Transcript Section 2.4 ~ Index Numbers

Section 2.4 ~
Index Numbers
Introduction to Probability and Statistics
Ms. Young
Sec. 2.4
Objective

To understand the concept of an index
number; particularly how the Consumer Price
Index (CPI) is used to measure inflation and
make comparisons over different time
periods.
Sec. 2.4
Index Numbers


We have all heard our elders say things like, “When I was
young, this same loaf of bread only cost a nickel” or “It only
cost me $8 to fill my tank in 1955”
One way to facilitate comparisons of the same thing over time
is by using index numbers


To calculate an index number for a certain product (gas, bread,
etc.), each value is compared to one reference value


Index numbers are numbers that provide a simple way to compare
measurements of the same thing made at different times or in different
places
Ex. ~ We might call the price in 1975 the reference value in which
case the index numbers come from the comparison of each year to
1975
The index numbers are the percentage of the comparison to
the reference value written without the percent symbol
index number =
value
100
reference value
Sec. 2.4
Index Numbers
reference value
year
price
Price as a % Price Index
of 1975
(1975=100)
1955
1965
1975
1985
1995
2000
2005
$ 0.291
$ 0.312
$ 0.567
$ 1.196
$ 1.205
$ 1.550
$ 2.310
51.3%
55.0%
100.0%
210.9%
212.5%
273.4%
407.4%
Percentages in
comparison to 1975
51.3
55.0
100
210.9
212.5
273.4
407.4
price index (percentages written
without the percent sign)
Sec. 2.4
Example 1

Suppose the cost of gasoline today is $2.79 per
gallon. Using the 1975 price as the reference value,
find the price index number for gasoline today.
value
index number =
100
reference value
index number 
currentgas price
100
1975gas price
$2.79
index number 
100  492.1
$.567

This means that the current price of gas is 492.1% of the
price in 1975 or 4.921 times the price in 1975

Or you could even say that it is 392.1% more than the gas price in
1975
Sec. 2.4
Making Comparisons with Index Numbers

The index numbers that are reported are in
comparison to the reference value, but you can use
the index numbers to compare other values as well

Ex. ~ Suppose we want to know how much more expensive
gas was in 1995 than in 1965, but 1975 is the reference
value.

Since both index values are in comparison to the same
reference value (the year 1975), you can just compare them to
each other
index number for 1995
index number for 1965

212.5
 3.86
55.0
This means that the 1995 price for gas was 3.86 times the 1965
price, or 386% of the 1965 price.
Sec. 2.4
Example 2

Use the gas index number table to answer the
following questions.

a. Suppose that it cost $7.00 to fill your gas tank in 1975.
How much did it cost to buy the same amount of gas in
2005?



The index number for 2005 is 407.4, which means that it was 407.4%
of the price in 1975 or 4.074 times the price in 1975
If the cost was $7.00 to fill your gas tank in 1975, then it was $28.52
($7 x 4.074 = $28.52) in 2005
b. Suppose that it cost $20.00 to fill your tank in 1995.
How much did it cost to buy the same amount of gas in
1955?

You must first compare the index number in 1955 to that in 1995
index number for 1955 51.3

 0.2414
index number for 1995 212 .5

The price in 1955 is .2414 times the price in 1995, so if it cost $20.00
to fill your tank in 1995, then it cost $4.83 ($20 x .2414 = $4.83) in
1955
Sec. 2.4
The Consumer Price Index

Inflation – when prices of goods, services, and housing costs increase
over time


Because of inflation, simply comparing the changes in prices for
certain items (such as gas) is not very meaningful unless you take
other factors into consideration


Deflation would be when prices decrease
Ex. ~ As we talked about earlier, the 1995 price of gas was 3.86 times the
1965 price. This may seem extremely high, but you have to remember that
this number is based solely on the price index and doesn’t account for any
wage increases
The Consumer Price Index (CPI) is a monthly report that represents
an average of prices for a sample of more than 60,000 goods,
services, and housing costs in comparison to a reference value
In other words, The Consumer Price Index reports the overall inflation
rate in comparison to a certain year (1982-1984 at this time)
 This allows us to compare overall prices at different times


Ex. ~ to find out how much higher typical prices were in 2005 than in 1995, you
divide the CPI’s for the two years (refer to p.77 for the report)
CPI for 2005 195.3

 1.28
CPI for 1995 152.4

Based on the CPI, typical prices in 2005 were 1.28 times those in 1995.
In other words, an item that cost $1,000 in 1995 would cost $1,280 in
2005
Sec. 2.4
Example 3

Suppose you needed $30,000 to maintain a particular standard
of living in 2000. How much would you have needed in 2006 to
maintain the same living standard?

Since the CPI is an index value, we can make comparisons the same way we
did before:
CPI for 2006 201 .6

 1.17
CPI for 2000 172 .2

The cost of living in 2006 is 1.17 times the cost of living in 2000, so a
$30,000 standard of living would cost $35,100 ($30,000 x 1.17) in 2006
Sec. 2.4
Adjusting Prices for Inflation

The CPI is also commonly used to make comparisons of salary
increases versus the inflation rate


This value can then be used to examine whether or not a company is paying
enough (or too much) in comparison to the average rate of inflation
Example 4
 In 1987, the mean salary for major league baseball players was
$412,000. In 2006, it was $2,867,000. Compare the increase in
mean baseball salaries to the overall rate of inflation measured by
the Consumer Price Index.

First compare the Consumer Price Indices for 2006 and 1987:
CPI for 2006 201 .6

 1.77
CPI for 1987 113 .6

Next, compare the average baseball salaries for the two years:
averagebaseball salary for 2006 $2,867,000

 6.96
averagebaseball salary for1987 $412,000

By comparing the two values, you can see that while the average
inflation rate rose about 77%, the baseball salaries rose about 600%
which tells us that the mean baseball salary far surpassed the average
rate of inflation!