What is an RSE?
It is important to understand that data from sample surveys are not exact but are statistical estimates with some associated sampling error in each direction—the result of generating estimates based on a sample rather than using the entire population.
So, what is a relative standard error (RSE)? First, a standard error is a measure of the reliability or precision of the survey statistic. The value for the standard error can be used to construct confidence intervals and to perform hypothesis tests by standard statistical methods. A RSE is defined as the standard error (square root of the variance) of a survey estimate, divided by the survey estimate and multiplied by 100. In other words, the RSE is the standard error relative to the survey estimate on a scale from zero to 100. The larger the RSE, the less precise the survey estimate is of the true value in the population.
The 95-percent confidence range for a given survey estimate can be determined with the RSE. To calculate the 95-percent confidence range:
1. Divide the RSE by 100 and multiply by the survey estimate to determine the standard error.
2. Multiply the standard error by 1.96 to determine the confidence error.
3. The survey estimate plus or minus the confidence error is the 95-percent confidence range.
For example, from the Floorspace - Living Space Characteristics by Total, Heated, and Cooled Floorspace table (HC1.2.1), the estimate for total floorspace for all housing units in the 2005 RECS is 256.5 billion square feet. The estimate's RSE shown in the RSE tab on the spreadsheet is 1.7 percent. The standard error is (1.7/100)*(256.5 billion square feet), which is approximately 4.4 billion square feet. The 95-percent confidence error is (1.96)*(4.4 billion square feet), or 8.5 billion square feet. Therefore, if the RECS had been conducted 100 times, we would expect that 95 times out of 100 the survey estimate of the true amount of floorspace in residential buildings in the U.S. in 2005 would have been contained in the interval 256.5 plus or minus 8.5 billion square feet, which is the range from 248.0 to 265.0 billion square feet.
Testing for Statistical Significance between Two Statistics
The difference between any two RECS survey estimates may or may not be statistically significant. An approximation of statistical significance between two values is computed as:
where S is the standard error, x1 is the first estimate and x2 is the second estimate. The result of this computation is to be multiplied by 1.96 (for a 95 percent confidence test) and, if this result is less than the difference between the two estimates, the difference is statistically significant.
For example, from the Floorspace - Housing Characteristics by Average Floorspace table (HC1.1.2), homes built in the 1960’s averaged 2,090 square feet with an RSE of 3.6, while homes built in the 1970’s averaged 1,988 square feet with an RSE of 3.4. They appear to differ by 102.0 square feet, but are these two numbers statistically different?
The standard error for homes built in the 1960’s (x1) is (3.6/100)* 2090 or 75.2 square feet. The standard error for homes built in the 1970’s (x2) is (3.4/100)*1988 or 67.6 square feet. So,
The simple difference, x1-x2, between these two estimates is 102.0 square feet.
(101.1) by 1.96 yields 198.2 square feet. Because 198.2 is greater than 102.0 above, the difference between the two estimates is NOT statistically significant. This means that although the average square footage for the 1960’s appears larger than the 1970’s, sampling error does not allow us to say it is statistically and significantly so.
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