-53- of of

On the Evaluations of Sum of Squares by Using the Range III

by

Toshihiko ^MENDORI, y oshihisa ^ZINNO and Hiroo ^FUKAISHI

(Received May 31, 2004)

Abstract

The sum of squares (briefly, s.s.) of the data plays an important role in statistics, but the calculation of the s.s. is more complicated than that of the data range. From this respect a number of research on statistical approximation of the s.s. by using the range has been done.

But we have extended the study of the mathematical handling for the g.l.b. and the l.u.b. of the s.s. in the case of variously constructed homogeneous data. In this paper we shall give the g.l.b.

and the l.u.b. of the s.s. for a stratified data set by using the range. We also give some illustrative examples.

§ 1. Introduction

In the previous paper [6] we gave a theorem which estimates the sum of squares for data with a frequency distribution by using the data range. In this paper we shall extend our study to the evaluation of the sum of squares for a stratified data set ( § 2). We shall also give some illustrative examples in the last section ( § 3).

Briefly we recall the notations and the theorems.

Let X = {x" x2 , ••• , xn }, n > 1, be a data set of n values. Define the data range R and the mean

x

of the data as follows :

R = maxx; - min X;

On the Evaluations of Sum of Squares by Using the Range III

1 ⁿ

x=-Ix1,

n ^i=I

respectively. Define the variance

s2

of the data and the unbiased estimate u² of the population variance as follows :

l ^{n _} 2

s²

=-I(x

1-x),

u

²

= - I ( x

^{l n} 1

-x) ,

²

n-1 ^i=I

n i=I

respectively.

Now define the function g (

x)

by the following :

n 2

g(x) = I(xi -x) .

i=l

Then the function g (

x)

takes the minimum at

x

⁼

x : g(x)

= n s² = (n-1) u^2•

Theorem 1 (Main Theorem [5; Theorem 2.1]) ~ The minimum of the sum of squares g

(x)

is evaluated

by

using the data range R as follows :

_!__R2 $

g(x)

$

{n _ l-(~lt}R2.

2 4 8n

Let 8 =

{c;i, c,,\, ... , c;

11

J,

m > 1, be a set of m class mark values satisfying the following conditions :

i)

c;i+

₁

-c;i =

R/(m-1) for 1$i$m-l, where R is the range of the values : R

= <;

₁₁₁^:....

<;

₁ ^,

ii) the frequency

h

of the data with the value c;i is positive for both extremes :

Ji

> 0,

J,n

> 0, and

111

iii)

I/1 =

n.

i=l

Then the mean

c;,

the variance

s2

of the data and the unbiased estimate u² of the population variance are given by

- 1 ¹¹¹

c; = -ItJ1,

n ^i=l

2

^{1 111 (}

-)2

s

= - I c;i - c; Ji,

n ^{i=l .}

respectively, and we have that

From Theorem 1, the minimum of g(x) is given by

g(~) =

ns2

=

(n-I)u2.

Theorem 2. ( [ 6; Theorem I] ) . The sum of squares g (

l)

for a frequency distri- bution is evaluated by the range R of the values as fallows :

{_!_+ 1+(-l)m.

n-2}R

²

^{~ g([) ~} {n

l-(-lt}R2.

2 4(m-1)² n 4 8n

§ 2. The sum of squares for a stratified data set 2.1. The case of a general stratified data set.

Let Xi

=

{xi" xi2 , ••• , xinJ, ni ~ 1, be a data set of ni values for eac;h i-th stratum in m strata X" X ₂^{, ••• ,} X ₁₁₁ ⁽ m ~ I ) . Define the data range R, the mean

x ,

^the

variance

s2

of the data and the unbiased estimate u² of the population variance as follows:

R

=

maxx;• -_{. . '}} minxi', _{. . 'J}

1 m n;

x =

-IIxiJ,

I, j I, j n i=I )=I

m

respectively, where n = ~ ni . For each of the i-th stratum, the data range

R,

the

1=]

mean xi, the variance si ² of the data and the unbiased estimate ui ² of the population variance are defined similarly :

Ri = maxxiJ - min xii,

J::,j '.:>n; J::,j '.:>n;

2

^{1 11; (}

_)2

S· _I

= -"'"'

_L.,. x .. _1J -X• _{I '}

ni ^J=I

2 }

^{11; (}

_)2

u;

= - -

1

I

^{xii- X; ,}

ni- )=I

respectively, for 1 ~ i ~ m.

Now define the function g

(x)

by the following:

m

g(x)

=

^{L9i(x), where gi (x) =} I(xiJ -x) . ^{11; 2}

i=l i=l

. On the Evaluations of Sum of Squares by Using the Range III

Then the function g (

x)

takes the minimum at

x = x:

g(x)

=

n

^{s2 =}

(n-1) u

^2•

Theorem 3. The minimum of the sum of squares g

(x)

is evaluated

by

using the data ranges R, Ri

(1

~ i ~ m) and R _ as follows :

[1 1 { m

2}]

max -R2, - "2:,R/ +R_ ~ g

(x)

2 2 ¹⁼¹

~ mm

. l{n

- - - - -

^1-(-lf}

R ,

² "I:, - - - - -

^{m {ni}

^1-(-It}

Ri

²

+ - - - - -^{ni

^1-(-If}

R_

²¹

,

4 8n i=I 4 8ni 4 8ni

where

R_

=

maxxi -minxi .

J.:;i.:;m J.:;i.:;m

Proof We have

=

n

(E{s/}+

Var{xi }), where

Then, from Theorem 1 we have

whereas from Theorem 2 we have

_!,_Ri² ~nisi²

~

^{{ ~ -}

l-(-lr }R/

2 4 8ni

for each i-th stratum, whence

_!,_R _{2 -}²

~

nVar{x-} ~-

{~-l-(-l)n}R

^{2 •}

I 4 8n -

•

Remark.

1. We have

g(x) =

^{0 iff} R

=

^{0, where X=X11}

=X12 = ...

^{=Xmnm· Then}

s2=

u²

=

^0.

2. max[-1

R^{2, -}1

(IR/+R_

²) ] : : ;

s

²

2n 2n i=t

. [{ 1 1 - (-1

r}

^{2 1 m}

^{n

^{i 1 - (- 1}

^{r }}

^{Rl-2 + { 1}

~ mm - ₂ R,

-I -

4

8n

n i=I 4

8ni ·

4

1-(-If }R 2]

8n

^{2 - '}

3. max [ ( 1

) R

2, (

¹ ) (

I

^{R/ + R _ 2}

)l ^~

^{u 2}

2 n - 1 2 n - 1 ^i=l

< mm - - -^{. [{ n} ---"---'--1 - (-1)^{11 }} R ² - -I

L

^m

{n.

^_!_ - - ' - - - ' - -^{I - (-1)11; }} R-² + - - -^{{ n}

- 4(n-l) 8n

^{(n-1) ' n-1 i=t}

4

^{8n; 1}

4(n-1)

1-(-lf

}R ^2].

8n(n-I) -

We give an evaluation without R_. Firstly, we consider the case of two strata, i.e., data which.consist of two groups.

Proposition 4. The minimum of the sum of squares g

(x)

is evaluated

by

using the data ranges R, Ri, R₂ as follows:

[ 1 ² 1 ( ^{2 2 ) ] ( - ) .}

[{n

^{1 - (-1}

f}

^{2 ]}

max

2

^{R ,}

2

^{R1 +R2 ~ g x ~ mm 4 Sn}

R,

M20 , where

M20 = max(M21, M22),

+{!!_:_ l-(-If}R12+{!!2 l-(-If2}R22'

4

8n1

⁴

8n2

M

22 - - -

_ n1n2 (R

- - ,+- 2 + - - · I + - - 2 ·

1 _ R 1 R )² n, -1 R ² n2 -1 R ²

n n 1 n2 n1 n2

On the Evaluations of Sum of Squares by Using the Range III

Proof Without loss of generality we can assume

Set

and

.X,

= ~(x

¹¹

+x

^{1, , ) , X,}

= ~(x

²¹

+x,,,).

In the proof of Theorem 1, we have

We have

by setting t=x21 -x11 , x=x,-x,,, y=x2-x21 . In the case of t=(R1-R2

)/2,

x=R,/2, y=Rz/2, i.e., x=x, =x2, we have the minimum of

g(x):

where at least one of R, or R2 is equal to R. We also have the maximum of

g(x)

as follows:

(I-1) _maxg

(x)

_{f (} x," ... 'x," x,nl' ... 'X1n1; X2p ... 'Xzp X2n2' ... ' X2n2 )

'----v---' ' - - - v - - - - ' ' - y - - - - 1 ~

n_{1 /} 2 n1 I ² n2 12 n2 12

n 1n 2 ( 1 1 )² n, 2 n2 . 2

= - -

R--R, --R0 +-R, +-R2.

n 2 2~ 4 4 '

for

x =

X1

=

Xz, both n₁ and n₂ even,

(I-2)

max

g

(x) =

^f

(x,I' ...

'x," x,n1' ... 'x,nl ;x21' ... 'X2p X2n2' ... 'X2n2)

(I-3)

max

^g

(x)

(I-4)

max

g

(x)

(11-1)

maxg(x)

~ ' - - - v - - ' '---v---' ' - - - v - - '

(n1±1)/2 (n1+1)/2 n2l2 n2l2

= n,n2

(R-

^n,

±1

^{R1 _}

^_.!_R2)2

⁺

(~--1-)

R/ +!!:l:_R/'

n 2n1 2 4 _4n1 4

f

(xii' ... '

X11, Xln1' ... 'Xln1 ;x21' ... 'X2p X2n2' ... 'X2n2)

~ ,._____,, '---v---' ' - - - v - - '

n1 /2 n1 /2 (n2+1)/2 (n2±1)/2

f (

x," ... ' x,"

x,n)' ... ' x,nl; X21' ... '~21' X2n2' ... ' X2n2 )

~ ,._____,, '---v---' ' - - - v - - '

(n₁±1)/2 (n1+1)/2 (n2+1)/2 (n2±1)/2

for

x = x

¹

⁼ x

^{2 , both n}1 and n₂ odd, or

_ n¹n 2

(R

^{1 R 1 R ) 2 -n1}

-1

^{R 2 n2 -1 R 2}

- - - - - I - - 2 + - - I + - - 2 ,

n n 1 n 2 n 1 n2

for x_{21 -} x₁₁

=

^{R, -} R, or (II-2)

maxg(x) =

f(x1J'···,X11'Xln1;x21'X2n2'""''x2n2)

Therefore we have

_ n 1n 2 (R 1 R 1 R ) ² n1 -1 R ² n2 -1 R ²

- - - - - , - - 2 + - - I + - - 2 ,

n n 1 n 2 n 1 n2

_.!_ (R/

⁺

R/) ~

^g

^(x) ~

^{M20 ,}

2

On the Evaluations of Sum of Squares by Using the Range III

where

M20 = max( M21,M2J,

+{!.'!]__ l-(-lf}R12+{n2 _ l-(-lf2}R22'

4 8n₁ 4 8n₂

M _ n¹n 2 (R 1 R 1 R )² n¹ -1 R 2 n 2 -1 R 2 22 - - - ^{1 - -} 2 + - - 1 + - - 2 ·

n n1 n2 n1 n 2

•

We consider the case of m strata, i.e., data which consist of m groups. Let (/1 ;

I)

be a partition of the index set / = { 1, 2, ... ,

m} ,

_{i.e. / 1} = { ii' i2 , ••• , ik } , / 2 = { }i, }2' ... , '},} be two subsets of / such that / = /₁ U/_{2 , /1}

n1

₂ = 0, /₁ =I= 0,

/ 2 =I= 0, in other words k + l = m, k

>o,

^l

>o.

Then we have the set P of all partitions of the index set /.

Theorem 5. The minimum of the sum of squares g

(x)

is evaluated

by

using the data ranges R, R;

(1

::s; i::;; m) as follows :

max [!R², _!_

I

^{R/] ::;; g}

^(x)

^{::s; min [{ n}

^l-(-ir

^{11 R2, Mmol,}

2 2 i=l 4 8n

where

M 1110

=

max(Mm1,M1112 ),

M

= -

1 max T ^"i +

m{n.

^__J_ ~ 1-(-lf} 2 ^{. ,}

ml n Ui ^{;!Je p} Ui ;IJ

~

^{4 8ni}

~

A 1 - " { ni - 1 + ( -1

r

^{n j - 1 + ( -1}

r }

²

T(I·I) - L.

n.n.

R _ _ _ _ _ -n_ - - - R .

I , o I J 2 _..~ 2 ^J

- ie1₁,je1₂ n; nj

{

ni -1+(-lf nj -1+(-lt }

2

+ "

n.n.

- - - - R _ - - - ' - - - R .

L,. ^I J 2 _..~ 2 ^J

i,jeI₁,i<j _ n; n1

1 ^{"2 111 (} 1 ) ² Mm2

= -

max

1c1 . I )

+

I

^{1- -} ~ '

n (/1 ; !Je p 1 ' ² i=I ni

1 1 ²

+

'°'

L., ^{n.n. I} J

(-R.

l

--R.)

J

i,jeI₂,i<j ni nj

Proof Set

where

for each of i

(1

~ i ~ m) . We can assume that xii ^~X;2 ~ ^... ~xi11;, without loss of generality. In the proof of Theorem. 1, we have

where

n; n_; 2

I;j = II(xik -xjl) ·

k=I 1=1

Set

If it holds

for 1 ~ i ~ m, then we have the minimum of g

(x) :

. I m

min g

(x)

= f

(x" x

2 , ... ,

x

111) = -

I R/,

2 ^i=l

where

Setting

we have

On the Evaluations of Sum of Squares by Using the Range III

XjJ - X i i , and

and R = maxK.

1:s;;:s;m

-(xii'"'' Xii' Xin;' "' 'Xill;)'

~ ' - - - v - - '

U; 11;-U;

T. _I}

=

u.u .t . .2+u. (n. -u. )(t .. _{I j I} I j j I)} + R.)_j ² + (n. -u.)u .. _{I I J}

(t..

_I} ^{-R. )}_I ^{2 +} (n. -u.)(n. -u. )(t .. - R. _{I I j j 1J I} + R. )_j ²

=

n.n.{t .. -(1-u;)R.+(1- uj)R-}

2

+(n.-u.)u. nj R.²+(n.-u.)u . .!!i...R.²

l j 1J I J I I I I J J J j

n; nj n; nj

and

Since

i) (n1-u1)u1R/+(n1-u1)u1~R/+···+(n 1-u1)u¹ nm R/

n1 n1

ii) ⁽ n- -u. u-- • _{1 I} ⁾ _I ⁿ¹ R _I 2 +··· + ( n- -u- u---R1 _{I I} ) _l n;-1 2 ⁺ ( · n- -u. u-R-_{I I} ) _{I I} 2

n; n;

( ) n;+1 R 2 ( ) nm R ., + n--U· _{/ I} u--- . _{I I} +···+ n--U· U · -_{l l l I} .-

n; n;

for each of z, 1

< i<

m, and

iii) (nm-um)u;n!!.2-Rm²+·--·+(nm-um)um nm-I Rm²+(nm-um)umRm²

nm nm

we have

g(x)

1 "" , ~ ,

- ~ T;j +~Si, n J:s;i<j5am i=I

(nm -um )um ..!!..._Rm ^2, nm

where

Therefore we have the maximum of g

(x)

as follows:

maxg(x)

⁼ f(xi,X2,···,xm) = Mm0' where

1

^{"i m}

{n. 1-(-lf}

²

M

= -

max T + _J_ - . ,

ml _n (!·/) _{1 • 2 E} p (11;!2)

~

₁₌₁ ^{4 8} _ni

~

,.. 1 _

L {

^{ni - 1 + ( -1}

r

^{n 1 - 1 + ( -1}

^{f }}

²

T(₁ ^{) -}

n.n.

R - - - - --n_ - - - R .

I;/, I j 2 ^{L ~} 2 ^j

- iEI_1• JEI₂ ni n.i

{

ni - 1 + ( -1

r

ⁿ 1 - 1 + ( - 1

f }

²

+ " _~

n.n. - - - -

^{-n_ -} ---'---R.

l j 2 ^{L ~} 2 ^j

i. JEI1 • i<.i ni n ₁

nj -1 + (-1

r }

²

_ . c . _ _ _ _ _ R. '

2n. ¹

J

1 " ^{m (} 1 )

M

= -

max T² + 1 - - :^{2 ,}

m2 _n (I . I ) P _{1, 2 E} (/1; 12)

L

_{1=1 ni}

~

On the Evaluations of Sum of Squares by Using the Range III

2.2. The case of a stratified frequency distribution.

Let {;,\, c;,\, ... ,

(mJ,

^m0 ~ 1, be a set of m₀ class mark values satisfying the fol- lowing conditions:

i) ^{(k+I - (k} ·,= LI for 1 ~ k ~ m_{0 -} 1 , i.e., the range of the values R

= (

m₀

-1)

LI,

mo

ii)

Ifk

= n.

k=I

Then the mean [ , the variance

s2

of the data and the unbiased estimate u² of the population variance are given by

respectively. Let fu be a frequency of the i-th stratum in m strata with the class mark value (_{1 ,} where fu ~ 0 for 1 ~ i ~ m, 1 ~ j ~ m₀ and m ~ 1. Then we have

for any integer k, 1 ~ k ~ m₀ •

The mean

?i ,

the variance

s/

of the data, the unbiased estimate

u/

of the population variance of the i-th stratum are given by

respectively, where

m0 m

ni

=

^{Ifu, n}

=

^{Ini ·}

j=I i=I

We assume that n;

>

1 for the sake of follows:

m

g(x) = I9i(x),

^where

i=l

We have the sum of squares g (

x)

as

From Theorem 1, the minimum of g

(x)

is given by -g ( [)

= ns

²

= (n-1) u

^{2• ,}

Theorem 6. The minimum g ([) of the sum of squares for a stratified frequency distribution is evaluated by the ranges of the values

R, R;,

^and

R_

as follows:

[{ I l+(-1r

⁰

n-2}R

²

~{I l+(-lri n;-2}R ^{2 1} R ^2]

max

-+---·-- 2

^4(m₀

-l)

^{2 n ' L.... i=I}

-+---"----"---·-- · +- 2

^{4(m;-1)2 n; 1}

2 -

,,,; g(x),,,;

min

l{n l-(-1r}R2, I {ni 1-(-lr }Rz+{ni 1-(-lr }R_z]'

l

^{4 8n i=I 4 8n; 4 8n;}

where

m₀ = R I i1 , m; = RJ i1 ,

R;

=

^max{tj lfj :;t:O, l~j~m₀}-min{tj lfj :;t:Q, l~j~m_{0 }} R _ = max { <;'i I 1 ::; i ::; m } -~in { <;; I 1 ::; i ::; m}.

Proof We have

=

ⁿ

(E{s/}

+ Var{tJ), where

1 ^m

E{s/} =

-In;s/, n i=I

{

- } ] m ( -

-)2

Var (;

=

-In; (;-( . n t=I

Then, from Theorem 2 we have

_!_+ l+(-1r

^{0, , .}

n-2 ~ g(~) ~ {n. ^1-(-lr}

^R2,

2 4(m₀

-1r

ⁿ

4

⁸ⁿ

whereas

for each of the i-th stratum, where m₀

=

R I i1 , m;

=

R; I i1 . Therefore

1

¹¹¹

{l l+(-lri ni-2} { 2} 1

¹¹¹

{ni 1-(-It} 2 -I-+

^{2 · - - ~ E S ; ~}

-I - _ ___;____:____

^R;'

n i=I 2 4(m; -1) n; n i=I 4 8n;

On the Evaluations of Sum of Squares by Using the Range III

and we also have

whence the conclusion of the theorem is given.

•

Remark.

1. g (~)

=

^{0 iff R}

=

0, where ;

=

^{<;; and R;}

=

^{R _} = 0 for all integer i, 1 :::; i :::; m.

Then s²

=

^u2

=

^0.

2. If R;

=

0 for each i, 1 :::; i :::; m, then R _

=

R and

0:::; g([) :::; {!!_ 1-(-lf}

Rz.

4 8n

3. We have

; j

= z

^{+ (}

^{;i - [)}

^{+ (; j -}

^{;i ) '}

where "[, ( <;; -[), ( <;₁ -

;i)

are considered as the general mean, the effect of the i-th stratum and the deviation within the i-th stratum.

Theorem 7. The minimum g ([) of the sum of squares for a stratified frequency distribution is evaluated by the ranges of the values R, R;, as follows:

l{ ^{l l+(-lr}

0

n-2}

²

~{1 1+(-lrj ni-2}

²

1

max - + - - - - ' - - - • - - R L.. -+---'---'---•-- R- 2 4(m0

-1)

^{2 n ' i=I} 2 4(m;-1)^{2 n; 1}

~

^g(X)

~

^min

l{:- l-tl)}'•

^Mmo

l

where

m₀ = R 111 , mi = R; I l1 ,

R;

=

max{;₁l.{;₁:;t0, t":::;j:::;m₀}-min{().{;

1:;t0, t:::;j:::;m₀} for 1:::; i:::; m, Mmo

=

max ( Mm1, M

mz),

1

^"i

m{n. 1-(-It}

²

M

= -

max T,

+

^{_J_ - . ,}

ml _n (!·!) 1 , 2 e p (11;!2)

~

1=1 ^{4 8} n;

~

{

n. -1 + (-1 )'1' n. - l + (-1

r }

²

= "

_~ nn. R- ¹ ll - ¹ R.

l J 2 _..~ 2 J

iEI1• jE/2 ni n j

nj-l+(-1)· R. ^n,

}2

2n. ¹

J

n j - 1 + ( _. -1

r }

_R. ²

2n. _J ¹

1 "2 m ( 1 ) 2

= -

max T(_{1 . 1 )} +

I

^{1- -} ~ ,

n (11; l2)E p 1 ' 2 i=l n.

l

,.. 2 (

^{I I )}

2 (

^{I I}

)2

T(_{1 1•} ^.₁₀ ⁾ = " _~ n.n. _{I J} R--R.--R. _{l )}

+ "

_~ n.n. _{l J} -R.--R. _{l J}

• iEI1,jEI2 ni nj i,jEI1,i<j ni nj

•

2.3. The case of a stratified discrete probability distribution.

Let

{?i,

<;;2, ••• ,

?mJ,

m₀ ~ l, be a set of m₀ class mark values and P; be an occurrence probability of the value <;; satisfying the following conditions:

i) <;k+I -l;k

=

L1 for 1 ~ k ~ ^m0 -1,

i.e., the range of the values R

= (

m₀

-1)

^L1,

mo

ii)

I

^Pk

⁼

^1.

k=l

Then the mean µ, the variance a² of the distribution are given by

mo 2 mo 2

µ =

I

A<;;k ' ^(J =

I

Pk ( <;;k - µ) '

k~ k~

respectively. Let {Pu I 1 ~ j ~ m;} be a set of occurrence probabilities of the i-th stratum in m₀ strata with the value <;;j, where pij ~ 0 for 1 ~ i ~ m₀ and m ~ 1.

Then we have

i=l

The mean A, the variance a/ of the values on the i-th stratum are given by

On the Evaluations of Sum of Squares by Using the Range III

"'o

A

= L?1Pu

^{1 ~'}

J=I

respectively, where

mo ^m

~

= ·rPu, I~ =

I.

}=I i=I

We define a sum of squares g

(x)

as follows:

m mo ?

g(x)

=

"'I9i(x),

where

9i(x)

=

I(?1-xYPu·

~I ~

From Theorem 3 in [6], the minimum of g (

x)

is given by g

(µ) = (J

^2•

Theorem 8. The minimum g

(µ)

of the sum of squares for a stratified probability distribution is evaluated by the range of the values as follows:

0 ~ g

(µ)

~ min -_[ ^I R ², -^{I (}

I~

^{111 R/ + R_ 2) ] ,}

4 4 1=!

where

Ri

=

max {

?

₁ I

Pu

^:;t: 0, I ~ j ~

mi}-

^{min {}

?

₁ I piJ t= 0, 1 ~ j ~

mJ

^{for I~ i ~ m,}

R_

=

^max{A

I

l~i~m }-min{A

I

l~i~m }.

Proof We have

111 mo 2 111 2

g(µ) = I~ I (

^(ij -µi) PiJ I pi+

I~

^{(µi -µ)}

i=I }=I i=I

= E{ (J/}

^{+ Var{µi },}

where

E{(J/} = I ^~(J/,

i=I

Then, from Theorem 2 we have·

whereas

0 <

- (JI -

_2 < _!_R_2 ^I 4

for each of the i-th stratum. Therefore '

and we also have

whence the conclusion of the theorem is given.

•

2.4. The case of a stratified continuous probability distribution.

Let p (

x)

be a bounded probability density function satisfying the following conditions:

i)

p(x)

^{= 0 for xE}

[a,

^{b ],}

i.e., the range of the variable x is given by

R =sup{ xl

p(x)=tO,

^{xE [a, b J}-inf { xl}

p(x)=tO,

^{xE [a,b ]},}

ii)

f

⁰

^{p(x)dx =}

^I.

Then the mean µ , the variance a² of the distribution are given by

µ = f_~ ^{xp(x)dx, a}

²

⁼ f_~ (x-µ)2 p(x)dx,

respectively. Let { P; (

x)

^{I 1} ~ ⁱ ~

m}

be a set of functions of the i-th stratum in

m

strata satisfying the condidions:

i)

p/x) =

0 forxE[a,b],

i.e., the range of the variable x is given by

Ri

= sup{

x

I

Pi (x )=t

0,

xE [a, b

]}-inf {

x

I

Pi (x )=t

0, xE

[a, b]}

for 1 ~ i ~ m,

m

ii)

p(x) = LPi(x).

i=l

The mean A, the variance ai² of the values on the i-th stratum are given by A

= Joo

^X

p/x) dx,

- 0 0 p

l

respectively, where

~ ⁼ f_~ Pi (x )dx,

m

I~=

^1.

On the Evaluations of Sum of Squares by Using the Range III

We define a sum of squares g(t) as follows:

· m [ 2

g(t) = t1gi (t ), where 9i (t) =

00

(x-t) Pi (x )dx.

From Theorem 3 in [ 6], the minimum of g

(t)

is given by g

(µ) = ri.

Theorem 9. The minimum g

(µ)

of the sum of squares for a stratified probability distribution is evaluated by the, range of the values as follows:

. [ 1 1 ^m

2)]

0 < g(µ) ~ ^{min -R2,}

-(I~R/+R_ ,

. 4 4 ^1=!

where

Ri = sup{ x

I

A (x ):;t: 0, xE [a, b ]}-inf { x

I

pi (x) :;t: 0, xE [a, b]}

R_

=

max{µ₁ll~i~m }-min{All~i~m }.

Proof We have

m

f

^{oo 2 pi (}

^X)

^{m 2}

g(µ)

^.=

ttpi _,( ?t -µt)

~ dx +

tt

^P1

^{(µi -µ)}

= E {

o-/} +

Var{µ1 },

where

E{o}} = f ~(J/,

i=l

Then, from Theorem 4 in [6] we have

I ^I 0 < g

(µ)

~

-

R ²,

4 whereas from Theorem 2 we have

2 1 2

0 < ^(Ji ~ -R;,

4 for each of the i-th stratum. Therefore

1 ^Ill 0 < E { ^{(J / }} ~

4

L

^~R/,

i=I

and we also have

0

~

^Var{µt}

~

^{: R_ 2 ,}

whence the conclusion of the theorem is given.

for 1 ~ i ~ m,

•

§ 3. Examples

Example 1. (Measurement of pinch power of an arm under rehabilitation, unit: kg.) Let us consider the data with the two strata Xi, X₂ as follows:

X₁ = { 3.2, 4.1, 4.0, 3.7, 3.0, 3.2, 3.5, 2.9, 5.1, 4.0}

x₂ = { 3.6, 4.3, 4.5, 5.o, 3.9, 3.8, 3.3, 3.8, 5.3, 3.5}

and

Then we have the following:

strata numbers max-min ranges means S.S.

XI 10 5.1 -2.9 2.2 3.7 3.961

Xz 10 5.3-3.3 2.0 4.1 3.920

X 20 5.3-2.9 2.4 3.9 8.806

The result of Theorem 3 states

max

(Lo, L1)

~ g

(x)

~ min (Mo, M ^{1 ),} where

R

=

^{4.1 -3.67}

=

^0.43,

L₀ = 0.5 X 2.4² = 2.88,

L, = 0.5 X (2.2²+2.0²+0.43²) = 4.51245, M₀ = 5 X2.4² = 28.8,

M, = 2.5 X 2.2² + 2.5 X 2.0² + 5 X 0.43² = 23.0245.

Therefore we have or

var.

0.3961 0.3920 0.4403

s.d.

0.6294 0.6261 0.6635

4.512 ~ g

(x)

~ 23.024,

0.2256 ~ ^s2 ~ 1.1512 (0.475 ~ s ~ 1.073).

On the other hand the result of Therem 4 states

max (Lo, L_{2 )} ~

g(x)

~ min (Mo, M20 ),

where

L₀ = 0.5 X 2.4² = 2.88,

L₂ = 0.5 X (2.2²+2.0²) = 4.42, M₀ = 5 X 2.4² = 28.8,

M₂₀ = max (M₂₁^, M₂₂⁾ = 27 .558,

M21 = 5X (2.4-0.5X2.2-0.5X2.0)²+0.25X (10x2.2²+10x2.0²) = 22.55,

On the Evaluations of Sum of Squares by Using the Range III

Therefore we have or

4.420 $ g

(x)

$ 27.558 0.2210 $ s² $ 1.3779

(in fact, g

(x)

= 8.806), (0.470 $ s $-1.174).

Example 2. (Examination point ratios of three groups, unit: %.) Let us consider the data X with the three strata X_1, X_2, X₃ in the frequency table as follows.

boundaries 20-30 30-40 40-50 50-60 60-70 70-80 80-90

(midpoints) (25) (35) (45) (55) (65) (75) (85)

XI 0 0 43 147 169 78 2

X2 0 8 86 183 65 3 0

X3 0 24 91 70 9 0 0

Then we have the following:

strata numbers max-min ranges means S.S. var. s.d.

XI 439 85-45 40 61.56 35310 80.42 8.97

X2 345 75-35 40 54.10 19220 55.71 7.46

X3 194 65-35 30 48.30 10890 56.13 7.49

X 978 85-35 50 56.30 91650 93.71 9.68

The result of Theorem 3 states

max ( L0 , L1 ) $ g (

x)

$ min

(M

O, M _{1 ),} where

R_

=

61.56-48.30

=

^13.26,

L₀ = 0.5 X 50² = 1250,

L, = 0.5 X (40²+40²+30²+ 13.26²) = 2137.9, M₀

=

244.5 X 50²

=

611250,

M₁

=

^{109.75 x402+86.25 X402}+48.5 x 30²+244.5 x 13.26²

=

400246.826.

Therefore we have or

2137.93 :s; g

(x)

:s; 400246.83 2.186 :s; s ² :s; 409.250

(in fact, g

(x) =

91650), (1.48 :s; s :s; 20.23).

On the other hand the result of Therem 4 states

max ( L0 , L2 ) ~ g (

x)

~ min (Mo, M 30 ), where

L₀ = 1250,

L₂ = 0.5 X (402+402+302) = 2050, M₀ = 244.5 X50² = 611250,

M30

=

max(M31, M 32)

=

603611.86,

X ( ^{,,...._l ,,...._l ,,...._l ) I} SI S l 3948 6

M31 ⁼ 0.001022 max T ^(i;23), ^{T (}2;31), ^{T (}3;12) + S1 + 2 + 3 = 4 .70,

,,..._I

T(1;23)

=

439X345X (50-O.4977X40-0.4971 X40) 2

,,...._]

+439 X 194 X (50-0.4977 X40-0.5 X 30) ²

+345Xl94X(0.4971X40-0.5X30)² = 36771600, T (2;31) = 33124400,

,,..._I

T (3;12) = 34688900,

S/ = (109.75-0.0005695) X40² = 175599,

Si

⁼ 137998,

s~

^{= 43650,}

M₃₂

=

0.001022 X max

(f~;

_23),

f~

₂^;31),

f~

1;12

))+

^{Si2 +Si+ S; =:=} 603611.86,

,,...._z

T (1;23) = 586335237 .23,

,,...._2

T(2JI) = 541027237.23,

,--..2

T (3;12) = 376346137.23, S1

2 = 1596.36,

s;

^{= 1595.36,}

S; = 895.36.

Therefore we have or

By Theorem 6 we have or

2oso ~ g

(x)

~ 603611.86, 2.096 ~ s² ~ 617.190 2099.49 ~ g

(x)

~ 400246.83,

2.147 ~ s² ~ 409.250

(1 .448 ~ s ~ 24.843).

(1.465 ~ s ~ 20.230),

On the Evaluations of Sum of Squares by Using the Range III

whereas by Theorem 7 we have

2099.49

s;

g

(x) s;

603611.86, or

2 .14 7 $ s ² $ 61 7. 190 (1 .465 $ s $ 24.843).

References

[ 1] C. F. Gauss: Theoria combinationis observationum erroribus minimis obnoxiae pars prior, Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores, 5 ( 1823):

(in the book C. F. Gauss: Gosa-ran, edited and translated into Japanese by T. Hida and T. Ishikawa, Kinokuniya Shoten, Tokyo, 1981.)

[2] M. J. R. Healy: Matrices for Statistics, Oxford University Press, New York, 1986.

[3] R. V. Hogg and A. T. Craig: Introduction to Mathematical Statistics, 4th ed., Collier Macmillan International Editions, 1978.

· [ 4] T. Kitagawa: Suisoku-tokei-gaku I (in Japanese), Iwanami Shoten, Tokyo, 1958.

[5] T. Mendori, Y. Zinno and H. Fukaishi: On the evaluations of sum of squares by using the range, Mem. Fae. Educ., Kagawa Univ.11, 53(2003), 1-12.

[6] T. Mendori, Y. Zinno and H. Fukaishi: On the evaluations of sum of squares by using the range II, Mem. Fae. Educ., Kagawa Univ. II, 53 (2003), 73-83.

[7] A. M. Mood, F. A. Graybill and D. C. Boes: Introduction to the Theory of Statistics, 3rd ed., McGraw-Hill, 1974.

[8] G. W. Snedec.or, W. G. Cochran and D. F. Cox: Statistical Methods, 8th ed., The Iowa State University Press, Ames, 1989.

Toshihiko MENDORI

Emeritus Professor of Kagawa University

1388-3 Nii, Kokubunji-cho, Kagawa, 769-0101, JAPAN E-mail address: mendori@hkg.odn.ne.jp

y oshihisa ZINNO

Faculty of Liberal Arts, The University of the Air Japan Kagawa Study Center c/o Kagawa University

1-1 Saiwai-cho, Takamatsu-shi, Kagawa, 760-8522, JAPAN E-mail address: 9722116933@ksv.s37a-unet.ocn.ne.jp

Hirao FUKAISHI

Department of Mathematics, Faculty of Education, Kagawa University 1-1 Saiwai-cho, Takamatsu-shi, Kagawa, 760-8522, JAPAN

E-mail address : fukaishi@ed.kagawa-u.ac.jp