A Comparison and Contrast of Some Methods for Sample Quartiles

Transcription

1 A Compaison and Contast of Some Methods fo Sample Quatiles Anwa H. Joade and aja M. Latif King Fahd Univesity of Petoleum & Mineals ABSTACT A emainde epesentation of the sample size n = 4m ( =, 1, 2, 3) is exploited to wite out the anks of quatiles exhaustively which in tun help compae anks fo quatiles povided by diffeent methods available in the liteatue. The citeion of the equisegmentation popety that the numbe of intege anks below the fst quatile, that between the consecutive quatiles, and that above the thd quatile ae the same, has been used to compae and contast diffeent methods. Fou segmentation identities can be obtained fo each method of quatiles which show clealy the numbe of obsevations in each of the fou quates if the obsevations ae distinct. The Halving Method and the emainde Method have been poposed fo the calculation of sample quatiles. The quatiles povided by each of these two methods satisfy the equisegmentation popety if the obsevations ae distinct. Moe inteestingly, in these two methods also epesents the numbe of quatiles having intege anks. Keywods:Quatiles; emaindes; Modulus; Quantiles. 1. Intoduction Quatiles, deciles, pecentiles o moe geneally factiles ae uniquely detemined fo continuous andom vaiables. A p th quantile of a andom vaiable X (continuous o discete) is a value x p such that P( X < x p ) p and P( X x p ) p. Let X be a continuous o discete andom vaiable with pobability function f (x) and the cumulative distibution function F( x) = P( X x). If the distibution is continuous, then P( X < x p ) = p and P( X x p ) = p since P ( X x ) =. Theefoe, fo the continuous case, F( x p ) = p. = p The quatiles Q 1 = x.25, Q2 = x. 5 and Q 3 = x. 75 fo a continuous andom vaiable with cumulative distibution function F (x) ae defined by F ( x. 25 ) =.25, F( x. 5 ) =. 5 and F( x. 75 ) =.75 espectively. Let X follow an exponential distibution with the pobability density function f x e x 1 x / β ( ) = β, > with the cumulative distibution function F( x) x / β = 1 e. Then 1 Q 1 / β / e = 1/ 4, 1 Q2 β = / e 2 / 4 and 1 e Q3 β = 3/ 4

2 so that Q 1 = β ln( 4 / 3), Q2 = β ln 2, Q3 = β ln 4. Howeve, fo the discete distibution, one has to use the basic definition. Conside the binomial distibution B ( n = 4, π = 1/ 2). The pobability mass function is given by 4 ( ) 4 x (1/ 2), x =,1, L,4; f ( x) = elsewhee. Then x 1, is the fst quatile of the distibution since.25 = P( X P( X < 1) = P( X 1) = P( X = ) = , = ) P( X = 1) = Similaly x 2, is the second quatile of the distibution since.5 = PX ( < 2) =.35.5, PX ( 2) = Note that the median is the same as.5-quantile o the 5 th pecentile, o the 5 th decile. It is not supising that the 6 th pecentile, x.6 = 2, since P ( X < 2) = and P( X 2 ) = Similaly it can be checked that the thd quatile is given by x = In case we have a sample (discete in natue), it is, howeve, difficult to define quatiles. One method, called the Hinge Method, is based on finding the median fst and then finding the medians of the uppe and lowe halves (including oiginal median in both halves) of the data. Done so, oughly 25% obsevations emain below the lowe quatile and 25% above the uppe quatile. A sample quantile is a point below which some specified popotion of the values of a data set lies. The median is the.5 quantile because appoximately half of all the obsevations lie below this value. The name factile fo quantile is used by some authos (see Lapin [6], p. 52). A emainde epesentation of the sample size n = 4 m ( =,1,2,3) is exploited to wite out the anks of quatiles exhaustively which in tun help compae anks fo quatiles povided by diffeent methods available in the liteatue. Some of them diffe only by vaious ounding notions of the coesponding anks fo quatiles. We compae and contast diffeent methods of quatiles in the light of equisegmentation popety that the numbe of intege anks below the fst quatile, that between the

3 consecutive quatiles, and that above the thd quatile ae the same. Fo each method of quatiles, fou segmentation identities ae obtained which clealy show the numbe of obsevations in each of the fou quates if the obsevations ae distinct. The Halving Method and the emainde Method have been poposed fo the calculation of sample quatiles. The quatiles povided by each of these two methods divide the odeed sample obsevations in fou quates with the same numbe of obsevations in each segment and povide the numbe of quatiles having intege anks if the obsevations ae distinct. 2. The Popula Method Thee ae many methods available fo calculating sample quatiles in diffeent elementay text books on statistics without any explanation. The most popula one, called the Popula Method heeinafte, is descibed below. The ank of the i(i = 1, 2, 3)th quatile is given by i ( n 1)/4 = l d, i = 1, 2, 3 (2.1) whee l is the lagest intege not exceeding i ( n 1) / 4. Then the Popula Method uses the following linea intepolation fomula fo the calculation of sample quatiles Q, (i = 1, 2, 3), (2.2) i = x( l) d ( x( l 1) x( l) ) = (1 d) x( l) d x( l 1) whee x (l) is the lth odeed obsevation (Ostle et al. [], p. 38). To wite out the anks exhaustively let us denote the sample size n( 4) by the following emainde-modulus epesentation n= 4 m = ( mod 4), ( =, 1, 2, 3), (2.3) so that the numbe of obsevations in each of the fou segments is given by m = ( n ) / 4. With this epesentation of the sample size, the anks and the quatiles of a sample will be denoted espectively by and Q ; i = 1, 2, 3; =,1, 2, 3. Though quatiles Q ; i = 1, 2, 3; =,1, 2, 3 ae usually denoted by Q i ; i = 1, 2, 3, we will not suppess as it plays an impotant ole in compaing the anks of quates given by diffeent methods. Let the numbe of obsevations in each segment be m i (i = 1, 2, 3, 4). Then the equisegmentation popety guaantees that m 1 = m2 = m3 = m4 if the obsevations ae distinct. In case, 1 n 3, the above fomulae can also be used to calculate quatiles with m =. It is inteesting to note that though the Popula Method is not based on good mathematical easoning, the equisegmentation popety is satisfied by the quatiles povided by this method fo all sample sizes n= 4 m ( m 1; =,1,3) if the obsevations ae distinct. Fo n= 4m 2 ( m 1), the numbe of obsevations in fou segments ae given by

4 m, ( m 1), ( m 1) and m espectively if the obsevations ae distinct. Thus it is essential to modify the fomulae of anks so that the equisegmentation popety is satisfied by quatiles povided by the Popula Method fo any sample size. It is obseved that wheneve n = 4 m 2, simple aithmetic ounding of anks povided by this method would satisfy the equisegmentation popety. Example 2.1 The sizes of the police foces in the ten lagest cities in the United States in 1993 (the numbes epesent hundeds) ae given below: (Bluman [1], p.7). We now calculate quatiles by the Popula Method. Hee the sample size is n = 1 = 4(2) 2 so that m = 2 and = 2. Fo = 2 we will denote the anks of quatiles by i 2 ( i = 1, 2, 3). The anks of the quatiles povided by the Popula Method ae (see equation 2.1) given by = n 1) / 4 = 2.75, = ( n 1) / 2 = 5.5, = 3( n 1) / 4 ( = so that by linea intepolation (see equation 2.2) the quatiles ae given by (8.25) (8) (9) 8.25 Q = x (2.75) = (1.75) x (2).75 x (3) =.25(1.9).75(2.) = Q = x (5.5) = (1.5) x (5).5 x (6) =.5(3.9).5(4.7) = 4.3 Q = x = (1.25) x.25 x =.75(7.6).25(.1) = To check the equisegmentation popety, we show the position of the quatiles by downwad aows in the sample: We obseve that thee ae 2 ( = m), 3 ( = m 1), 3 ( = m 1) and 2( = m) obsevations in the fou segments, i.e. the quatiles do not satisfy the equisegmentation popety fo n = 4 m A eview of the Well-known Fomulae of Sample Quatiles In this section we suvey the fomulae fo quatiles available in the liteatue. We povide algebaic expessions fo quatiles by all existing methods in the liteatue. The use of emainde allows us to figue out the decimal pat of the fomulae of anks fo quatiles fo a paticula sample of size n. Let m = ( n ) / 4, n = 4m 4, and be the ank of i th quatile with m obsevations in each segment. Then the ank of the i th quatile is given by (4m ) 1 = i = im i( 1) / 4 = im [ u ] d / 4 (3.1) 4

5 whee i and ae integes with1 i 3, 3, [ u ] is the lagest intege less than o equal to u = u = i( 1) / 4 and i ( 1) = d (mod 4). The quatiles can then be calculated by the simple linea intepolation as Q (3.2) = ( 1 d / 4) x( im [ u]) ( d / 4) x( im [ u] 1) whee x (i) is the ith odeed obsevation, u = u( i, ) = i( 1) / 4, [u] is the geatest intege not exceeding u and d = d = 4( u [ u]). The above method will be called the Popula Method. Method 1 (Popula Method) The anks fo sample quatiles povided by the Popula Method can be witten out exhaustively as (see 3.1): 1 = m 1/ 4, 2 = 2m 2 / 4, 3 = 3m 3/ 4, = m / 4, = 2m 1, = 3m 1 2 / 4, 1/ 4, 2 = m 3/ 4, = 2m 1 2 / 4, = 3m 2 = m 1, = 2m 2, = 3m 3. The anks fo diffeent sample sizes povided by this method ae tabulated below: n = 4m n = 4 m 1 n = 4 m 2 n = 4 m 3 = = 1 = 2 = 3 1 m 1/ 4 m 2 / 4 m 3/ 4 m m 2 / 4 2 m 1 2 m 1 2 / 4 2 m m 3/ 4 3 m 1 2 / 4 3 m 2 1/ 4 3 m 3 Segmentation identities ae given by m m m m 1 ( m 1) 2, 1, ( m 1) , A ank appeaing as = 1 in the segmentation identity implies that the ank is an intege, and a ank appeaing as = implies that the coesponding ank is not an intege. It is seen that the equisegmentation popety is satisfied by the Popula Method fo =, 1, 3 but not fo = 2. Note that the Lapin Method (Lapin [7], 45-46) is a epesentation of the Popula Method accommodating simple linea intepolation.

6 Method 2 (Popula Method with Aithmetic ounding) This method is based on aithmetic ounding applied to the anks offeed by the Popula Method. The anks fo diffeent sample sizes povided by this method ae tabulated below: n = 4m n = 4 m 1 n = 4 m 2 n = 4 m 3 = = 1 = 2 = 3 1 m m 1 m 1 m m 1 2 m 1 2 m 2 2 m m 1 3 m 2 3 m 2 3 m 3 Segmentation identities ae given by ( m 1) m m m = 4m 1 It is seen that the equisegmentation popety is satisfied by the Popula Method only fo = 3. Method 3 (Mendenhall and Sincich Method) This method suggests to ound up the ank of the fst quatile povided by the Popula Method if the ank is halfway between two integes. It also suggests ounding down the ank of the thd quatile if the ank is halfway between two integes. It is easy to see that the suggestion by Mendenhall and Sincich ([9], p. 54) only applies to samples with size n = 4 m 1. Fo othe sample sizes anks offeed by the Popula Method do not lie exactly in the halfway between two integes, and as such those anks ae the same in both the Popula Method and the Mendenhall and Sincich Method. The anks fo diffeent sample sizes povided by this method ae tabulated below: n = 4m n = 4 m 1 n = 4 m 2 n = 4 m 3 = = 1 = 2 = 3 1 m 1/ 4 m 1 m 3/ 4 m 1 2 2m 2/4 2m 1 2 m 1 2 / 4 2m m 3/ 4 3 m 1 3 m 2 1/ 4 3m 3 Segmentation identities ae given by m m m m 1 ( m 1) 2 ( m 1)

7 It is seen that the equisegmentation popety is satisfied by the Mendenhall and Sincich Method fo =, 3 but not fo = 1, 2. Method 4 By this method, the anks of quatiles ae given by α = αn/4, α = 1, 2, 3. Sepaate the lagest intege ( i ) not exceeding α, and decimal pat ( d ) of α and wite α = i d. The α th ( α = 1, 2, 3) quatile is finally given by Q α, ( α = 1, 2, 3) = x( i) d ( x( i 1) x( i) ) = (1 d) x( i) d x( i 1) whee x (i) is the ith obsevation. This method is a slight vaiation of the Popula Method discussed in Section 2. The anks fo diffeent sample sizes povided by this method ae tabulated below: n = 4m n = 4 m 1 n = 4 m 2 n = 4 m 3 = = 1 = 2 = 3 1 m m 1/ 4 m 1/ 2 m 3/ m 2 m 1/ 2 2 m 1 2 m 1 1/ m 3 m 3/ 4 3 m 1 1/ 2 3 m 2 1/ 4 Segmentation identities ae given by ( m 1) m m m 1 ( m 1) 2 ( m 1) = 4m 1, ( m 1) = 4m 2, ( m 1) 3, ( m 1) = 4m 3. It is seen that the equisegmentation popety is not satisfied fo any =,1,2, 3. Method 5 (Hines and Montgomey [2], p. 18) anks of quatiles ae given by α = αn/4.5, α = 1, 2, 3. Sepaate the lagest intege (i) not exceeding α, and decimal pat ( d ) of α and wite α = i d. The α th ( α = 1, 2, 3) quatile is finally given by Q α, ( α = 1, 2, 3) = x( i) d ( x( i 1) x( i) ) = (1 d) x( i) d x( i 1) whee ) (i x is the ith obsevation. This method is a slight vaiation of Method 4. The anks fo diffeent sample sizes povided by this method ae tabulated below:

8 n = 4m n = 4 m 1 n = 4 m 2 n = 4 m 3 = = 1 = 2 = 3 1 m 1/ 2 m 3/ 4 m 1 m 11/ m 1/ 2 2 m 1 2 m 1 1/ 2 2 m m 1/ 2 3 m 1 1/ 4 3 m 2 3 m 2 3/ 4 Segmentation identities ae given by m m m 1 ( m 1) ( m 1) = 4m 3 It is seen that the equisegmentation popety is satisfied by this method fo =, 1, 2 but not fo = 3. Method 6 (Johnson [5], p. ) The anks of the quatiles ae given by α = αn/4, α = 1,2,3. Sepaate the lagest intege (i) not exceeding α, and decimal pat (d) of α and wite α = i d. If n / 4 is not an intege, ound it up to the next intege and find the coes- ponding odeed obsevation. If n / 4 is an intege, calculate the mean of the (n / 4) th and the next obsevation. The anks fo diffeent sample sizes povided by this method ae tabulated below: n = 4m n = 4 m 1 n = 4 m 2 n = 4 m 3 = = 1 = 2 = 3 1 m 1/ 2 m 1 m 1 m m 1/ 2 2 m 1 2 m 1 2 m m 1/ 2 3 m 1 3 m 2 3 m 3 Segmentation identities ae given by m m m m 1 2 3, 2, 3. 1, It is seen that the equisegmentation popety is satisfied by this method fo =, 3 but not fo = 1, 2.

9 Method 7 (Hinge Method) An inteesting method to find exteme quatiles is based on finding the median fst, and then finding the medians of uppe and lowe halves of the data. The tadition is to count the median in both halves (Maye and Sykes [8], p. 25). Tukey ([16], pp. -35) called them hinges. Fo n= 4m, it follows that the ank of the median is = (1 4m) / 2 = 2m 2 / 4. 2 Then by the Hinge Method = [1 (2m 2 / 4)]/ 2 = m 3/ 4 and = [(2m 2 / 4) m ]/ 2 = 3m 1/ 4. Fo n = 4 m 1, it follows that the ank of the median is = [1 (4m 1)]/ 2 = 2m 1. Then by the Hinge Method = [1 (2m 1)]/ 2 = m 1 and = [(2m 1) (4m 1)]/ 2 = 3m 1. Fo n = 4 m 2, it follows that the ank of the median is = [1 ( 4m 2 )]/ 2 = 2m 1 2 / 4. Then by the Hinge Method = [1 (2m 1 2 / 4)]/ 2 = m 1 1/ 4 and = [(2m 1 2 / 4) (4m 2)]/ 2 = 3m 1 3/ 4. Fo n = 4 m 3, it follows that the median is = [1 (4m 3)]/ 2 = 2m 2. Then by the Hinge Method = [1 (2m 2)]/ 2 = m 1 2 / 4 and = [(2m 2) (4m 3)]/ 2 = 3m 2 2 / 4. The anks fo diffeent sample sizes povided by this method ae tabulated below: n = 4m n = 4 m 1 n = 4 m 2 n = 4 m 3 = = 1 = 2 = 3 1 m 3/ 4 m 1 m 11/ 4 m 1 2 / m 2 / 4 2 m 1 2 m 1 2 / 4 2 m m 1/ 4 3 m 1 3 m 1 3/ 4 3 m 2 2 / 4 Segmentation identities ae given by m m 1 ( m 1) ( m 1) 2 3, 1, ( m 1) = 4m 2, ( m 1) = 4m 3. Clealy the equisegmentation popety is satisfied by the Hinge Method only fo =. Method 8 (Vinning Method) The fomulae given by Vinning ([17], p. 44) can be simplified as ( n 3)/4 th obsevation if n is odd Q1 = ( n 2)/4 th obsevation if n is even Q 3 (3n 1) / 4 th obsevation if n is odd = (3n 2) / 4 th obsevation if n is even The example he povides with n = 35 divides the odeed sample obsevations into fou segments with 9, 8, 8 and 9 obsevations among them. The median has an intege ank

10 namely the 18th position. The anks fo diffeent sample sizes povided by this method ae tabulated below: n= 4m n= 4m 1 n= 4m 2 n= 4m 3 = = 1 = 2 = 3 1 m 2/4 m 1 m 1 m 1 2/4 2 2m 2/4 2m 1 2m 1 2/4 2m 2 3m 2/4 3m 1 3m 2 3m 2 2/4 3 Segmentation identities ae given by m m m 1 ( m 1) ( m 1) = 4m 3 Clealy the equisegmentation popety is satisfied by the Vinning Method only fo = and = 2. Milton and Anold Method ([], pp ) suggested the anks of exteme quatiles to be 1 = ([( n 1) / 2] 1) / 2 and 3 = n 1 1 but it tuns out that they ae exactly the same as the anks of exteme quatiles given by the Vinning Method. Method 9 (Siegel Method) Siegel ([14], p. 7) suggests the anks of exteme quatiles to be 1 = ([( n 1) / 2] 1) / 2 and 3 = n 1 1 while, unlike any othe method, he suggests the ank of the median to be 2 = [( n 1) / 2] whee [ a ] is the lagest intege not exceeding a. The anks fo diffeent sample sizes povided by this method ae tabulated below: n= 4m n= 4m 1 n= 4m 2 n= 4m 3 = = 1 = 2 = 3 1 m 2/4 m 1 m 1 m 1 2/4 2 2m 2m 1 2m 1 2m 2 3m 2/4 3m 1 3m 2 3m 2 2/4 3 Segmentation identities ae given by m ( m 1) m ( m 1) ( m 1) 1 m ( m 1) 2 ( m 1) m ( m 1) = 4m 3 Clealy the equisegmentation popety is not satisfied fo any value of.

11 Method 1 (Smith Method) The fomulae povided fo pecentiles by Smith ([15], pp ) can be specialized to quatiles as n 2 th obsevation if n / 4 is not an intege 4 Q1 = 1 n n 4 th th obsevation if n / 4 is an intege Q 3 3n 2 th obsevation if 3 n / 4 is not an intege 4 = 1 3n 3n 4 th th obsevation if n / 4 is an intege He suggests ounding the anks to the neaest intege. The example he povides fo n = does satisfy the equisegmentation popety with m = 3. The anks fo diffeent sample sizes povided by this method ae tabulated below: n= 4m n= 4m 1 n= 4m 2 n= 4m 3 = = 1 = 2 = 3 1 m 1/2 m 3/4 m 1 m 1 1/4 2 2m 2/4 2m 1 2m 1 2/4 2m 2 3m 2/4 3m 1 1/4 3m 2 3m 2 3/4 3 Segmentation identities ae given by m m m 1 ( m 1) 2 3, 1, 2, ( m 1) = 4m 3. Clealy the equisegmentation popety is satisfied by the Vinning Method fo =, 1, 2 but not fo = 3. Method (Shao Method) It is supising that the method poposed by Shao ([], 1976, pp ) is the only method in the liteatue that enjoys the equisegmentation popety. a) If the sample size is divisible by 4, the quatiles can be easily detemined. When a quatile is located between two values, the mid point of these two values is consideed to be the quatile. b) If the sample size is not divisible by 4, the quatiles can easily be detemined in thee steps: (1) If the sample size is even, Q 1 is the median obtained fom the lowe 5% values of the sample.

12 (2) If the sample size is odd, Q 1 is the median obtained fom the lowe 5% values of the sample afte having discaded the median of the complete sample. (3) Locate Q 3 by the methods stated in (1) and (2) except that the uppe 5% of the values of the sample ae used in the pocess. The anks fo diffeent sample sizes povided by this method ae tabulated below: n = 4m n = 4 m 1 n = 4 m 2 n = 4 m 3 = = 1 = 2 = 3 1 m 2 / 4 m 2 / 4 m 1 m 1 2 2m 2/4 2m 1 2 m 1 2 / 4 2m m 2 / 4 3 m 1 2 / 4 3m 2 3m 3 Segmentation identities ae given by m m m m 1 2 3, 1, 2, 3. Obseve that the equisegmentation popety is satisfied by this method fo any value of. 4. Suggested Methods We discuss below two methods namely the Halving Method and the emainde Method each of which satisfies the equisegmentation popety. 4.1 The Halving Method We obseve that Method 7 guaantees the equisegmentation popety if the median of the whole data set is always ignoed in the calculation of the lowe and uppe quatiles. Method 1 with this kind of adjustment will heeinafte be called the Halving Method (Joade [3]). Example 4.1 We calculate below the quatiles of the data in Example 2.1 by the Halving Method. The ank of the median is = (1 n) / so that = Q = x = (1.5) x.5 x =.5(3.9).5(4.7) = 4.3. (5.5) (5) (6) The fst quatile is the median of the obsevations below the median of the whole sample, i.e. = (1 5) / 2 3 so that Q = x (3) = 2. The thd quatile is the median of the = obsevations above the median of the whole sample i.e. = (6 1) / 2 8 so that Q x (8) 7.6 = =. =

13 To check the equisegmentation popety, we show the position of the quatiles by downwad aows in the sample: We obseve that thee ae 2( = m) obsevations in each of the fou segments, i.e. the quatiles do satisfy the equisegmentation popety. The anks fo diffeent sample sizes povided by this method ae tabulated below: n= 4m n= 4m 1 n= 4m 2 n= 4m 3 = = 1 = 2 = 3 1 m 2/4 m 2/4 m 1 m 1 2 2m 2/4 2m 1 2m 1 2/4 2m 2 3m 2/4 3m 1 2/4 3m 2 3m 3 3 It may be emaked hee that the fst quatile is the median of the smallest n/2 obsevations if n is even, and that of the smallest ( n 1)/2 obsevations if n is odd. Similaly the thd quatile is the median of the lagest n/2 obsevations if n is even, and that of the lagest ( n 1)/2 obsevations if n is odd. 4.2 The emainde Method We obseve that each of the anks 1 and 3 given by the Halving Method is smalle than that given by the emainde Method by 1 / 4. We also obseve that the anks of the quatiles given by the Popula Method satisfy the equisegmentation popety if the ank is ounded down fo ( = 2, d = 1) and ounded up fo ( = 2, d = 3). A special kind of ounding applied to the anks povided by the Popula Method fo quantiles of even ode has been discussed by Joade [4]. The anks obtained this way, called the emainde Method, satisfy the equisegmentation popety. Let [u] be the lagest intege not exceeding u, and u the smallest intege exceeding u. Again let u = u = i( 1) / 4 and ( 1) = d (mod 4). Then we have the following theoem. Theoem 4.1 Let m = ( n ) / 4, n = 4m 4, and be the ank of the i th quatile with m obsevations in each segment. Then the anks given by im [ u ] = im u im u if (, d) A and d 2, if (, d) A and d > 2, if (, d) A (4.1a) (4.1b) (4.1c)

14 whee i and ae integes with 1 i 3 and 3, and A= {(, d) : = 2, d = 1,3}, satisfy the equisegmentation popety. If (, d) A, then the quatiles can be calculated by the simple linea intepolation as Q, = ( 1 d / 4) x( im [ u]) ( d / 4) x( im [ u] 1) whee x (i) is the ith odeed obsevation. Example 4.2 Let us now calculate the quatiles fo the sample in Example 2.1 by the emainde Method. Hee n = 1 = 4(2) 2 i.e. m= 2, = 2. Since u = 1(2 1) / 4 = 3/ 4 (i.e. = 2, d = 3 > 2), the ank of the fst quatile is = 1( m) u = 2 3 / 4 = 3 (see 4.1b). Again since u = 2(2 1) / 4 = 1 2 / 4 (i.e. = 2, d = 2 2), the ank of the second quatile is = ( m) u = 2(2) 1 2 / (see 4.1 c). Finally since u = 3(2 1) / 4 2 = = 2 1/ 4 (i.e. = 2, d = 1 2), the ank of the thd quatile is = 3( m) [ u] = 6 2 [1/ 4] = 8 (See 4.1a). So the quatiles ae Q = x (3) = 2., = (1.5) x.5 x = ( ) / 2 = 4.3, and = x (8) = 7.6. (5) (6) To check the equisegmentation popety, we show the position of the quatiles by downwad aows in the sample: We obseve that the equisegmentation popety is satisfied hee with m = 2. The anks of the quatiles fo diffeent sample sizes given by the emainde Method ae tabulated below: n= 4m n= 4m 1 n= 4m 2 n= 4m 3 = = 1 = 2 = 3 1 m 1/4 m 2/4 m 1 m 1 2 2m 2/4 2m 1 2m 1 2/4 2m 2 3m 3/4 3m 1 2/4 3m 2 3m 3 3 The Halving Method as well as the emainde Method satisfies the equisegmentation popety. It is woth noting that in each of the two methods the value of ( n = 4m ) is the numbe of quatiles with intege anks. The Shao Method, howeve, doesn t have algebaic expession fo the anks and hence may not be suitable fo using it o genealizing it to othe quantiles. Though the Halving Method is the simplest one and satisfies the equisegmentation popety, it seems to be difficult to genealize the notion to quantiles in geneal. Note that the emainde Method fo quatiles happens to be the Popula Method with aithmetic ounding fo oute quatiles when = 2. The emainde Method is genealized to quantiles of even ode by Joade [4].

15 It emains open to check the equisegmentation popety fo samples with ties. Finally it is woth emaking that fo a sample of lage size, the empical cumulative distibution function may be used to calculate sample quatiles (Mendenhall et al. [1], Section ) o oss, S. (1987, Section 4). Acknowledgements The authos acknowledge King Fahd Univesity of Petoleum and Mineals, Saudi Aabia, fo poviding excellent eseach facilities. The authos ae gateful to Pof. M. M. Ali, Ball State Univesity, USA, fo constuctive suggestions that have impoved the quality of the pape. efeences [1] Bluman, A. G. (). Elementay Statistics: A Step by Step Appoach. McGaw Hill, New Yok. [2] Hines, W. and Montgomey, D. C. (199). Pobability and Statistics in Engineeing and Management Sciences, New Yok: John Wiley. [3] Joade, A. H. (). The halving method fo sample quatiles, Intenational Jounal of Mathematical Education in Science and Technolog, 34(4), [4] Joade, A. H. (). The emainde method fo sample quantiles of even ode, Technical epot No. 274, King Fahd Univesity of Petoleum and Mineals, Saudi Aabia. [5] Johnson,. (2). Mille and Feund s Pobability and Statistics fo Enginees. Pentice Hall. [6] Lapin, L. L. (1975). Statistics: Meaning and Method, New Yok: Hacout Bace Jovanovich, Inc. [7] Lapin, L. L. (1997). Moden Engineeing Statistics, Duxbuy Pess. [8] Maye, A. D. and Sykes, A. M. (1996). Statistics, London: Edwad Anold. [9] Mendenhall, W. and Sincich, T. (1995). Statistics fo Engineeing and the Sciences. Englewood Cliffs, NJ: Pentice Hall. [1] Mendenhall, W., Beave,. J. and Beave, B. M. (). A Bief Intoduction to Pobability and Statistics, United States: Duxbuy. [] Milton, J. S. and Anold, J. C. (). Intoduction to Pobability and Statistics, New Yok: McGaw-Hill. [] Ostle, B., Tune, K. V., Hicks, C.. and McElath, G. W. (1996). Engineeing Statistics: The Industial Expeience, New Yok: Duxbuy Pess. [] Shao, S. P. (1976). Statistics fo Business and Economics, Columbus, Ohio: Chales

16 E. Meill Publishing Co. [14] Siegel, A. F. (). Pactical Business Statistics, New Yok :McGaw-Hill. [15] Smith, P. J. (1997). Into Statistics, Singapoe: Spinge-Velag. [16] Tukey, J. W. (1977). Exploatoy Data Analysis, eading, MA: Addison Wesley. [17] Vinning, G. G. (1998). Statistical Methods fo Enginees, New Yok: Duxbuy Pess. oss, S. (1987). Intoduction to Pobability and Statistics fo Enginees and Scientists. New Yok: John Wiley.

17 A Compaison and Contast of Some Methods fo Sample Quatiles Anwa H. Joade and aja M. Latif King Fahd Univesity of Petoleum & Mineals Thee ae about a dozen methods to find sample quatiles. The emegence of so many methods is due to non-igoous definition of quatiles. In this talk we pobe the issue, and suggest a new citeion of equisegmentation to detemine quatiles. The existing methods have then been checked in the light of this citeion and found that only Shao Method satisfies it. Two new methods have been poposed and illustated.