THE SUMMATION NOTATION Ʃ


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1 Sngle Subscrpt otaton THE SUMMATIO OTATIO Ʃ Most of the calculatons we perform n statstcs are repettve operatons on lsts of numbers. For example, we compute the sum of a set of numbers, or the sum of the squares of the numbers, n many statstcal formulas. We need an effcent notaton for talkng about such operatons n the abstract. Sngle Subscrpt otaton Sngle Subscrpt otaton Lst ame Subscrpt The symbol s the lst name, or the name of the varable represented by the numbers on the lst. The symbol s a subscrpt, or poston ndcator. It ndcates whch number n the lst, startng from the top, you are referrng to. 4 Sngle Subscrpt otaton Sngle Subscrpt otaton 4 Sngle subscrpt notaton extends naturally to a stuaton where there are two or more lsts. For example suppose a course has 4 students, and they take two exams. The frst exam could be gven the varable name, the second Y, as n the table below. Chow s score on the second exam s observaton Y
2 Sngle Subscrpt otaton Student Y Smth 87 8 Chow Benedett 8 90 Abdul 9 97 Double Subscrpt otaton Usng dfferent varable names to stand for each lst works well when there are only a few lsts, but t can be awkward for at least two reasons. In some cases the number of lsts can become large. Ths arses qute frequently n some branches of psychology. When general theoretcal results are beng developed, we often wsh to express the noton of some operaton beng performed over all of the lsts. It s dffcult to express such deas effcently when each lst s represented by a dfferent letter, and the lst of letters s n prncple unlmted n sze. 7 8 Double Subscrpt otaton j Double Subscrpt otaton The frst subscrpt refers to the row that the partcular value s n, the second subscrpt refers to the column. 9 0 Double Subscrpt otaton Test your understandng by dentfyng n the table below. 4 4 Sngle Summaton otaton Many statstcal formulas nvolve repettve summng operatons. Consequently, we need a general notaton for expressng such operatons. We shall begn wth some smple examples, and work through to some that are more complex and challengng.
3 Sngle Summaton otaton Many summaton expressons nvolve just a sngle summaton operator. They have the followng general form stop value summaton ndex start value Rules of Summaton Evaluaton. The summaton operator governs everythng to ts rght, up to a natural break pont n the expresson.. Begn by settng the summaton ndex equal to the start value. Then evaluate the algebrac expresson governed by the summaton sgn.. Increase the value of the ndex by. Evaluate the expresson governed by the summaton sgn agan, and add the result to the prevous value. 4. Keep repeatng step untl the expresson has been evaluated and added for the stop value. At that pont the evaluaton s complete, and you stop. 4 Evaluatng a Smple Summaton Expresson Suppose our lst has just numbers, and they are,,,,. Evaluate Answer: = 7 Evaluatng a Smple Summaton Expresson Order of evaluaton can be crucal. Suppose our lst s stll,,,,. Evaluate Answer: ( ) = 7 = 89 The Algebra of Summatons Many facts about the way lsts of numbers behave can be derved usng some basc rules of summaton algebra. These rules are smple yet powerful. The frst constant rule The second constant rule The dstrbutve rule The Frst Constant Rule The frst rule s based on a fact that you frst learned when you were around 8 years old: multplcaton s smply repeated addton. That s, to compute tmes, you compute ++. Another way of vewng ths fact s that, f you add a constant a certan number of tmes, you have multpled the constant by the number of tmes t was added. 7 8
4 The Frst Constant Rule Symbolcally, we can express the rule as: y x a = ( y x + ) a The Frst Constant Rule (Smplfed Verson) Symbolcally, we can express the rule as: a = a 9 0 The Frst Constant Rule (Applcaton ote) The symbol a refers to any expresson, no matter how complcated, that does not vary as a functon of, the summaton ndex! Do not be msled by the form n whch the rule s expressed. Expand and evaluate the sum: Soluton: ( ) = = ( ) Express the sum usng summaton notaton: Soluton: + = Express the sum usng summaton notaton: n 4 8 Soluton: n = n 4 4
5 Evaluate (a) (b) Soluton (a) k 4 ( + ) = ( + ) + ( + ) + ( + ) + ( + ) (b) k = j= k = k ( + ) j j= a ( ) ( ) = = a = a + a + a + a 4 j Use the summaton propertes to 40 4 evaluate (a) (b) (c) ( ) Soluton (a) 40 = 40() = 00 Soluton Exercse Evaluate the followng: (b) (c) ( + ) = = = ( ) = = 4(4 + )( 4 + ) = 4() = Exercse Smplfy the followng: ( ) j The Second Constant Rule The second rule of summaton algebra, lke the frst, derves from a prncple we learned very early n our educatonal careers. When we were frst learnng algebra, we dscovered that a common multple could be factored out of addtve expressons. For example, x + y = ( x + y) 9 0
6 The Second Constant Rule The rule states that Agan, the rule appears to be sayng less than t actually s. At frst glance, t appears to be a rule about multplcaton. You can move a factorable constant outsde of a summaton operator. However, the term a could also stand for a fracton, and so the rule also apples to factorable dvsors n the summaton expresson. a = a The Second Constant Rule (s) Apply the Second Constant Rule to the followng: y The Dstrbutve Rule of Summaton Algebra The thrd rule of summaton algebra relates to a another fact that we learned early n our mathematcs educaton  when numbers are added or subtracted, the orderng of addton and/or subtracton doesn't matter. For example ( + ) + ( + 4) = ( ) The Dstrbutve Rule of Summaton Algebra So, n summaton notaton, we have ( + ) = + Y Y Snce ether term could be negatve, we also have ( ) = Y Y 4 Defnton: The Sample Mean and Devaton Scores The sample mean of scores s defned as ther arthmetc average, = The orgnal scores are called raw scores. The devaton scores correspondng to the raw scores are defned as dx =
THE SUMMATION NOTATION Ʃ
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