Section 2.4 ~ Index Numbers

Introduction to Probability and StatisticsMs. Young

Objective

Sec. 2.4

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.

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 Index numbers are numbers that provide a simple way to compare

measurements of the same thing made at different times or in different places

To calculate an index number for a certain product (gas, bread, etc.), each value is compared to one reference value 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

Sec. 2.4

valueindex number = 100

reference value

Index Numbers

Sec. 2.4

year price Price as a % of 1975

Price Index (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%

51.3

55.0

100

210.9

212.5

273.4

407.4

reference value

price index (percentages written without the percent sign)

Percentages in comparison to

1975

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.

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

valueindex number = 100

reference value

100price gas 1975

price gascurrent numberindex

$2.79index number 100 492.1

$.567

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

This means that the 1995 price for gas was 3.86 times the 1965 price, or 386% of the 1965 price.

Sec. 2.4

212.53.86

55.0

1965for number index

1995for number index

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

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

2414.05.212

3.51

1995for number index

1955for number index

The Consumer Price Index Inflation – when prices of goods, services, and housing costs

increase over time Deflation would be when prices decrease

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 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)

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

CPI for 2005 195.31.28

CPI for 1995 152.4

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:

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

17.12.172

6.201

2000for CPI

2006for CPI

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:

Next, compare the average baseball salaries for the two years:

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!

Sec. 2.4

77.16.113

6.201

1987for CPI

2006for CPI

96.6000,412$

000,867,2$

1987for salary baseball average

2006for salary baseball average