s t o d e te rm in e th e res t , as
t h e y a r e co n n e c ted b y n um er ica l, r e la t io n s . C o n s
e q u e n t ly a C u r v e o f P r o b a b i l i t y is d e te rm in e d w h en
w e k n o w A a n d o n e o f th e fo u r cy rj, r , e. T h e
a c c om p a n y in g fig u r e w ill h e lp to s h o w w h a t th e s e
m a th em a t ic a l s ym b o ls r ep re sen t . T h e fig u r e is th a t
g o t fo r th e V e r t i c a l In d e x o f t h e M a le s o f th e S ix th to
T w e l f th D y n a s t ie s , an d th e v a lu e s o f th e q u an t it ie s a r e
sh ow n o n C h a r t I, F . a .
A is th e a r e a co n ta in e d b e tw e e n th e c u r v e an d
th e a x i s o f x . T h e o th e r s ym b o l, c, is c a lle d th e
m o d u lu s o f t h e cu r v e , a n d h a s m a n y im p o r ta n t p ro p e r tie s , b u t th e s e c a n o n l y b e e x p la in e d b y co n s id e r in
g th e p ro p e r t ie s o f t h e c u r v e ; it is o f c o u r s e th e r a t io o f th e c ir cum fe r en c e o f a c ir c le to its d iam e te r .
W h e n th e C u r v e o f P r o b a b i l i t y t a k e s th is fo rm th e o rd in a te , y , a t a n y p o in t, x , d o e s n o t r ep r e s e n t th e
c h a n c e th a t a n o b s e r v a t io n s h a ll g iv e a v a lu e r ep r e s e n ted b y th e p o s itio n , x , o f th a t p o in t . W e a r e n o
lo n g e r p r o c e e d in g b y d iffe r e n c e s m e a su r ed in u n its , b u t n ow a n y v a lu e w h a te v e r (w ith in r e a so n a b le
lim its, o f co u r s e ) is p o s s ib le . F o r e x am p le , in th e e x p a n s io n o f th e b in om ia l ( 1 + x ) n th e o rd in a te s
a r e su p p o s ed to b e u n it d is ta n c e a p a r t ; n ow t h e y a r e c lo s e t o g e th e r , a n d th e a r e a b e tw e e n tw o
o rd in a te s r ep re sen ts th e ch an c e th a t a n o b s e r v a t io n w i l l g iv e a v a lu e b e tw e e n th o s e r ep r e se n ted b y th e
fe e t o f th e o rd in a te s . S u p p o s e w e t a k e tw o o rd in a te s v e r y n e a r a n d v e r y n e a r ly e q u a l, o f h e ig h t y ,
an d su p p o s e th e d is ta n c e s o f th e ir fe e t from th e o r ig in o f a b s c is s a e a r e x a n d x + d x , w h e r e d x rep re -
A **
s en ts a sm a ll in c r em e n t t o x : th e n y d x o r — - e ,2 d x rep r e s e n t s th e c h a n c e t h a t an o b s e r v a t io n
e V it
g iv e s a v a lu e b e tw e en x a n d x + d x , o n th e u n d e r s ta n d in g t h a t c e r ta in t y is r ep r e s e n ted b y A ; in o th e r
w o rd s , A is th e to t a l n um b e r o f o b s e r v a t io n s co n s id e r e d . T h e c h a n c e t h a t a n o b s e r v a t io n s h o u ld g iv e
a v a lu e l y in g b e tw e en x 1 a n d x 2 is t h e sum o f a l l th e e x p r e s s io n s y d x b e tw e en x x a n d x 2, w h ich
is g e n e r a lly w r it te n J y d x , o r
—=rC c* d x . T h i s th e r e fo r e is th e n um b e r o f t im e s w e m a y
e x p e c t to g e t v a lu e s b e tw e e n x x an d x 2 w h e n w e t a k e A ca se s . I f w e t a k e th e lim it s t o b e a t
in fin ity in o p p o s i te d ir e c t io n s th e r e s u lt o f th e sum m a t io n is , o f co u r s e , A .
N o w co n s id e r th e v a lu e g iv e n b y a n y o b s e r v a t io n : s u p p o s e i t d iffe r s from th e m e an o f a ll
th e m e a su r e s b y 8 ; th e n 8 is th e ab s c is s a , th e v a lu e o f x , fo r th is m e a su r e . T h e e r ro r o f m e an s q u a re
is d e fin ed a s th e s q u a re r o o t o f th e a v e r a g e o f th e sq u a re s o f th e e r ro r s , a n d c o n s e q u e n t ly is th e v a lu e
o f d2, w h e r e 2 d e n o te s th e sum m a tio n fo r a l l th e o b s e r v a t io n s w h ic h a r e n in n um b e r . F o r
v « A *
o b s e r v a t io n s f i t t in g th e c u r v e y = — y=.e~ th is is, in th e la n g u a g e o f th e I n t e g r a l C a lc u lu s , th e cVir ’
s q u a re r o o t o f
S m J H K e e* d x . x 2
* -/S T h u s , c a l lin g € th e e r r o r o f m e a n s q u a re , o r th e
s ta n d a rd d e v ia t io n « = T h e a v e r a g e o f a l l d iffe r en c e s , 8, from th e m e an , m u s t , o f co u r s e , b e
z e ro , from th e d e fin itio n o f th e w o rd * m e an .’ I f , h ow e v e r , a l l th e s e d iffe r e n c e s a r e s u p p o s e d p o s i t iv e
w e g e t a n a r i thm e t ic a l a v e r a g e o f th e e r ro r s , w h ic h is — — =
■
J 0 <
V TT
c a l lin g th is a r i thm e t ic a l a v e r a g e o f th e e rro r s 77, w e h a v e tj = T h e p ro b a b le e rro r r is vn
d e fin e d t o b e su ch t h a t h a l f th e c u r v e (i. e. h a l f th e e x p e c te d o b s e r v a t io n s ) lie w ith in a d is ta n c e r o n
e ith e r s id e o f th e m e an . F o r th i s w e m u s t h a v e / — r ~ d‘ d x = \ A , an d th e v a lu e o f r is fo u n d
J -r cVir
b y t r ia l an d in te rp o la t io n t o b e -476 9 c. W e h a v e th u s fo u r q u a n t it ie s , c, e, rj, an d r , c o n n e c ted
b y d e fin i te n um e r ic a l r e la t io n s : w h en a n y o n e o f t h e s e is fo u n d th e o th e r th r e e c a n b e c a lc u la t e d . A s
is sh ow n in th e n e x t n o te , w e s h a ll fin d rj th e a r i thm e t ic a l a v e r a g e o f t h e e rro rs .
T h e n c, th e m o d u lu s = rj f i t — 1*7725 r j;
e, th e e r r o r o f m e a n sq u a re , o r th e s ta n d a rd d e v ia t io n , = = v ? : 1 * 2 5 3 3 7 ,;
an d r , th e p ro b a b le e rro r = *4769 c = *8453 V — ‘ 6 7 4 5 «•
A s a co n se q u en c e o f this, tw o q u an t it ie s , A an d o n e o f c, ij, r , a n d e, c om p le t e ly d e te rm in e a n y
C u r v e o f P r o b a b i l i t y . T h e q u e s t io n n ow is h o w t o fin d th e v a lu e s o f th e s e w h e n w e a r e g iv e n a n y
s e r ie s o f o b s e r v a t io n s ; i. e . h ow a r e w e t o fin d th e C u r v e o f P r o b a b i l i t y c o r r e sp o n d in g t o a n y C u r v e
o f F r e q u e n c y . I t w ill a t o n c e b e se en t h a t A r e p r e s e n t s th e n um b e r o f o b s e r v a t io n s in th e s e r ie s ;
th e tw o c u r v e s th e n co r r e sp o n d in a r e a . T h e y a r e d r aw n in su ch p o s itio n s o n th e p a p e r th a t th e
m e an fo r th e tw o o c c u r s a t th e s am e v a lu e . I t is fo u n d t h a t i t is, in g e n e r a l, b e s t t o m a k e th e
m o d u li (d e n o ted b y c) o f t h e tw o cu r v e s , o r r a th e r th e s ta n d a rd d e v ia t io n s , from w h ic h th e m o d u li
c a n b e d e r iv ed , th e s am e fo r th e tw o cu r v e s . T h e p r a c t ic a l m e th o d b y w h ic h th i s is d o n e w i l l b e
e x p la in e d in th e n e x t n o te .