What does an index measure and how do you calculate it?
As previously mentioned, an index allows us to track the change in a given variable over time. The way an index tracks change over time is to utilize a baseline time period, which is a point in time at which all future values will be compared. By comparing all future values to a given point in the past, one is able to simply track how the variable is changing from this point in time.
In my principles of macroeconomics classes, I taught my students how to calculate a simple index in three easy steps. First, define a basket of goods and determine the quantity of each good sold and at which price over a period of time. It is important to take account of both the quantity and the price because a rise in the total value sold could be a factor of rising prices, rising quantities sold, or a combination of the two.
Once you have the quantities and prices of each good sold over time, you next calculate the total value sold by multiplying each good's price by each good's quantity and summing these products across all goods in the basket.
The last step is to define a base (reference) year and calculate the index by taking the ratio of the total value of the basket from a given year to the total value of the basket from the base year. For scaling reasons, this ratio is then multiplied by 100.
For example, let's say you live in Sports Town where two goods are produced and sold: Basketballs (B) and Footballs (F). The following table gives the quantities and prices of each of each of these goods, as well as the total value of the basket from 2006 to 2010:
Year | Quantity | Price | Total Value of the Basket |
2006 | 10B and 5F | B: $2.00 and F: $4.00 | (10*$2.00)+(5*$4.00) = $40.00 |
2007 | 12B and 6F | B: $2.25 and F: $4.10 | (12*$2.25)+(6*$4.10) = $51.60 |
2008 | 12B and 8F | B: $2.50 and F: $4.25 | (12*$2.50)+(8*$4.25) = $64.00 |
2009 | 9B and 12F | B: $2.50 and F: $4.50 | (9*$2.50)+(12*$4.50) = $76.50 |
2010 | 14B and 10F | B: $2.90 and F: $4.50 | (14*$2.90)+(10*$4.50) = $85.60 |
Now that the total value of the basket is calculated, we must choose a base year and can then calculate the index. Let's choose 2006 as our base year, which means the value of the basket in the base year is $40.00. The following table gives the value of the index for Sports Town from 2006 to 2010:
Year | Total Value of the Basket | Index |
2006 | $40.00 | ($40.00/$40.00)*100 = 100 |
2007 | $51.60 | ($51.60/$40.00)*100 = 129 |
2008 | $64.00 | ($64.00/$40.00)*100 = 160 |
2009 | $76.50 | ($76.50/$40.00)*100 = 191.3 |
2010 | $85.60 | ($85.60/$40.00)*100 = 214 |
Once the index is calculated, we can utilize it to calculate the percentage change in the value of Sports Town's basket. By having a base year value of 100, a simple examination of the index values allows us to see how the basket's total value has changed since the base year. For example an index value of 129 in 2007 means the value of the basket increased 29% between 2007 and 2006, while an index value of 214 in 2010 means the value of the basket increased 114% between 2010 and 2006.
In addition to comparing the value of the basket to the base year's value, one key use of an index is to calculate the change in value from one year to another. For example, say we want to see how much prices changed from 2009 to 2010. To do this, we simply calculate the percentage change in the index from 2009 to 2010: (214 - 191.3)/191.3 = 0.12, or 12%. Therefore, the value of the basket increased 12% between 2009 and 2010.
That concludes the calculation of a simple index and the meaning behind the numbers. Next we will discuss two real world examples of an index and what they measure.
The Consumer Price Index (CPI) vs. The PayScale Index
An index commonly used to discuss the health of our economy is the Consumer Price Index (CPI), which is calculated by the Bureau of Labor Statistics (BLS), the same government organization responsible for tracking unemployment.
The CPI is a more complicated version of the simple index described above. Basically, it measures the value of a basket of goods and services produced and sold in the urban U.S. The goods and services included in the CPI's market basket are determined by detailed expenditure survey data collected by the Consumer Expenditure Surveys and fall into the following categories (Note: taxes are not included in the basket):
- Food and Beverages
- Housing
- Apparel
- Transportation
- Medical Care
- Recreation
- Education and Communication
- Others Goods and Services (e.g. Tobacco, personal services like haircuts, etc.)
Once the BLS determines the goods and services included, they collect the prices of these goods and services by calling or visiting thousands of retail stores, service establishments, rental units and medical offices all over the U.S.
Once the prices are collected and verified, the CPI is calculated for urban consumers (as determined by Census surveys) in two steps. First the BLS calculates the CPI for each good and service akin to the method described above: price x quantity for a given year / price x quantity for the base year (1982-1984 currently).
Next the total CPI for the entire market basket is calculated as the weighted sum of each individual good/service's CPI, where the weights are based on the good/service's importance in total family expenditures. In other words, the more a family spends on a particular good out of their total spending, the higher the weight.
Now that we know what an index is and how it is calculated, we can accurately analyze the following chart, which graphs the annual CPI since 2000:
The CPI is currently around 220. The base period is 1982-1984, which means the average of the CPI values from 1982-1984 is 100. Therefore, the CPI is up about 120% from the base period. This means things today cost 2.2x more than they did back in the early 80's.
Like the CPI, the PayScale Index shows the change in the value of a market basket over time. However, the market basket for the PayScale index is the value of full-time private industry employees as measured by their total cash compensation. Therefore, where the CPI shows changes in the prices of goods and services purchased in the urban U.S., the PayScale index shows changes in the prices of labor purchased in the U.S. and thus measures the change in pay.
Below is a chart showing the quarterly PayScale Index since Q1 2007, where the numbers on the vertical axis are the PayScale Index Values. The Q4 2010 value of the national PayScale Index is 104.1. The base period for the PayScale Index is 2006, which means its value in 2006 is 100. Therefore the price of labor (i.e. wages) have increased just over 4% since 2006.
Change in the CPI vs. Change in the PayScale Index
One key use of the CPI is to calculate the inflation rate. The inflation rate is calculated as the percentage change in the CPI for all urban consumers. Inflation can be calculated as month-to-month, quarter-to-quarter, or year-to-year.
For example, the CPI in December of 2010 was ~219 and the CPI in December of 2009 was ~216. Using these values, we can calculate how prices changed from December 2009 to December 2010 (the annual inflation rate): (219 - 216)/216 = 0.015, or 1.5%. Therefore the value of the market basket rose 1.5% between 2009 and 2010.
The chart below graphs the annual (12-month) percentage change in the CPI from 2000. This chart shows the annual rate of inflation from 2000 to 2010. As you can see from the chart, the CPI was falling in 2009 to the point of negative inflation (deflation), meaning the value of the market basket fell in 2009. Note that this fall in value could have been a result of falling prices, decreasing demand or a combination of the two.
Similar to the annual change in the CPI, we also report an annual change in the PayScale Index, as seen in the chart below.
From this chart you can see wages have not really changed between the end of 2009 and the end of 2010 as the annual changes for the last three quarters are ~0%.
For both the CPI and the PayScale Index, the Index Charts and the 12-month Percentage Change Charts are interesting. The Index Charts makes it easier to see when prices (CPI) or wages (PayScale Index) were at their overall highest and lowest. The Percentage Change Charts are useful for figuring out how much more people are spending today versus a year ago (CPI) or how much more they are earning today versus a year ago (PayScale Index).
Pay trends are useful for understanding the general labor market, but do you know where your pay fits into the labor market? When you want powerful salary data and comparisons customized for your exact position or job offer, be sure to build a complete profile by taking PayScale's full salary survey.
Regards,
Katie Bardaro
Research Analyst, PayScale, Inc.