We will present some examples based on the real data to demonstrate the use of the formulas.
Calculating the Standard Error from the Confidence Interval
The estimated number of males, never married is 41,011,718 from summary table B12001 for the United States for 2008. The margin of error is 93,906.
Standard Error = Margin of Error / 1.645
Calculating the standard error using the margin of error, we have:
SE(41,011,718) = 93,906 / 1.645 = 57,086.
Calculating the Standard Error of a Sum
We are interested in the number of people who have never been married. From Example 1, we know the number of males, never married is 41,011,718. From summary table B12001 we have the number of females, never married is 34,893,327 with a margin of error of 81,141. So, the estimated number of people who have never been married is 41,011,718 + 34,893,327 = 75,905,045. To calculate the standard error of this sum, we need the standard errors of the two estimates in the sum. We have the standard error for the number of males never married from example 1 as 57,086. The standard error for the number of females never married is calculated using the margin of error:
SE(34,893,327) = 81,141 / 1.645 = 49,326.
So using the formula for the standard error of a sum or difference we have:
Caution: This method, however, will underestimate (overestimate) the standard error if the two items in a sum are highly positively (negatively) correlated or if the two items in a difference are highly negatively (positively) correlated.
To calculate the lower and upper bounds of the 90 percent confidence interval around 75,905,045 using the standard error, simply multiply 75,444 by 1.645, then add and subtract the product from 75,905,045. Thus the 90 percent confidence interval for this estimate is [75,905,045 - 1.645(75,444)] to [75,905,045 + 1.645(75,444)] or 75,780,940 to 76,029,150.
Calculating the Standard Error of a Percent
We are interested in the percentage of females who have never been married to the number of people who have never been married. The number of females, never married is 34,893,327 and the number of people who have never been married is 75,905,045. To calculate the standard error of this sum, we need the standard errors of the two estimates in the sum. We have the standard error for the number of females never married from example 2 as 49,326 and the standard error for the number of people never married calculated from example 2 as 75,444.
The estimate is (34,893,327 / 75,905,045) * 100% = 45.97%
So, using the formula for the standard error of a proportion or percent, we have:
To calculate the lower and upper bounds of the 90 percent confidence interval around 45.97 using the standard error, simply multiply 0.05 by 1.645, then add and subtract the product from 45.97. Thus the 90 percent confidence interval for this estimate is
[45.97 - 1.645(0.05)] to [45.97 + 1.645(0.05)], or 45.89% to 46.05%.
Calculating the Standard Error of a Ratio
Now, let us calculate the estimate of the ratio of the number of unmarried males to the number of unmarried females and its standard error. From the above examples, the estimate for the number of unmarried men is 41,011,718 with a standard error of 57,086, and the estimate for the number of unmarried women is 34,893,327 with a standard error of 49,326.
The estimate of the ratio is 41,011,718 / 34,893,327 = 1.175.
The standard error of this ratio is
The 90 percent margin of error for this estimate would be 0.00233 multiplied by 1.645, or about 0.004. The 90 percent lower and upper 90 percent confidence bounds would then be [1.175 - 0.004] to [1.175 + 0.004], or 1.171 and 1.179.
Calculating the Standard Error of a Product
We are interested in the number of 1-unit detached owner-occupied housing units. The number of owner-occupied housing units is 75,373,053 with a margin of error of 224,087 from subject table S2504 for 2008, and the percent of 1-unit detached owner-occupied housing units is 81.8% (0.818) with a margin of error of 0.1 (0.001). So the number of 1-unit detached owner-occupied housing units is 75,373,053 * 0.818 = 61,655,157. Calculating the standard error for the estimates using the margin of error we have:
SE(75,373,053) = 224,087 / 1.645 = 136,223
and
SE(0.818) = 0.001 / 1.645 = 0.0006079
The standard error for number of 1-unit detached owner-occupied housing units is calculated using the formula for products as:
To calculate the lower and upper bounds of the 90 percent confidence interval around 61,655,157 using the standard error, simply multiply 120,483 by 1.645, then add and subtract the product from 61,655,157. Thus the 90 percent confidence interval for this estimate is [61,655,157 - 1.645(120,483)] to [61,655,157 + 1.645(120,483)] or 61,456,962 to 61,853,352.
Users should be cautioned that all methods in this section are approximations. They may be overestimates or underestimates of the estimates standard error, and may not match direct calculations of standard errors or calculations obtained through other methods
For more detailed information on the calculations of standard errors and the reasoning for the formulas please reference the Chapter 6: Accuracy of the Data. Document.
