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# Numbers, reciprocals, averages

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### Numbers, reciprocals, averages

1. 1. Basic math tools:Numbers, reciprocals, and averages juliohuato@gmail.com September 8, 2011
2. 2. TopicsNumbersReciprocalsAverages
3. 3. Numbers Economic data is reported in diﬀerent numeric formats. Examples: the BEA estimated the U.S. 2007 GDP in 14.42 trillion dollars, the BEA also estimated U.S. 2007 imports of goods and services to be approximately one sixth (1/6) of GDP, the BLS estimated that, as of November 2008, the unemployment rate was 6.7%, and the CB reported that in 2007, the ratio of money income of the richest ﬁfth of U.S. households to the poorest ﬁfth was approximately 15:1.
4. 4. Numbers In using numbers, we should be able to go back back and forth between the decimal, fraction, ratio, and percentage formats. Fractions, ratios, and percentages are also known as proportions, although expressed in three alternative ways. Examples: 20.00 = 20 = 200/10 = 20 : 1 = 2, 000% 1.0 = 1 = 12.5/12.5 = 1 : 1 = 100% 0.5 = 1/2 = 25/50 = 1 : 2 = 50% 0.333 = 1/3 = 300/900 = 1 : 3 = 33.33% 0.25 = 1/4 = 25/100 = 1 : 4 = 25%
5. 5. Reciprocals We often need to take reciprocals. Taking a reciprocal is dividing a number into 1. For example, here are diﬀerent forms to express the reciprocals of 50, 5, 0.5, and 0.01, respectively: 1/50 = 0.02 = 2% 1/5 = 0.2 = 20% 1/0.5 = 2 = 200% 1/0.01 = 100 = 10, 000%
6. 6. Reciprocals Note that: When we divide any number by a small number (a number lower than 1), the result is a larger number. Contrariwise, when we divide any number by a large number (a number greater than 1), the result is a smaller number. The reciprocal of zero is undeﬁned. In fact, any number (positive or negative) divided by zero is undeﬁned. The result is a number so large that it cannot be deﬁned as a number. Conventionally, it is called “inﬁnity” (∞). That is: x/0 = ∞, where x may be 1 or any other number, positive or negative.
7. 7. Simple average We often have data on a variable and need to ﬁnd its typical or representative value. Averages come in handy for this. Example: A group of four students ﬁnd in their pockets the following amounts of cash (in dollars): {25, 15, 10, 30}. What is the amount of cash in the pocket of a typical student in the group? Note that no particular individual needs to have exactly that amount. We may have learned in middle school to calculate the simple or arithmetic average of the data given: 25 + 15 + 10 + 30 80 = = 20 4 4 The typical amount of cash in the pocket of an individual in this group is \$20.
8. 8. Simple average Let us generalize the results. Let x be any variable of interest for which we have data {x1 , x2 , . . . , xn }, where n is the number of values of x. In statistics, n is called the “sample size” or the “number of observations” of the variable. Now, let x be the simple ¯ or arithmetic average of the data (a.k.a. “arithmetic mean”). Then: x1 + x2 + . . . + xn x= ¯ n n If we let i=1 xi = x1 + x2 + . . . + xn , the formula can be simpliﬁed to: n 1 x= ¯ xi n x=1 This reads as: “the simple mean of x is the sum of the data values of x, from the ﬁrst to the last, divided by the sample size (or multiplied by the reciprocal of the sample size).”
9. 9. Weighted average Let y be the cash in the pocket of each person in another group (in dollars): {5, 17, 8 }. Clearly, y = (5 + 17 + 8)/3 = 30/3 = 10. ¯ Now suppose we are given the averages of each group: x = 20 and ¯ y = 10 and ask to ﬁnd the typical value for both groups taken ¯ together. We cannot just take the average of the simple averages: (20 + 10)/2 = 15. That gives the same “weight” to each of the averages in determining the average of averages. However, the ﬁrst group has four people and the second group only three. As a result, each individual in the second group would be given more importance in inﬂuencing the total average. The correct average requires that each individual has the same “weight” regardless of group. Happily, we have the data for all individuals in both groups: {25, 15, 10, 30, 5, 17, 8}. The simple average for the two groups merged as a single total group is: 25 + 15 + 10 + 30 + 5 + 17 + 8 = 15.7 7
10. 10. Weighted average What if we don’t have the data for each individual, but only the averages and the sample sizes of the two groups? In that case, we can take the ’textbfweighted average (a.k.a. “weighted mean”): z = wx x + wy y ˜ ¯ ¯ where z is the weighted mean of the means, i.e. the simple mean ˜ of the two groups merged as one, and wi = ni /n is the “weight” of group i given by the sample size for group i as a fraction of the entire merged sample. Note that wx + wy = 1. In this case: z = wx x + wy y = (4/7) 20 + (3/7) 10 = 15.7 ˜ ¯ ¯
11. 11. Weighted average For m groups (where m is any arbitrary number of groups): x = wa xa + wb xb + . . . + wm xm ˜ ¯ ¯ ¯ where wi = ni /n and m wi = 1. i=a Example: In three towns, the average (\$/per bag) price of oranges is, respectively, (4, 2, 6). The population in each town (in thousands) is, respectively, (12, 14, 18). The average for the three towns taken together, i.e. the weighted average, is given by: xa = 4, xb = 2, xc = 6; ¯ ¯ ¯ 12 12 14 18 wa = = = .27, wb = = .32, wb = = .41 12 + 14 + 18 44 44 44 x = wa xa + wb xb + wc xc = (.27 × 4) + (.32 × 2) + (.41 × 6) = 4.2 ˜ ¯ ¯ ¯
12. 12. Weighted average Note the following: The weights sum to 1: wa + wb + wc = .27 + .32 + .41 = 1.00 It was possible to use thousands as the units of the sample sizes, because the “weights” are the sample sizes (the population in each town) as a fraction of the entire merged sample size (the population of the three towns added together). In the “weight” formulas, the thousands in the numerators cancel out the thousands in the denominators. The notation used in the formulas is mixed. That should help you get comfortable with diﬀerent symbols used to denote the same mathematical objects. Diﬀerent textbooks use diﬀerent notations, and sometimes the same book has to change notation from chapter to chapter or section to section.
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