numbers, reciprocals, averages
DESCRIPTION
Basic math tools weTRANSCRIPT
Topics
Numbers
Reciprocals
Averages
Numbers
Economic data is reported in different numeric formats. Examples:
I the BEA estimated the U.S. 2007 GDP in 14.42 trillion dollars,
I the BEA also estimated U.S. 2007 imports of goods andservices to be approximately one sixth (1/6) of GDP,
I the BLS estimated that, as of November 2008, theunemployment rate was 6.7%, and
I the CB reported that in 2007, the ratio of money income ofthe richest fifth of U.S. households to the poorest fifth wasapproximately 15:1.
Numbers
In using numbers, we should be able to go back back and forthbetween 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%
Reciprocals
We often need to take reciprocals. Taking a reciprocal is dividing anumber into 1. For example, here are different forms to expressthe 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%
Reciprocals
Note that:
I When we divide any number by a small number (a numberlower than 1), the result is a larger number. Contrariwise,when we divide any number by a large number (a numbergreater than 1), the result is a smaller number.
I The reciprocal of zero is undefined. In fact, any number(positive or negative) divided by zero is undefined. The resultis 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.
Simple average
We often have data on a variable and need to find its typical orrepresentative value. Averages come in handy for this.Example: A group of four students find in their pockets thefollowing amounts of cash (in dollars): {25, 15, 10, 30}. What isthe amount of cash in the pocket of a typical student in thegroup? Note that no particular individual needs to have exactlythat amount. We may have learned in middle school to calculatethe simple or arithmetic average of the data given:
25 + 15 + 10 + 30
4=
80
4= 20
The typical amount of cash in the pocket of an individual in thisgroup is $20.
Simple average
Let us generalize the results. Let x be any variable of interest forwhich we have data {x1, x2, . . . , xn}, where n is the number ofvalues of x . In statistics, n is called the “sample size” or the“number of observations” of the variable. Now, let x be the simpleor arithmetic average of the data (a.k.a. “arithmetic mean”).Then:
x =x1 + x2 + . . . + xn
n
If we let∑n
i=1 xi = x1 + x2 + . . . + xn, the formula can besimplified to:
x =1
n
n∑x=1
xi
This reads as: “the simple mean of x is the sum of the data valuesof x , from the first to the last, divided by the sample size (ormultiplied by the reciprocal of the sample size).”
Weighted averageLet 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 andy = 10 and ask to find the typical value for both groups takentogether.We cannot just take the average of the simple averages:(20 + 10)/2 = 15. That gives the same “weight” to each of theaverages in determining the average of averages. However, the firstgroup has four people and the second group only three. As aresult, each individual in the second group would be given moreimportance in influencing the total average. The correct averagerequires that each individual has the same “weight” regardless ofgroup. Happily, we have the data for all individuals in both groups:{25, 15, 10, 30, 5, 17, 8}. The simple average for the two groupsmerged as a single total group is:
25 + 15 + 10 + 30 + 5 + 17 + 8
7= 15.7
Weighted average
What if we don’t have the data for each individual, but only theaverages and the sample sizes of the two groups? In that case, wecan 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 meanof the two groups merged as one, and wi = ni/n is the “weight” ofgroup i given by the sample size for group i as a fraction of theentire merged sample. Note that wx + wy = 1. In this case:
z = wx x + wy y = (4/7) 20 + (3/7) 10 = 15.7
Weighted average
For m groups (where m is any arbitrary number of groups):
x = waxa + wbxb + . . . + wmxm
where wi = ni/n and∑m
i=a wi = 1.Example: In three towns, the average ($/per bag) price of orangesis, respectively, (4, 2, 6). The population in each town (inthousands) is, respectively, (12, 14, 18). The average for the threetowns taken together, i.e. the weighted average, is given by:
xa = 4, xb = 2, xc = 6;
wa =12
12 + 14 + 18=
12
44= .27,wb =
14
44= .32,wb =
18
44= .41
x = waxa + wbxb + wc xc = (.27× 4) + (.32× 2) + (.41× 6) = 4.2
Weighted average
Note the following:
I The weights sum to 1: wa + wb + wc = .27 + .32 + .41 = 1.00
I It was possible to use thousands as the units of the samplesizes, because the “weights” are the sample sizes (thepopulation in each town) as a fraction of the entire mergedsample size (the population of the three towns addedtogether). In the “weight” formulas, the thousands in thenumerators cancel out the thousands in the denominators.
I The notation used in the formulas is mixed. That should helpyou get comfortable with different symbols used to denote thesame mathematical objects. Different textbooks use differentnotations, and sometimes the same book has to changenotation from chapter to chapter or section to section.