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A Primer on Summation NotationGeorge H Olson, Ph. D.

Doctoral Program in Educational LeadershipAppalachian State University

Spring !!"

Summation Operator

#he summation operator $%& 'Gree( letter, capitalsigma) is an instruction to sumover a series o* values. +or instance, i* e have the set o* values *or the varia-le, X 'X1,

X2, X3, X4, X5), then

=

=

/

0

n

i

iX X1+ X2+X3+ X4+ X5

Literally, the e1pression, =

=

/

0

n

i

iX , says2 -eginning ith i0 and ending ith i/,

sum over the variablesXi. As an e1ample, let

Xi ", X2 0!, X3 00, X4 0/, X5 03.

#hen n / 'the num-er o* cases), and

=

=

/

0

n

i

iX " 4 0! 4 00 4 0/ 4 03

3!.

5n many conte1ts, it is clear that the summation is over all cases and e do not

need the superscript over the summation operator. +urthermore, in most conte1ts it is

assumed that the summation -egins ith i 0. Hence, the notation, i

iX is ta(en to

imply=

=

/

0

n

i

iX . 5n most situations, here the varia-le has only one su-script, as inXi, the

su-script can -e omitted. 5n these situations, X implies =

=

/

0

n

i

iX .

5n other conte1ts, the varia-leXmay have more than one su-script, e.g.,Xij. #his occurs,*or instance, hen individuals -elong to to or more su-groupings or cross6

classi*ications. 7e might have a situation as shon -elo in #a-le 0.

#a-le 0

Group 0 Group Group 8

900, 90, 980, 9:0 90, 9, 98, 9:, 9/, 93 908, 98, 988, 9:8, 9/8

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Here e have three groups, each ith a di**erent num-er o* cases. 7e denote the

ith case in thejth group ith the sym-ol,Xij. #o sum all the cases, over all three groups,

e ould use the *olloing, dou-le summation operator,

8

j

n

i ij

j

X ,

hich instructs us to sum over the three groups $j0, , and 8& and, ithin each group,

sum over the num-er o* cases in the group $i0, , 8, : *or Group 0; i0, , 8, :, /, 3 *or

Group ; i0, , 8, :, / *or Group 8&. +or simplicity, e o*ten rite the summatione1pression as,

ijX ,

here it is assumed that e are to sum over all groups and all cases ithin each group.+or e1ample, lets su-stitute the *olloing num-ers *or the sym-olic values given a-ove.

#a-le

Group 0 Group Group 8

0!, ", 0, 08 3, 00, ", 0!, ", 0 0:, 3, 3, 0!,