On the relationship between the approximate moments of an observed distribution and its functional
moments
by Chester H Scott
A THESIS Submitted to the Graduate Committee in partial fulfillment of the requirements for the
degree of Master of Science in Mathematics
Montana State University
© Copyright by Chester H Scott (1949)
Abstract:
The traditional development of Sheppard's formula is as fol- lows. Choose the class interval as a unit of
x and lat xt be the mid-abscissa of the tth class, then the approximate moments as directly computed
from the observed distribution are defined by (Formula not captured by OCR) where g(x) is the
observed frequency, N is the total frequency, and K is the class width. The functional moments are
(Formula not captured by OCR) where f (x) is the theoretical relative frequency. The solution involves
a relationship between v'n and u'n which is (Formula not captured by OCR) SInce this approach
involved relatively complex mathematics, it seemed desirable to simplify the process. The writer did
this by use of the moment-generating function concept. Consider a frequency distribution f (x) divided
into any convenient number of classes whoe class mark is x1 and class width K. And further let ς be the
error in using the class mark x1 instead of the var- iable x. Then (Formula not captured by OCR) were
x and ς are independently dis- tributed, x is the variable before grouping and x1 is the vari- able after
grouping. Then (Formula not captured by OCR) After expending and simplifying the above reduces to
results identical to that which Sheppard obtained. t\y s
i p p a o m m r m . m m m s ' : # a# o g # m w ~
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S IS T OF TABLES
TABLE
PAGE'
c l.
Ten T h o u san d T im es t h e A rea u n d e r a " P e r s o n ..
Type I i I C urve (sit-h<X g ~ '0-*5)
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IX .
T en 'T h o u san d Time's t h e A re a u n d e r -a F e a rso m
Type ’ I I I C urve (w ith x . g.
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I I I i ■ One H undred T housand T im es, t h e A rea u n d e r, a
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C o m p ariso n o f t h e M n c o r r e c te d 5 C o r r e c t e d ^ T h e o ^ •:
i e t i c a l - l y E x p e c te d Sfomehts f o r Tables" I 5- I I 5 '
I l f . . . ... ...
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13
LIST. OF F I G M B m
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FIGURE
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F re q u e n c y D i s t r i h u t i o n ; . » . * . . . . . . . . . . . . . . . . . . . . . . . . .
, PAGE
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ACKNOWLEDGMENTS.■
It gives= me a great deal of pleasure to 'acknowledge the as^
distance rendered by my many, friends, in the preparation of, this
paper*
I wish particularly to thank Dr* J* J 0 Livers, Montana
State College,, for his -encouragement and assistance.:
S
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marmegr
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e W the-errs# && %s.l%g .## ISjLgueie-TKsaadbL %* aaeteaa -sf 1%» vaa^
SSaKbdLet %». Then ae* * %
c * .where x a#&'-% are
t e j W W & y % &&' t w variable W W f e
t m 3Eg. Ie tee 9arl«*
WKle after g % w * 6 % . .%hea '
'- .'
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M
#
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after eoewaaiag eaG
# o v e retWeg to ^eaol-W'
Meatleal to that #Magh.Shegopard ^talae^L.
% &&e
f&oeea# Gf IstezyeeB lag th e Sate o f a
aeap le, oa» fla a # I t aeairah le t o eh ta ia some o f I t s s t a t i s t i c
&el aomeatsv m e a Gaallae with IaBga g o a a tltlo e o f &at% I t I s
o fte a ooBtealaat to erreags t&la date Iato fragaaaay te h lo s %&tb
a aoa&ey o f sgaal c la s s la teevala*
(Tbls la Goae to make th e
o e lo a la tlo a o f aoaeato l e s s laborious*)
I t I s th e ^reotlGs o f
a t e t l s t i e l a a s t o regard eaoh element o f a 0I&9# as h sela e She
value o f the rnldmabseless o f th a t olaas#
She pzeeeaure is*Q&ve$
lB tr # u s » s some egeaMw In IKtie f i n a l reaulte*
I t I s the
ebdeatlifa lie develop fo rau iee to eoBaeet t h is error*#0* moat date errangM l a faeguanay tab les, 6%&@*i6&*e In*
a is e t m th a t one Sbouia use fro* 1# to 20 slasbea*
W
femar
would lead to a lo s s o f aoeuraey? ehereaa more e@ul& only te&a
to . uaaeaesasr&ly laer&aa# th e a r itb a o tle lav8&#e&*
I t 085 1b<&geld th a t say group o f data one ooasldara I s dust
a portion o f 4* Tnswxf l8 % e' population* fwajOdedL the p srea t popular
tic s *
TSSte e r a # o h a r o u t a r l s t l o s o f t h i s l a r g e r g ro u p m y n o t Tbet
a#o*m* b u t I t s&a Bometlmoo Ibet re p z e se & te a by e t h o o r e t l u o l f r e *
queaey d lstn lb u tlo a o f a om tlaaoue variable a aa& Ibe le a o te l Ibgr
fts).
I t l a l e f l u e a a s t h a t JOaaierbjLsdt Jfoar W blsh / " f W & e at
where a ana Ta are Wo v a lu es (RE" at w ith a<b @n& PlA^a<b] represonta
the p ro b a b ility o r th eo ro tlo a l r e la t iv e frequency w ith which %
'w ill f a l l between a end bs providing th e date or gasples have been
chosen a t random.
TBbe t o t a l are# under th e graph o f SOwt dihtrlbu^
tloa fgactioa f{%)
above the
Is a e f i & M as tu&lty*
therefore many fu&etlona &oal& sot aayve as BBthemetleal ato&ele
for an observes distribution*
. - -_ .
Zor classified ■data,^ the &th sample moment about -Wm origin
b.
-i
h
.
■
Ie defined by m. * - Z _ %(f** where a is the total fregue&oy* h
&
S 1*1 I a
la the number of intervals, % ia the class mark of the Ith class
interval* and f Is the freqqsB&y for tbs Ith latsrvel* $he the* ■*■
"
^
oretioal Ktb moment abeot the orl$la* denoted by s. is defined
-
by 8* #
where f(z) is the theoratleel relative fre&
*
,
,
queasy and is defined over the Interval (a»bv*
■
.
='
Sba first moment* a maaaere or oentrel ten&ea&y* is oal.leSF
the mean end may be denoted by
vend
I
or % If the data Is eiesaified
If fro# a theoretical distribution; m* or % e & ]&_
b
l
b
i%l
end al » Trxf (x}dk* '
A
a
'
the second moment Is e measure of variation*
v
. t.
I& the latter
ease it is often ooBvenlsst to red&o* this measure to the same
unit as that of the data*
Bherefors f#" a a is usually nsed,
where mu la d efin ed by a* = -
* This so&Gally 1& the
*
—
S n
^=«1 “■'
aeoo&d moment about the mean* The qaantlty s Ie oalled the stand­
ard deviation*
defined by a . -
The segood theoretidal moment about the mean Is
b
«
/ (x -e l I'CfffiEjdsK* % e at@nd#rd d ev ietlo a applied
^
a
here i®cr& r&g*
x
,
";
' •
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■
>
9&e third moment about the mean m y be used as a measure of
skewness of the distribution*
Bhaar as aymmetrloel distribution this.
moment Is sero, beoouse* for each deviation ZTaea; the right of the
TTlT
a»ea there la
equal devietlgB from the l e f t o f the mean, ao
th at i#ea th ese tie via 11 one are @mb@# a©6' m ultiplied, hy the el-sss
frequaBcy they #111 eaaeel ea&b ot#br l a the agwaatioa*
%W%ver*
I f the I l s t r l W tip© has a ion# r&&k& t e l l ( t h is i s #@i& t o he
Skemaesa to the righ t)* the W ilrl aameat #111 he p oeitlT s heeawae
th ese la##* p o s itiv e ^evlatio&s when @##a @a& m tlti# lie 4 by th e ir
Glees frequencies w i l l mare than avepbaleag# the r e la t ly e ly sm ell*
e r (ImriBtiOhs on the l e f t *
I a order th at th is meaeore he IodepeBd-
e a t o f the u n it o f z aod a ls o l&aepeBda&t o f th e Gholoe of o r lg lo
B ke*aaas er«A * may be d e f in e d
aa&cKg #
& % _ * . * S g f o r grouped d a t a
fo r the model f i x ) .
Q wltafoftea the fourth aom&Bt about the mesa i s used to d etarjalma the ext&Bt o f pea&ednes# o f th e graph.
I f two d lstrlb utioB S
have the same standard d eria tlo a but one of them has # very la r g e
pereeatage of I t s data ooaGeatreted about the mesa* t&ea th at oae
m ill u su a lly have considerably lo a g er t a i l s to cempaesate fo r th e
egaeeatretiom o f data about the a ea a .
Because o f th e heavy GeBtrl-
button o f th e fourth powers o f these t a i l s in the fourth moment?
the peaked d istr lb u tlo a w ith le&g t a i l s w ill tend to here a r e la ­
t iv e ly la rg er fourth aoment about the mesa.
f o l i o s t h a t t h i s i s alw ay s tru e *
I t does net n sc se se r lly
D i s t r i b u t i o n f u n c tio n s e e a be
constructed fo r which t h is la not true*
however* the fourth moment
la u se fu l la the great majority o f eases tp deters&ae a measure o f
peskedaesa*
In order to obtain & measure o f paakedneas? often
spekea o f as kwrt&sia, which la lndepeadest o f th e GO&eiderad s a l t
'o f
d iir id e s t i l l s f m r t h m & m t W c r * T f ^ . re**
12
4
p r e s s m ts t h i s m ssstire o f k&ztoa&p*
2% r
fo r o le s s lfle d
4'
S
d a ta EKBKlc*,, s
jg % f o r t&e t k o o r e t l o a l f r s q a s s o y o sr im * JEti m ost
* ' n& ^ 4
•'
'
elomsstary trea^mmts. of the aabjeot writers maim, at# eqnarat# 1#*
torprototlo&s Of momeats beyond t&& fomMSk*
l e t o s esmamo tk@s Ibke d&ta ke&ag ook aM ^ rad % ss a r r a n g e d
into freq%$&oy tabled# Ia aaoh & aaee the ramge o f vsrlatloa o f a:
i$oiadLd be divided Into a mamber o f eqaal elea# Intervald^ 38a# elm^
of 0@l#)let&ena o f moments i& I s the prsotloo to regard
seek freqaenoy o f the olass as having the valw o f the elee# mark
<xf the lntarimi la #&le& i t faIls^
error l a th e f in a l r e m its ^
$bie method Intro&aae# sow
I t I n t h e p n r^ oae o f SB ia t b e e l e t o
develop formulae *16 oorreot th is errors
#H@re h a s b&ea o o a e ld a r a b le w o r k d o n e o # t h e S u b jaot^ b a t i n
most IaetBacea the developmabt o f th@@e formulae has Involved" re­
l a t iv e ly oomple% m tb m a tia s*
I t ILat IbiMt wrltnr^n In te n tlm to
d e v i s e si s im p le r en d m ors u n d ers t e n d s b l e e ^ r o a e k *
MPbia at&Jeotf which la o f prim# l&jpcxrtance lie s # mMcretaMlo g o f the theory o f fre^aeney dilBtrlhutlohS he# Ixaaai treated &ai
coirnl der&hle d e t a il in jpe#er8& Ibgr jPeaBem* Fllon^ .and
320 the I lte r e tm ^ .relatlohahljp# between the corrected and a&oor*
reeted moments o f & gronpM. d la tr llm tlc # are knowa a s TBheppaza*#
o o r r a c r tim a * a f t e r %, # * .S h iw a r d ^ . M g * B ^ o a 4& ike -%e 3prohl6#
-
la rotghly. a# follow s* . l e t as choose the ela a s In terv a l as is
malt o f _& and l e t aL Tbws th e ^ a ^ W e ls a a o f th e t t h olaas^ IWaeas
th e a#^03&Lm@te momenta a s d ir e c tly oo^pnted from th e observed
's *
a ie t r im t lo a are defined bar
r w%
? Z _ 3:91(37' j&(a*k;aby %&ere g ta )
*%
;
.
la: th e Oheerved fragnehOy* if 1# th e to ta l dPiyasgaasategr BBdrk list the
<*
o le s s w idth. The ftm ctional momehts are
#Wre
f(%) i s th e th e o r e tic a l r e la t iv e freghahcy^ The m ln tlo n IBet the
problem Involves a rela tlo h sh lp between. & aad v * :
h
** -
.
‘
I f &(%} C8B ha expended by Taylor*# theorems
a C x ^ h i s JLo
&
S^ H xt U
■
Pcsrson, K». 4*8EL the Probable Bf&ore o f fregaig&ey OOhstenta*"
S l o a s t r l k B . vo&* 2 ( 19034* %&* 2 7 3 *81*
BsaTGOBf %«-$ 5 B& B l l e a * &* B* & .?
B rp b K tle O fr sz S o f
Freoaency C onstants and oh th e I n f lmeaee o f -Banlosa Seise*tlo n on T e r le tlo a and G orrelatlaaf^ P h il, Trans* A**
v o l* 191 (1398), BB* S & M ll. .
. Sheppard. #*F*? *0% the A pplication o f the Theory of Bcror
to Oehes o f BOrml M8%]#%atlo& cad-B orm l Gorrslatloaa**
P h il* T r sn o . A*.*, v o l * 1&2 ( 18# ) * PB*. 101*6 7 »
Si
S ie tz , Hi K , Banabepb of M h e B a ti2S i S t e t t B t t ^ , S , 95, '
Somghton 3&HIinH5o.*, ksw %orb$: 1924*
j- .JV-gr
•-
. CO-
.
r-
J t:
K
X J ' \.
m # tw % J W w
».
i?a
5 - "c36* C w
SS1 C s t i . ) /
S Z
5J
sJ * - m M ' ■^ t
t ® -60-
l a ##&& th & t i t *#& a l& i t * d e r i v a t i v e s -Vcmiah a #
% a iiv ip th e n Sgy th e MaGiaarlm 8 w& f a r ^ o ia
/
OS
^
*
-1=0 ,
(# 1+1)11 -«o
%Hg
W 4&7 *
ISgr s # 6eea a i# & i n t e g r a t i o n b y
J
-,e O
• CO
CO
■
%rW^X
* V * L (* :X
'
% *- - , S i
* T% ^
#&&" a b o v e #%a
' -
'
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'
>- e' I * TkBBa a r e a qzwie# i f k )
.- ^
*eWtw
3«. W ith s t t i - f i c i e l i t r e a t r i e t i o n s o n ' t h e f u n c t i o n s in v o lv e d o n e
ca n c h a n g e ’ t,be o r d e r o f .sum m ation.
'
13.«*
■**
tBxe prim ed moments are alsottb any arbitrary O r lg in a If- the
-siean i s chosen e s that origin* the p ris m s may be dropped^.
Mo*
meats np to the six th are
% 2 ^
-
% z v4
3^
mini W
TTif,.
2
«
Jda*=*
8#
******m = *x= B
6
5h, Ic2
3 ewk4
Jc6 '
ILiUIg flC
lI
5
4
16
44S
-Qertainly i t would be d esirab le to know whether these so
ca lled ^corrections^ corrected and I f so* how w e ll„ • Tables Xs
II* III and IV will ten# to Illustrate this*
Ibble XV shows the
results of applying the *corrections* to the momenta Of the
frequency distributions in tables I*. XX and XXX*- ,
TABLE I
TEM T m u iS S -^ ilE S THE MEA UMBER A PEARSG1 TYPE I I I OORtE'
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TABLE H
W TMoa&&& ikkBSss Tens #%e& im m 1 imsGOB m s # 1 m m s
I ^
■ ft
'iSiIuL
- 9
5
77
"^05
18#
1750
2027
1#Q8
1302
#6
■
7:
. *!• 5 '
■'*" H..
” X
3 ;.
7
■:- -■
'','203'
:'. -35
19 %
17
1% ^
U
T/
— 693 ■'' .."
- 5^ 0 .'/::
-2027 :'■'
#
#
39%.--'V
l&LS 960
^5
1#0
:
605
6397
19045
367%
15750
3927
IS #
1171#
19900
163#
m s
2700
H 56
10%
lo&ooo
V l
V i
163*396
- 56193
-138915
-133625
-4 7 2 5 0
»
2027
1# #
99154
99500
145775
14729#
'' 117126
7W 5
40990
V
i
73209
50^197
973405
668125
W 50
2027
lSOO
10946%
49%00
1020435
1325922
1288406
999635
607500
. :3.
^
k
.
- 805355/'
-4546772
*6866% 5
-3340682
- 423250' =SwodKr',. 1 # ^
. .3163#/"2487500/
W W :
V l
8S57805
40920957
47647949
16703125':
1275750
8027'
- -a # !:- ''-,'94915$'':
12437500
50000889
11927W
107351088
141724#
12995255
9112500
125#7368
16893^31■ fe? .
136687500;
7426297
59*337*762
141137643;
9 8 9 ^ 5 ^ 6 " ..
965i§0274
3340#
20577
9 1 9 ^
s 0+# 5
"rr
if.,
^
'
,- • •
>-i '■
OOBaaaaed f r o *
'S W o tlo a * %%
Iu o-I^fL «u.zt.^-U
Um
^Jfca
^**.*,&
- t .Wkl
-i. Il -H
-- SW. g -S^Sr "193®f
W
a
tb
e
tib
a
l
M
e
t
i
a
t
i
M
T
o
l
4^»«' -‘-4^
■ -- ■-.'>•:
: •’
-Tv'.
TABLE LU
r
# E HONDBED THOUSAND.'TIMES THE ABEA UNDER A'SlfW a C tM
'i'
•~4 eB
tA
36*0 '
9
- 171.5
49
- '720.0
-3VQ
240
- 2310,0
—2*5
.924
27S4 ■ - 5568iO
- 9838.3
^ 5
655#
- 1 .0
12090 ... - 12098.0
-0*5
^ 8733*0
1##'
c.. 19%^ - ,, ,,0 .0
. 0,@
' 8733.0
o*.'5
- ^ # 1:: ' 1209KO
IvO .
.98^8.5 ,
1,5, /
'5568.0
3*0'
- 231^,0
2,,^'..
'v^.92.4
r \.\vv240 ' .! .720.0
171,3
:?/. 49.
,3 6 ,0
Total : . i o o ^ #
0,0
•
44
■ 2 V ':
tIf A -: ■ '' t i t i
2304, 00.. - - 9216.00
576,00
'144.00 • ■
7353*0.6 - - 25735.72 '
- 2100.88
. 600^25
2160 vOO,-' 2.T 6480.00
19440, 00 ; - - 53320.00
,5775,002 '■-14437.50
36093.75
- 90234.38
-22272.00
11136,00
-890 88,-00
44544.^0
33204494; - 49867,41
14757*75 : ,-2 2 I 3 6 .62
12098,00 . . . - 12098.00
12098,00 , '-12098.00
1091.62i 7 ^ 543^81
4366(50 / /-" 2183.25
;00.00 ... -y ; . 00,00
00,00
■0G.W
'■ 10%* 62
' 4366(50
545. 3%
'12098.00 '
- '12098*00 ' ■ 12098.00 ;
12098; OO
' '22136^.62 ' .. 33204. 9 4 ,
. 34757.75
.49807,41 '■
. 11136.00
890# . OO .
... 22272, 0 0 ,'
44544.00
5775.00 , . r i 4437. 50.
36093.75
90234.38
-19440.00 ■ 58320*00
2160.00 , . ' 6480, 00: : :
2100 88
6600^25
7353.06
25735,72
: 144.00 •• y v5'76.O0’ ;•
9216.00
. '2304.00
102,075.00 ' ■
00.00
00.00 . ■ 312, 258,74
't If I
•'
.
'
' '
,
.
tIfI
"36864,00
90075.02
174960.00
225585(94
178176.00
74711.11
12098.00 I
272.,%
' ■ . 00.00
272*91
12093*00
74711.11
.178176*00
'22#5 $ * 9 4
,',.174960(00
: . \ 9d m ,o 2
',36864^0
I , 585^485,94
......
-V -
; '•
. (KXAWWK QP TRRSS GNKtGBKBlBBfSB),. OQMgSD*. Tiw m m 3%%#G% TKaGSGKfBBBk MRSBiaMSLBDB W&BS I , 3%, 3#A
Table IT
%AA* I
n
%
Voa
%
I
■ 't
X
0
X
0*0094
. 0 .0004.
9
16.005
a.
16*3384.
32. 0# , . 33*o6 .
3 ■
^ 4.91
4
5 - 960W 746. $4# . 9D.
5304
6 im *m aO $4 4 . ^6,
%*64D .
%
%
Tbm W
%
I
I
I
.
0.0002 ,
.0 *0 X8 . O
. 16
. 16.3296 . . 16*00 .
3%
31*96
31*96#
893*3W . 861, 1#
#64
3#4
3933*??68
952? . #
96*
#
98, 939f89?& % % & *#
6* Tb* % r6# 086i* s iweq^<WWd #pm ata, T%#
tai&cwpatdLaakllar SeapgKpbgba 8# 9n&8«
%n
I
'I
0. #
O+Q0 .:
' w w o ; 1*02
0 .0 0
0 .0 0
3*13#
3*00
&:<#
i g . # # 14*91
-
;
%
I - I
■
’■ *09
t
' . 1* #
0*09
IaW
0*90
13# # >
<K^e*Wa abomasa* and % *%*psm»***mlw;
16
-
-
From th e p re v io u s d em o n stratio n , o f th e d e r iv a tio n o f Shep­
pard *s c o r r e c t io n s i t would seem d e s ir a b le to have a le s s complex
approach to th e problem .
To a id in t h i s ap p ro ach th e moment-
g e n e ra tin g f u n c tio n co n cep t w i l l be in tro d u c e d h e r e .
As th e name
im p lie s , i t i s a f u n c tio n w hicl>-'generates moments and i s d e fin e d
b
by M^(9) - J e
f (x )d x .
Exp ndlng e
Atc
in to a power s e r i e s one
a
o b ta in s
ex
— I + 0X
Q2X^
8 !
5 /
th e r e f o r e
M (e) = /
[ i * ex * QfeX2
a
J
5 !
f (x)dx
I n te g r a tin g term by term one g e ts
.b
2
b
f
b
M (9) = / f (x )d x +
* Q
x f (x)dx ♦ ~ / x fe:f (x)dx -»■
e
X
c
D
£• a
a
a
=
+ Iije +
. 9s
§?
U3 3 ! + * * •
O bserve t h a t th e c o e f f i c i e n t o f — i s th e k th moment about th e
kl
o rig in .
A lso u s e f u l i n t h i s ap p ro ach i s th e f u n c tio n
g ( x ^ ,x Sj) - x^
b u te d .
X0 where x^ and Xg a re in d e p e n d e n tly d i s t r i ­
Independence im p lie s t h a t f ( x ^ , x g) - f ( x ^ ) * f ( x g ) .
One
can g e n e r a liz e th e m om ent-generating f u n c tio n d e f i n i t i o n to
h o ld f o r th e v a r ia b le g ( x ) , th e r e f o r e M_(9) * / e Qg^ f ( x ) d x .
6
a
By ex ten d in g t h i s to th e ca se where g i s a f u n c tio n of two v a r ­
ia b l e s one h as M (9) = J* *
/
1 e 9g ^X1*x£ ^ f (x ^ , x g ) dx^dx^.
iI i i e
n
-s / i
-48
" v; #g
&
s - '
'81#^e' eatsb @rl tke.^ ia fe g ra iiS ' on tjhs. 'r i g h t 'IS, #h* S o m e B ^ g ^ ^ l-a f*
ing function of the Indicated variable^ then
M
*'
'C'-//
(G) &'
ObnsIdBi1 -a- Snequeneyl distribution f (x) divided- into M tj
convenient n ta ib e e o f e l a t - s e s whose e i a e s m aih i t
' .'
w id th
/
K, Instead o f "the '#&riaW.e.'
• See Fig.’« -2*
-Bre-
; " ■.
;
K------- K-------
Q _
0
@n$ c l a s s
%
." J'
: ' '
I
. ' / ;4- /" ' . - '
And -fteth&r- [Id**: c Tbs* t h e .Gafz-CRe im n s ia # t W (#asn9
F i g e:- 1»
... X
F req u en cy ; D i s t r I b u t i o n -
!Then one has the relation sh ip -Xj. '^ X * 6 ^ 'where ^x and € -are
.W eFohdently dibtribdte(%r ^ ±b. the v^yiable [ b # o ^ ' gFoWing'
is the variable ,Sftbt .g-riette6#; ^
tributed because ? Cx
.
F
a W f .Epe i#ependentiy die-
X * (x + dx)] '.=?' f (x) dx and. ..........
' -. '
'
' -.-b: / '
[6 6 6 6 (e ^ 4 )]
Shen
been proved that 3^.(6)
--1. + u^B +- 1*g.
.,
^
It W
gy
p^d It
• ' "XS""
*
'
'
’
- *
. . I p ,
*
«
,
„
fo llo w s t h a t Mx ' (6) = I 4 r[e * ^ 2 -.Zl + vt | t 4*
I s th e n
X
S•
v 3•
l e f t t o f in d th e ex p a n sio n o f Mf (S ) =. The d e f i n i t i o n , o f Moment-
g e n e ra tin g f u n c tio n t e l l s one t h a t
' , k /2
M (6)' -
a
"
e f6 a f „.
Upon i n t e g r a t i n g
I - C j b J k/ 2
O '"
-
5
Le
1 r
J _k /2 - 5k u
ait/2 - S-6fc^ ]
3i n h e k /2
E x p a n d in g i n t o a p o w er s e r i e s an d s i m p l i f y i n g
^ 6 ) = I e E s^
- M
■** X 2 /
5!
* I *
**
”^ F T
J
. o& + a t . Q& + k S . 3$ 4
12
2 1
SO
41
448
6
By substitution* m u l t i p l i c a t i o n a n d d r o p p in g p r im e s t o i n d i c a t e
m om ents a b o u t t h e m ean o n e o b t a i n s
,
'r
,
Kx1 W I
..
.» ....
■ .
1 + U1B + Co2 - n ; s _ 4
Cu, ■- -XL—
4
2
( u . *> 5u 4kz
;6
4
^
w k4, o4
~ {Uti5
SO 4
6. .
96
3uok4 +
"***■
448 6 /
16
16
s£ *
5.'
-Ct
-
Io m en ts up t o t h e s i x t h a r e
U^ •-
.
~ 0 '
% , - iT^
IB
- g
-
%L
7% a * * . -
• '-
4
^
13-ltistra fclGns. hava beea given in tM-s p@p@^ to .deBOssi^ate
how ^Slieppardea oorreotions" a e th a lly futietion-,
Also presented
wag a sim plified, Ssrelopment o f the ferMnlee for-, these ^eorrae**tleaa* by use o f the moment-genera tin g fnnetlon
The
m i t e r b e lie v e s that th ie epproeoh s im p lifie s th e tr sd itio n e l
s o lo tle tu
-
For the n u aerisel em ap les chosen In t h is pd;poB the aCorr a e t i one® So g i v e very g o o d B S in B tm e n ts t o t h e ' lo w er" i p p r o r i m t ©
moments^
B o w e v erjh they se e m t o g i v e ■i n s i e o a a t e - ■e o x ^ rso tI o a s t o
t h e o b s e r v e d m om ents h i g h e r S M m t h e f o u r t h *
T M s - I f t p l i e t no
M s l e w rong In t h e "oorre&tions*; but t h e m e th o d o lo g y em ployed
in grouping the date may bays been at fault o r the quantity of
d a t a may h av e been I n s u f f i c i e n t *
data M m
T ab le ^I F I n d ie a tM t h a t th e .
used seems to bear out both of the above suppositions.
r i s i n g out of this p&per* then, is the possible problem of
further Investigating the adjustments to higher moments by vary­
ing the method of grouping as well as varying the quantity of data
used.
Blnoe
is not always a reliable Indez of peakadness, one
oould study that problem with the purpose of finding # new and
valid measure*
JSt present the writer does not h e # the opportu-
"
■.
-4
M t y for further investigation of the above mentioned problems*
m am m as m w
jump
B oel, Petil G*,*. Ibtroduoti OR Ike m t& eaatleal SteM etloa*
Be* .York:
John Wiley ana, Sons3. Ino,»..$■ 1947#
#"@er#G&* %** "On t h e P ro b a b le S r r o r e o f BragKeooy O oaateota**
m oaetrlksi. y o l. 2 (1903)» 3 9 . 273-81*
Bearaon* K* aad P llon * I , Bi #** *0a the Probable Srrora of
Breqoeacy Constants ant on th e -ln flaan ee o f ItemSos
S electio n on T eristio n ami Correlation*" P h il, Tr&im*
A ,, vol* 191 (1898), PP. 229-311*
H e t s g H* Lag. M i to r -ln -ch i ef* ' .Eamdtook o f Msthemet lo a l S ta tis ­
tic a l
lew Tbrfci
Honghton M ifflin Ge., 1924#
S eiro se# E, Ltijt Annals of Mathematical S t e t i s t l e e *. v o l. I*
Bo. 2 , Bay 1930*
Sbegpari3 # . #.* "On the A pplication o f the Theory o f Error to
Oesea o f formal D iatriW tlon and Rbr&Bl Oorreletlona*"
'
.
P h il*
A*, vol* 192 (1898), pp, 101-67*.,
Steffanaea* J ti 9.* In terp olation *
Baltimore:
.
The B lllla a s and
Wilkins coapeny* 1947*
Waugh* Albert 3*, Mboz^tory IBBaituwaJL aa& Problems jet?*;
*&
S t a t is t ic a l Method*. Row l&r&? tk#raw -B lll IkSHsBf Ooapany, Inc** 1938*
WeatMrhurn5 Ge Eti5 A F ir s t Course In. Mathematical Q ta fistlo s*
Oaznhridgs:
Cambridge C a iw r s ity P ress, 1947»
Whittsfcer5 32. %* and WatBbn5 (I* R.* a OonrM o f Mod^rs.. .Analysis
American e d itio n .
Be* York#
Cambridge:
Cambridge C alveraity Press*
The Maoalllaa Company* 1947-
MONTANA STATF i lurxvcBc-T-rw .
_____
3 1762 10015456 4
m
ScoGlo
92609
cop.2
Scott, Chester H.
—
-
U 37r
Scitlo
<10P.2.
9 2 6 0 1