Basic math tools we

1. Basic math tools:
Numbers, reciprocals, and averages
juliohuato@gmail.com
September 8, 2011

2. Topics
Numbers
Reciprocals
Averages

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. 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. 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. 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. 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. 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. 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. 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. 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. 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.