Sampling Error
Because the estimates are based on a sample, exact agreement with results that would be obtained from a complete enumeration of merchant wholesale firms represented on the sampling frame using the same enumeration procedures is not expected. However, because each firm represented on the sampling frame has a known probability of being selected into the sample, it is possible to estimate the sampling variability of the survey estimates.
The particular sample used in this survey is one of a large number of samples of the same size that could have been selected using the same design. If all possible samples had been surveyed under the same conditions, an estimate of an unknown population value could have been obtained from each sample. These samples give rise to a distribution of estimates for the unknown population value. A statistical measure of the variability among these estimates is the standard error, which can be approximated from any one sample. The standard error is defined as the square root of the variance. The coefficient of variation (or relative standard error) of an estimator is the standard error of the estimator divided by the estimator. Note that measures of sampling variability, such as the standard error and coefficient of variation, are estimated from the sample and are also subject to sampling variability. (Technically, we should refer to the estimated standard error or the estimated coefficient of variation of an estimator. However, for the sake of brevity, we have omitted this detail.) It is important to note that the standard error and coefficient of variation only measure sampling variability. They do not measure any systematic biases in the estimates. Table 3 provides the minimum, maximum, and median coefficients of variation for estimates of monthly sales and end-of-month inventories for each kind of business. The ranges and medians shown in Table 3 are based on final MWTS estimates for January 2004 through December 2004. Coefficients of variation for estimates of annual sales, end-of-year inventories, purchases, gross margins, and gross margins-to-sales ratios for each kind of business are provided in Table 4. These coefficients of variation are based on 2003 AWTS data, adjusted to results of the 2002 Economic Census. (All coefficients of variation are expressed as percents.) The Census Bureau recommends that individuals using estimates contained in this report incorporate this information into their analyses, as sampling error could affect the conclusions drawn from these estimates.
The estimate from a particular sample and the standard error associated with the estimate can be used to construct a confidence interval. A confidence interval is a range about a given estimator that has a specified probability of containing the result of a complete enumeration. Associated with each interval is a percentage of confidence, which is interpreted as follows: If, for each possible sample, an estimate of an unknown population value and its approximate standard error were obtained, then:
1. For approximately 90 percent of the possible samples, the interval from 1.65 standard errors below to 1.65 standard errors above the estimate would include the result of a complete enumeration.
2. For approximately 95 percent of the possible samples, the interval from two standard errors below to two standard errors above the estimate would include the result of a complete enumeration.
To illustrate the computation of a confidence interval for an estimate of total sales, assume that an estimate of total sales is $10,750 million and the coefficient of variation for this estimate is 1.8 percent, or 0.018. First obtain the standard error of the estimate by multiplying the total sales estimate by its coefficient of variation. For this example, multiply $10,750 million by 0.018. This yields a standard error of $193.5 million. The upper and lower bounds of the 90-percent confidence interval are computed as $10,750 million plus or minus 1.65 times $193.5 million. Consequently, the 90-percent confidence interval is $10,431 million to $11,069 million. If corresponding confidence intervals were constructed for all possible samples of the same size and design, approximately 9 out of 10 (90 percent) of these intervals would contain the result obtained from a complete enumeration.
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Source: U.S. Census Bureau
Service Sector Statistics Division
Last Revised: April 8, 2005
