A Primer on Summation Notation George H Olson, Ph. D. Doctoral Program in Educational Leadership Appalachian State University Spring 2010

Transcription

1 Summato Operator A Prmer o Summato otato George H Olso Ph D Doctoral Program Educatoal Leadershp Appalacha State Uversty Sprg 00 The summato operator ( ) {Greek letter captal sgma} s a structo to sum over a seres of values For stace f we have the set of values for the varable = { } the 5 = Lterally the expresso says: begg wth = ad edg wth =5 sum over the varables As a example let = 8 = 0 3 = 4 = =5 5 = 6 The = 5 {the umber of cases} ad 5 = = 60 I may cotexts t s clear that the summato s over all cases ad we do ot eed the superscrpt over the summato operator Furthermore most cotexts t s assumed that the summato begs wth = Hece the otato 5 mply s take to I most stuatos where the varable has oly oe subscrpt as the subscrpt ca be omtted I these stuatos mples I other cotexts the varable may have more tha oe subscrpt eg Ths occurs for stace whe dvduals belog to two or more subgroupgs or crossclassfcatos We mght have a stuato as show below Table Table Group Group Group

2 Here we have three groups each wth a dfferet umber of cases We deote the th case the th group wth the symbol To sum all the cases over all three groups we would use the followg double summato operator 3 whch structs us to sum over the three groups (= ad 3) ad wth each group sum over the umber of cases the group (= 3 4 for Group ; = for Group ; = for Group 3) For smplcty we ofte wrte the summato expresso as where t s assumed that we are to sum over all groups ad all cases wth each group For example lets substtute the followg umbers for the symbolc values gve above Table Group Group Group The = = [0+8++3] +[ ] +[ ] = = 43 A more complex stuato occurs whe cases are grouped to crossclassfcatos Table 3 represets a stuato where cases are cross-classfed by sex ad age-category

3 Table 3 Age Category Chld Adolescet Adult Male Sex Female I the table the otato k deotes the th dvdual the th row (Sex category) ad kth colum (Age Category) Hece 43 s the last case the Adult- Female cell To dcate summato over all the cases the table we would use the otato k k where t s assumed that the summato s over all cases over all rows ad all K colums k Dot otato It s ofte easer to deote aggregates usg dot otato I a expresso such as the dot ( ) represets aggregato or summato over the mssg (dotted) subscrpt For stace the example gve earler where we had 5 = 60 we could smply wrte = 60 {read -dot = 60} Usg dot otato we would represet the cell aggregates the Sex by Age Category table above as show Table 4 Table 4 Age Category Chld Adolescet Adult Sex Male 3 Female 3

4 where for the Male-Adolescet cell = If we wated to deote the sum of the values for all the males we would wrte ; for all females Smlarly the sum of the values for all chldre s ; all adolescets ; ad all adults 3 Ad the sum of the values for all the cases ote that = k = Rules of summato = k k ad k Summato of a costat Let c be some costat value The c c I other words summg a costat tmes s the same as multplyg the costat by Hece f c = 5 the c 5 60 Ths rule ca be exteded to double summato Thus c= c = c As a example cosder the stuato volvg the three groups gve earler Table If all cases all groups have the costat value 0 the = 0 = [ ] + [ ] + [ ]

5 = (4 0) + (6 0) + (5 0) = 5 0 = 50 Multplcato by a costat If all the values of a varable are multpled by the same costat c the c c c For example let the set of 5 values of the varable be { } assume that each s multpled by the costat The = (3)+(9)+(5)+(7)+(0) = ( ) = 34 = = c Aga ths ca be expaded to multple summatos: ad so o k c c = c c c = c k k As a example aga cosder the three group stuato gve earler Table If all cases were multpled by 5 the 3 5( ) = [(5 0) + (5 8) + (5 ) + (5 3)] +[(5 6) + (5 ) + (5 8) + (5 0) + (5 8) + (5 )]

6 +[(5 4) + (5 6) + (5 6) + (5 0) + (5 9)] = 5( ) =5 3 = 0(43) = 75 ( ) I some stuatos the values dfferet groups are multpled by dfferet costats For stace suppose the values Group ( the example ust gve) were multpled by the costat c the values Group by c ad the values group 3 by c 3 The the sum of all the cases would be gve by c I ths stuato t s ecessary that the costats rema wth the summato operator ow suppose that the Sex by Age Category example gve earler we have the values as show below Table 5 Table 5 Age Category Chld Adolescet Adult Male Sex Female If all the male values are multpled by the costat 5 ad all the female values are multpled by the costat 0 The lettg c = 5 ad c = 0 the sum over all cases rows ad colums s gve by k c = c k k k = 5(4+0+7) + 0(+6+5)

7 = 5(3) + 0(3) = 475 ote that the rght-had sde of the summato otato above could have bee wrtte as c k k wthout the paretheses Order of operatos The order of operatos whe usg summato operators s the same as that for arthmetc That s operatos volvg the values a summato operato are dcated by mathematcal puctuato Whe dcated by puctuato operatos are to be carred out o values pror to summato Some examples should suffce ( 3 ) ad log log log log O the other had ( 3) ad log log( ) ote that these rules apply eve whe we have multple summatos For example lettg captal represet the umber of groups

8 I words wth each group : sum the values over cases the square the sum After squarg the sums wth all of the groups sum the squared sums over groups For example cosder the data gve Table earler = = 6899 Dstrbutve rule of summato The summato operator s dstrbutve whe the value beg operated upo s tself a sum (or dfferece) For example Y Y ote that the last step ths equato follows from the rule pertag to summato over a costat gve earler Hece Summato volvg two or more varables If each case has values o two varables ad Y say the

9 Y Y Istead Y = ( Y + Y + + Y ) These rules ca be exteded qute logcally to stuatos volvg more tha two varables Exercses I each of the followg exercses use symbols to exted the expresso gve the summato operato For example 7 3 = ( ) 6 Y l l Y Y Z Reduce the followg exteded expressos to a expresso usg the summato operator For example

10 ( ) would be wrtte as Y + Y + 3 Y Y Y Y 6 8 Y + Y + + Y 9 ( 5 + Y 5 ) + ( 6 + Y 6 ) + ( 7 + Y 7 ) + + ( 0 + Y 0 ) 0 Y Y Y Y Y5