N O T E IV .
A n A c co u n t o f a l l th e S ym b o ls employed in th e M a th em a tic a l C a lcu la tion s belonging to th e C u r v e s o f
P r o b a b ility , i llu s t r a t e d by th e complete d e ta ile d W o r k in g o u t o f a S e r ie s .
T h e s ym b o ls w h ic h w e u s e a r e A , M , r\, r , a n d c. A n o th e r o n e h a s b e e n m en t io n ed in th e t e x t , e.
A is th e a r e a o f th e C u r v e o f F r e q u e n c y , o r th e C u r v e o f P r o b a b ilit y , th e s e tw o c u r v e s h a v in g
th e s am e a r e a . L o o k e d a t from a n o th e r p o in t o f v iew , A is th e n um b e r o f s p e c im en s in
th e s e r ie s u n d e r c o n s id e r a t io n , s om e t im e s c a lle d n.
M is th e a r i thm e t ic m e a n o f th e ser ie s .
ij is th e a v e r a g e e rro r o f th e s p e c im en s ; i. e . i f w e t a k e a ll th e d iv e rg e n c ie s o f t h e s p e c im en s from
th e me an , a s p o s itiv e , a d d th em to g e th e r an d d iv id e b y n , th e n um b e r o f sp e c im e n s , w e g e t q.
« is th e e rro r o f m e a n squ a re , o r th e s ta n d a rd o f d e v ia t io n , a n d it s sq u a re , m u lt ip lie d b y n , = th e
s um o f th e squ a re s o f t h e e r ro r s , o r th e d iv e r g e n c ie s from th e m e an .
r is th e p ro b a b le e rro r , an d is s u c h t h a t e x a c t l y h a l f th e s p e c im en s l ie b e tw e e n th e lim its
M — r an d M + r .
S t r i c t l y s p e a k in g , c , a n d co n s e q u en t ly — ^=, a r e n o t in tr in s ic p a r ts o f th e C u r v e o f F r e q u e n c y ,
• cwn
b u t o n ly o f th e C u r v e o f P r o b a b i l i t y , a n d t h e y s e r v e t o d e te rm in e i t s sh ap e . A s sh ow n in th e
A p p e n d ix , N o t e I I I , q, e, r , a n d c a r e co n n e c ted b y a r i thm e t ic a l r e la t io n s in th e c a s e o f th e C u r v e
o f P r o b a b i l i t y . C o n s e q u e n t ly w e d o n o t a t t em p t t o fin d c u n t il w e h a v e fo u n d s om e o f t h e co n s tan t s
o f th e C u r v e o f F r e q u e n c y .
A s a l r e a d y s ta ted , w e t r y t o fin d th e C u r v e o f P r o b a b i l i t y w h ic h is n e a r e s t t o th e C u r v e o f
F r e q u e n c y , an d fo r th is w e s e e t h a t th e m e an is th e s am e , th e a r e a is th e sam e , an d th e p ro b a b le e rro r
o r th e s ta n d a rd d e v ia t io n (o r e rro r o f m e a n sq u a re ) is th e sam e . W h e n th e s e co n d it io n s a r e s a t is fied
i t is fo u n d th a t th e a v e r a g e e r ro r is a ls o v e r y n e a r ly th e s am e fo r b o th , i f th e n um b e r o f s p e c im e n s in
th e s e r ie s is a t a l l co n s id e r a b le . C o n s e q u e n t ly th e n um e r ic a l r e la t io n s w h ic h w e fo u n d in th e c a se
o f th e C u r v e o f P r o b a b i l i t y a ls o h o ld v e r y a p p r o x im a t e ly in r e s p e c t t o th e c o n s ta n t s o f a C u r v e
o f F r e q u e n c y .
N o w in a n y p a r t icu la r c a s e th e a v e r a g e e r r o r a n d th e s ta n d a rd d e v ia t io n c a n b e c a lc u la t e d fo r th e
C u r v e o f F r e q u e n c y , a n d from e ith e r th e p ro b a b le e r ro r c a n b e d e r iv ed b y m e an s o f o n e o f th e
n um e r ic a l r e la t io n s sh ow n in th e A p p e n d ix , N o t e I I I . I t w i l l b e fo u n d th a t th e r e su lt s o f th e tw o
m e th o d s a r e a lm o s t id en t ic a l.
T h e n um e r ic a l r e la t io n s a r e e = — n — - 7 = r = *4 7 ^ 9 c »
v 2 V ir
a n d th e r e fo r e c = 1 - 7 7 2 5 q, e = 1-253217, r = -8453 77, r = -674 5 e.
S t ie d a s a y s t h a t th e s e fo rm u la e c a n b e r e g a rd e d a s a c cu r a te fo r s er ie s o f n o t le s s th a n ten . C o n s e q
u e n t ly t h e y a r e a lw a y s a p p lic a b le in th is w o r k , a s w e r e g a rd a sm a lle r n um b e r th a n th a t a s q u ite
in a d e q u a te to fo rm a s e r ie s , an d w e u s e n o c u r v e s fo r s e r ie s o f le s s th a n th i r t y spe c im en s .
A n e x am p le w i l l m a k e th e a p p lic a t io n o f th is c le a r . I n th e s e r ie s fo r th e N a s a l In d e x , M a le s ,
R o y a l T om b s , th e m e a su r em e n ts a r e :
4 2 4 3 4 4 4 5 4 6 4 7 48 4 9 5 ° 5 * & 53 5 4 5 5 5 6 57
1 2 0 2 2 2 1 1 4 0 2 1 o 1 o 1
M = 48-50.
«a = ^ - = {6•I2+»x5•l2 + *x3•5|i!+axa•32+axI•31+IX•5|^
+ 1 x-j|2 + 4xi-3a + ^x3-5l2 + i x4-3' + i x6-j|!!+i X8-5I1}
»o
•5 + ! 3 ’5 + 13-5 + 4 9 + + 84-5 + 7™ 5 + 6°-5 •
••• « = 3-95
••• r = 3-93 x '6745 = s -67
H e r e
A g a in
_ 28 _ 6-5 + 2X5-5 + 2X3-5 + 2X2-5 + 2Xi -5 + -5 + -5+4 x i -5 + 2X3-5 + i x 4-5 + i x 6-5 + i * 8-5
n 20
i + 9 + 5 + U + 4-5 + 1 1 + i 3 + 8-5
.-. r = 3-3o x -8453 = 2-79.
In th is e x am p le th e v a lu e s o f r fo u n d b y th e tw o m e th o d s d if fe r b y le s s th a n 5 °/0 o f th e i r v a lu e .
A s th e n um b e r o f e x am p le s in th e s e r ie s in c r e a se s , th e d iffe r e n c e b e tw e e n th e tw o v a lu e s o f r
d e c r e a se s , an d th e la b o u r o f fin d in g r b y th e fir s t m e th o d in c r e a se s e n o rm o u s ly . C o n s e q u e n t ly
in th is w o r k w e h a v e c o n te n t e d o u r se lv e s w ith u s in g r a s fo u n d b y th e s e c o n d m e th o d .
W e w i l l n ow g iv e a n a c c o u n t o f th e p ro c e d u re a d o p t e d in th is w o r k to find th e n um e r ic a l v a lu e s o f
th e s ym b o l s e x p la in e d a b o v e .
I t w ill b e a d v is a b le t o s ta r t w ith a s h o r t r f c u m t o f th e p ro c e d u r e a d o p t e d , an d th e n t o g iv e
a n e x am p le fu l l y w o r k e d ou t. W e f ir s t d e te rm in e th e m e an , M , o f th e s e t o f o b s e r v a t io n s u n d e r
c o n s id e r a t io n , th e n um b e r o f in d iv id u a ls in th e s e r ie s b e in g n. T h e n c a lc u la t e 77 = — , an d from th is
fin d r — -8453 V anc* c = 1 - 7 7 2 5 77. T h e n fin d — 7= = b s a y , a n d R = —7= .
c v it * Jn
T h e t ru e v a lu e o f th e m e an is th e n g iv e n a s M + R , an d th e v a lu e s o f M + r a r e c a lc u la te d ,
b e tw e e n w h ic h h a l f th e in d iv id u a ls c om p o s in g th e s e r ie s s h o u ld lie . T h e e q u a tio n o f th e co r r e sp o n d in
g C u r v e o f P r o b a b i l i t y is y = b e ~ * . A few p o in ts o n th is c u r v e a r e fo u n d , a n d th e c u r v e d r aw n in
th e o rd in a r y fo rm . F r om th is th e c u r v e in b lo c k fo rm is d r aw n , a n d to g e th e r w i th th e F r e q u e n c y
C u r v e is rep ro d u c e d . T h e tw o cu r v e s a r e com p a r ed , a n d th e p e r c e n t a g e o f m is fi t c a lcu la ted .
T h e e x am p le w o r k e d o u t h e r e is t a k e n from th e V e r t i c a l I n d e x o f th e M a le s o f th e S i x t h
to T w e l f th D y n a s t ie s . T h e s p e c im e n s a r e a r r a n g e d th u s :—
6 6 6 7 68 69 70 7 1 7 2 7 3 , 7 4 7 5 7 6 7 7 78 7 9 80 8 1 82
i i 2 6 7 1 0 2 5 2 2 2 3 2 7 2 1 10 6 3 5 i i ( 1 7 1 ) -
The mean, M , = {1 x 66+1 x 67 + 2 x68 + 6 x 69 + 7 x 70+10x 71 + 25 x 72 + 22x 73 + 23x 74
+ 27 x 75 + 21 x 76 +10 x 77 + 6 x 78 + 3 x 79 + 5 x 80 +1 x 81 +1 x 82}
-5-171 = 1—265—0 1671 ' 171 = 73—/Ji7i
= = 73.98.
if, the average error,
= { I X 7-98 + I X 6-98 + 2 X 5-98 + 6 X 4-98 + 7 X 3-98 + IO X 2-98 + 2 5 X 1-98 + 22 X -98 + 2 3 X -02
+ 27 X 1-02 + 2 1 X 2-02 + 10 X 3-02 + 6 X 4-02 + 3 X 5 -0 2 + 5 X 6-02 + I X 7*02 + I X 8-0 2 } - 5-171
_ 18 5 -5 2 + 18 4 .9 4 v
171
The reason that the sum of the errors in excess is not exactly equal to the sum of the errors in
defect is because the mean, M , is not exactly 73-98.
We have, then, rj = = 2-166 = 2-17 say.
Consequently r, the probable error, = 2-166 x -8453 = 1*831 = 1-83 say.
c, the modulus, = q f it == 3-839 = 3*84 say.
And R H n H g j 1 .140.
wn V171
A 1 171
c V i=t = — 5 (since c — tj y 7ri) S3 2■*■ 166'— xir = 205- 130.