Exercise in Summation Skills In this class, we assume that you are able to work with summation algebra, and that you are comfortable with basic mathematical notation. Work through this handout to familiarize yourself with the notation and calculations. If you have questions or have difficulty completing this worksheet, please talk to the instructor or TA. We will use a small data set with 7 observations. For each unit i, values for each of 3 variables (x, y, z) are recorded. Note that for this data set, i = 1, 2, . . . , 7. Also notice that for i = 1, y1 = 7 and for i = 7, x7 = -4. i 1 2 3 4 5 6 7 yi 7 8 6 5 4 2 3 xi 3 3 3 3 -4 -4 -4 zi 4 2 3 1 5 2 4 Basic rules of summation 1. Add up all the numbers to get a sum. 76543211yyyyyyyynii++++++=∑= 35324568771=++++++=∑=iiy Exercise #1: sum the x’s and z’s; recall that you can add negative numbers (i.e., subtraction). 2. Sum a set of constants. What happens when you sum a set of n observations, each having the same value, k? nkkni=∑=1 213333333371=++++++=∑=i Exercise #2: What is the sum when n = 7 and k = 1/7. 3. Sum a set of numbers, each of which is multiplied by a constant. ∑∑===++++++=niiniiykkykykykykykykyky176543211 10535*3)3*3()2*3()4*3()5*3()6*3()8*3()7*3(371==++++++=∑=iiy. A general rule: algebra always uses the “inside out” rule: do the operation inside the parentheses before doing the operations outside the parentheses (i.e., multiply each pair of numbers before summing). Another general rule: you can think of division as multiplying inverses; i.e. ynny⎟⎠⎞⎜⎝⎛=1. Exercise #3: What well-known measure of central tendency would obtain if 711==nk? Exercise #4: What is the mean (average) of the x’s (x) and the z’s (z)?4. Sum a set of numbers, each with a constant added to it. ∑∑==+=+++++++++++++=+niiniinkykykykykykykykyky176543211)()()()()()()()( 562135)33()32()34()35()36()38()37()3(71=+=+++++++++++++=+∑=iiy. Exercise #5: What would you obtain if k = -5; that is, if ∑=−+71))5((iiy = ∑=−71)5(iiy? Exercise #6: Obtain ∑=−71)(iixx and ∑=−71)(iizz. Subtracting the mean from each data point is called “centering the data” or “correcting for the mean.” 5. Sum the products of pairs of numbers. ).*()*()*()*()*()*()*(776655443322111yxyxyxyxyxyxyxyxniii++++++=∑= Exercise #7: Show that ∑=71iiiyx= 42. Exercise #8: What is ∑=niiizx1and ∑=niiiyz1? 6. The sum of squares (uncorrected). Square each number and then sum. 2726252423222112yyyyyyyynii++++++=∑= 20394162536644932456872222222712=++++++=++++++=∑=iiy. Exercise #9: Show that 271712∑ ∑= =⎥⎦⎤⎢⎣⎡≠i iiixx. Notice that rule 6 is a special case of rule 5 (where yi2 = yi*yi), and that the sum of squares is not equal to the square of the sum (or the sum, squared). 2112⎥⎦⎤⎢⎣⎡≠∑∑==niiniiyy where .1225)35(2271==⎥⎦⎤⎢⎣⎡∑=iiy 7. The sums of squares (corrected for the mean) is calculated using rules 4 and 6: ∑∑==−=−+−+−+−+−+−+−=−niiniiynyyyyyyyyyyyyyyyyy1222726252423222112)()()()()()()()( Exercise #10: What is ∑=−712)(iixx and what is
View Full Document
Unlocking...