Statistical Reasoning: Measurement and Index Numbers

Introduction to Index Numbers

• Definition: An index number provides a standardized and simplified method to compare measurements taken at different points in time or across different geographical locations.

• Reference Value: To create an index, one specific time or place must be designated as the reference value or base value. All other measurements are then expressed relative to this reference point.

• General Formula: The index number is typically calculated by dividing the current value by the reference value and then multiplying by $100$.

• Primary Purpose: The main function of index numbers is to facilitate easy comparisons across varying datasets, especially when dealing with changes over long periods or different economic climates.

Table 2.1: Average Gasoline Prices and Index Development

• The following table presents the average price of gasoline in the United States at 10-year intervals. These are raw, unadjusted prices from the respective years.

• Reference Year: 1985 (Price = $1.20; Index = 100.0).

• Source: U.S. Department of Energy.

Example 1: Finding an Index Number

• Problem: Suppose the cost of gasoline today is $2.70$ per gallon. Using the 1985 price as the reference value, find the price index number for gasoline today.

• Reference Value: $1.20$ (from 1985).

• Calculation:   $\text{Current Index Number} = \frac{2.70}{1.20} \times 100 = 225$

• Interpretation: The index number for the current price is $225$. This indicates that the current gasoline price is $225\%$ of the price in 1985.

Making Comparisons with Index Numbers

• Simple Comparison: In 2005, the price index was $192.5$. This means gas was $192.5\%$ of the 1985 price, or simply $1.925 \text{ times}$ the price in 1985.

• Non-Reference Comparison: Comparisons can also be made between two years where neither is the chosen reference year by utilizing their respective index numbers.

Example 2: Using the Gas Price Index

• Scenario A (Forward Comparison): Suppose it cost $\$16.00$ to fill a gas tank in 1985. How much did it cost to buy the same amount of gas in 2015?

  • Solution: The 2015 index (1985 = 100) is $210$.

  • Calculation:     $\text{2015 Price} = \$16.00 \times 2.10 = \$33.60$

• Scenario B (Backward Comparison): Suppose it cost $\$20.00$ to fill a gas tank in 2005. How much did it cost for the same amount of gasoline in 1965?

  • Data: Index for 2005 = $192.5$; Index for 1965 = $25.8$.

  • Calculation (Ratio of Indices):     $\text{Ratio} = \frac{25.8}{192.5} \approx 0.134$

  • Final Cost Calculation:     $\text{1965 Cost} = \$20.00 \times \frac{25.8}{192.5} = \$2.68$

The Consumer Price Index (CPI)

• Overview: The Consumer Price Index (CPI) is a critical economic indicator computed and reported monthly.

• Data Source: It is based on price tracking for a sample of more than $60,000$ goods, services, and housing costs.

• Reference Base: The CPI commonly uses the 3-year period of 1982–1984 as its base period ($1982\text{--}1984 = 100$).

Table 2.2: Average Annual Consumer Price Index Data

Example 3: CPI Changes (Standards of Living)

• Question: If an individual needed $\$30,000$ to maintain a certain standard of living in 2011, how much would they need in 2021 for the same standard?

• Indices: CPI (2011) = $224.9$; CPI (2021) = $271.0$.

• Comparison Calculation:   $\text{Ratio} = \frac{271.0}{224.9} \approx 1.205$

• Result: Typical prices in 2021 were approximately $1.205 \text{ times}$ those in 2011.   $\text{Required Income} = \$30,000 \times \frac{271.0}{224.9} \approx \$36,150$

• Conclusion: To maintain the same standard of living, the individual would need $\$36,150$ in 2021.

The Inflation Rate

• Definition: The annual inflation rate is defined as the relative change in the CPI from one year to the next.

• General Formula:   $\text{Inflation Rate} = \frac{\text{CPI}_\text{newer} - \text{CPI}_\text{older}}{\text{CPI}_\text{older}} \times 100$

• Example (2019 to 2020):   $\text{Inflation Rate} = \frac{258.8 - 255.7}{255.7} \approx 0.012 = 1.2\%$

• Result: The inflation rate from 2019 to 2020 was approximately $1.2\%$.

Example 4: Baseball Salaries and Infation Adjustment

• Objective: Compare the growth of mean Major League Baseball (MLB) salaries against the rate of inflation measured by CPI.

• Data:

  • Mean Salary 1987: $\$412,000$

  • Mean Salary 2021: $\$4,410,000$

  • CPI 1987: $113.6$

  • CPI 2021: $271.0$

• Step 1: Compare CPI Increase:   $\text{CPI Ratio} = \frac{271.0}{113.6} \approx 2.4$   (Overall consumer prices rose by a factor of about $2.4$).

• Step 2: Compare Salary Increase:   $\text{Salary Ratio} = \frac{4,410,000}{412,000} \approx 10.7$   (Mean baseball salaries rose by a factor of more than $10$).

• Conclusion: Mean MLB salaries rose more than four times as fast as the overall rate of inflation between 1987 and 2021.

Qualitative and Specialized Index Numbers

• Producer Price Index (PPI): Unlike the CPI, which measures retail costs for consumers, the PPI measures the prices that manufacturers or producers pay for the goods they purchase.

• Consumer Confidence Index: This is a qualitative index based on surveys designed to gauge consumer attitudes. It helps businesses determine if the public is likely to spend or save money in the near future.

• Diverse Applications: New indices are frequently created by various organizations to simplify comparisons across complex datasets or to track evolving economic trends.