Variance is a measure of the uncertainty caused by the use of a sample of retail prices, instead of the complete universe of retail prices. Each month the U.S. Bureau of Labor Statistics collects prices from a sample of approximately 90,000 commodities and services (C&S) quotes in approximately 21,500 outlets around the United States for the Consumer Price Index (CPI). In addition, we collect approximately 8,000 housing quotes each month.
The most commonly used measure of sampling variability is the standard error of the estimate—the square root of the variance. The standard error of the CPI’s change can be used to construct confidence intervals to determine whether the change for a particular CPI series is significantly different from zero. This information should help users determine which index changes are significant.
Tables 1V through 5V show the median values of 12-month percent changes, as well as the median values of the standard errors. Table 1V shows this information for U.S. city average, and tables 2V through 5V show the same information for the Northeast, Midwest, South, and West regions of the country.
For example, from January through December 2016, the 1-month changes in the U.S. city average all items index had a median value of 0.15 percent. The standard errors of those 12 estimates had a median value of 0.03 percent. Margins of error are usually expressed as a statistic’s point estimate plus or minus two standard errors, so the margin of error on this CPI’s 1-month change is approximately 0.15 percent plus or minus 0.06 percent. Therefore, in a typical 1-month period, the true change in the CPI was probably somewhere between 0.09 percent and 0.21 percent. The tables also show median percent changes and standard errors for 2- and 6-month intervals and for the full year 2016. Margins of error can be calculated for these intervals in the same way as for a 1-month period.
Analyzing the data reveals three significant observations. First, standard errors increase as one moves from the U.S. city average to individual regions of the country and from all items to individual item categories. Second, standard errors differ between item categories. Third, the standard errors decrease on a relative basis (standard error divided by price change), as the price change interval gets longer.
The primary reason standard errors increase as one moves from the U.S. city average to individual regions of the country is that sample sizes differ. In general, smaller sample sizes lead to larger standard errors. For example, the U.S. city average all items index is computed each month from approximately 82,000 prices throughout the United States, and its median standard error for 1-month changes is 0.03 percent. By contrast, the Northeast region all items index is computed from approximately 17,000 prices, and its median standard error is 0.07 percent. Regional indexes have larger standard errors because their sample sizes are smaller.
One can observe this same effect moving from the all items index to individual item categories. Again, the U.S. city average all items index is computed each month from the prices of approximately 82,000 selected items, and its median standard error is 0.03 percent. By contrast, the U.S. city average recreation index is computed from about 5,700 prices, and its median standard error is 0.12 percent, or four times as large. Again, smaller sample sizes lead to larger standard errors.
There are two reasons that standard errors differ between item categories. First, item categories differ in sample size. For example, the U.S. city average food and beverages index is computed from approximately 35,000 prices each month, while the U.S. city average recreation index is computed from approximately 5,700 prices. Therefore, it is not surprising that the recreation index has larger standard errors. Second, there are real differences in item category price behaviors caused by different selling practices, seasonal influences, and consumer demand. This is especially true for the apparel category, in which it is common for the prices of individual items to fluctuate by 50 percent or more each month. As a result, standard errors for apparel indexes are comparatively large.
The third observation is that standard errors decrease, on a relative basis, as the price change interval gets longer. For the U.S. city average all items index, the median standard error divided by the median percent change is 0.03/0.15 = 0.20 for 1-month changes, 0.05/0.29 = 0.172 for 2-month changes, 0.05/0.81 = 0.062 for 6-month changes, and 0.06/1.09 = 0.055 for the 12-month change between December 2015 and December 2016. This shows that the relative accuracy of percent changes in the CPI generally improves as the price change interval gets longer. On an absolute basis, standard errors increase, but at a decreasing rate.
Users should exercise caution when using CPI estimates to make inferences about index changes for relatively short time periods, for individual goods and services, or for local areas. The standard errors of those estimates may be on the same order of magnitude as the estimates themselves; and, thus, few inferences about them are reliable.
For technical information about the calculation of CPI variance estimates, see the CPI section of the BLS Handbook of Methods.
Last Modified Date: February 10, 2023