stat_notation_contd

stat_notation_contd - Some Statistical Notation(contd 12...

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Page 13 12. SD(x + y) = ? Let x, y be lists having the same number of entries. If someone tells you the average of x, the average of y, but nothing more about the lists, can you find the average of x + y? Yes, because property (iv) of average tells you that adding the two averages will give you the aver- age of x + y. What about SD? If someone tells you the SD of x, the SD of y, but nothing more, can you get the SD of the sum, that is, the SD of x + y? No. There is no formula into which you can plug the two SDs and get the SD of the sum. The next example, which involves two hypotheti- cal stock portfolios, owned by Ann and Ed, shows why you can’t look for such a formula. Ann’s stock portfolio is small: she owns one share of stock in company A and one share of stock in company B. Ed’s portfolio is also small: one share in company A (like Ann) and one share in company C. The columns in the table on the next page show the prices, in dollars, of Some Statistical Notation (cont’d).
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Page 14 the various shares over the first five months of a year. So A was sell- ing for $2 at the end of January, $5 at the end of February, and so on. Ann’ s Portfolio Ed’s Portfolio A 2 5 9 11 13 B 3 9 11 17 20 Month Jan. Feb. Mar. Apr. May A 2 5 9 11 13 C 20 17 11 9 3 Now suppose, in what follows, you don’t know the numbers in the table. You have two tasks. In both, a reliable person is going to give you some information, and you have to calculate an SD using that in- formation. For the first task, the reliable person will tell you the SDs of the two columns in Ann’s portfolio—that is, the SD of the A column and the SD of the B column—and ask you to work out the SD of A + B. For the second task, the same person will tell you the SDs of the two columns in ED’s portfolio, and ask you for the SD of A + C. Suppose there were a formula for SD(x + y) in terms of SD(x) and SD(y). Then, as soon as the reliable person tells you the two SDs for Ann’s portfolio, you could plug them into the formula, do the calcula- tions specified in the formula, and get the SD of A + B. And for Ed’s portfolio, you could do the same thing with the SDs of the A and C columns to get the SD of A + C. In the first task, the formula is supposed to give the SD of the A + B column below. Month Jan. Feb. Mar. Apr. May A 2 5 9 11 13 B 3 9 11 17 20 Ann’ s Portfolio A + B 5 14 20 28 33 In the second task, the formula is supposed to give the SD of the A + C column—see the top of the next page.
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Page 15 Month Jan. Feb. Mar. Apr. May A 2 5 9 11 13 C 20 17 11 9 3 Ed’ s Portfolio A + C 22 22 20 20 16 The reliable person knows the numbers in the tables, and can calcu- late: SD of the A + B column = 9.9 (to one place of decimal) SD of the A + C column = 2.2 (to one place of decimal) Here is the punchline. No formula can deliver the 9.9 and the 2.2. The explanation is just a matter of looking closely at the data for the two portfolios. First, the A columns in the two portfolios are the same: Ann’ s Portfolio A 2 5 9 11 13 Month Jan.
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