Chapter 3: Numbers in the Real World
Lecture notes
Math 1030 Section D
Section D.1: What Is an Index Number?
Definition of index number
An index number provides a simple way to compare measurements made at different times or in different
places. The value at one particular time (or place) must be chosen as the reference value. The index number
for any other time (or place) is
index number =
value
× 100.
reference value
Ex.1 Finding an index number.
The price of gasoline in 1965 was 31.2 cents. Suppose the current price of gasoline is $3.20 per gallon.
Using the 1975 price (= 56.7 cents) as the reference value, find the price index number for gasoline today
and in 1965.
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Chapter 3: Numbers in the Real World
Lecture notes
Math 1030 Section D
Making Comparisons with Index Numbers
Ex.2
(1) Suppose it cost $7.00 to fill a gas tank in 1975. How much would it have cost to fill the same tank in
2005? (price index for 2005 = 407.4, reference value = 1975 price).
(2) Suppose it cost $20.00 to fill the gas tank in 1995. How much would it have cost to fill the same tank
in 1955? (price index for 1995 = 212.5, price index for 1955 = 51.3, reference value = 1975 price).
Changing The Reference Value
Ex.3
Suppose the current price of gasoline is $3.20 per gallon. Using the 1985 price (= $1.196) as the reference
value, find its price index number. Compare this answer to the answer in Example 1, where 1975 was the
reference year.
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Chapter 3: Numbers in the Real World
Lecture notes
Math 1030 Section D
Section D.2: The Consumer Price Index
Inflation and Consumer Price Index
We have seen that the price of gas has risen substantially with time. Most other prices and wages have also
risen, a phenomenon we call inflation. Thus, changes in the actual price of gasoline are not very meaningful
unless we compare them to the overall rate of inflation, which is measured by the Consumer Price Index
(CPI).
The Consumer Price Index is based on an average of prices for a sample of more than 60, 000 goods, services,
and housing costs. It is computed and reported monthly.
The Consumer Price Index is just an index and the reference value is an average of prices during the period
1982 − 1984.
Ex.4
To find out how much higher typical prices were in 2005 than in 1995, compute the ratio of the CPIs for
the two years using the shorthand CPI2005 to represent the CPI for 2005 and the shorthand CPI1995 to
represent the CPI for 1995:
Ex.5 CPI changes.
Suppose you needed $30, 000 to maintain a particular standard of living in 2000. How much would you
have needed in 2005 to maintain the same standard of living? Assume that the average price of your
typical purchased has risen at the same rate as the Consumer Price Index (CPI).
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Chapter 3: Numbers in the Real World
Lecture notes
Math 1030 Section D
The Rate of Inflation
Rate of Inflation
The rate of inflation from one year to the next is usually defined as the relative change in the Consumer Price
Index.
Ex.6
The rate of inflation from 1980 to 1981 was
Ex.7
Find the inflation rate from 1998 to 1999 (CPI1998 = 163.0, CPI1999 = 166.6).
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Chapter 3: Numbers in the Real World
Lecture notes
Math 1030 Section D
Adjusting Prices for Inflation
Adjusting prices for inflation
Given a price in dollars per year X ($X ), the equivalent price in dollars for year Y ($Y ) is
CP IY
price in $Y = ( price in $X ) ×
CP IX
where X and Y represent the years, such as 1992 and 2005.
Ex.8 Baseball Salaries.
In 1987, the mean (average) salary for major league baseball players was $412, 000. In 2005, it was
$2, 633, 000. Compare the two salaries in ”2005 dollars“ ($2005 ).
How does the rise in baseball salaries compare to the overall rise in prices with inflation? (CPI2005 = 195.3,
CPI1987 = 113.6)
Ex.9 Computer Prices.
According to computer performance tests, a Macintosh computer that cost $1, 000 in 2005 had computing
power equivalent to that of a supercomputer that sold for $30 million in 1985. If computer prices had
risen with inflation, how much would the computing power of the 1985 supercomputer have cost in
2005? What does this tell us about the cost of computers?
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Chapter 3: Numbers in the Real World
Lecture notes
Math 1030 Section D
Section D.3: Other Index Numbers
Ex.10 Health Care Quality Index.
Statisticians at Health Risk Management, Inc. invented the Health Care Quality Index to compare the
quality of health care in different states. The index represents an average of 46 different measures of
health care quality, such as spending on public health, the fraction of children receiving immunizations,
and the quality of public drinking water. They chose the average score among all the states as the reference
value. Thus, the average score is defined to be 100 on this index.
Minnesota had the highest index value of 118, while Lousiana had the lowest value at 83. Discuss the
meaning of these index values.
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