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 39,401,560 from summary table B12001 for the United States for 2006. The margin of error is 99,234.
Standard Error = Margin of Error / 1.645
Calculating the standard error using the margin of error, we have:
SE(39,401,560) = 99,234 / 1.645 = 60,325.
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 39,401,560. From summary table B12001 we have the number of females, never married is 33,385,649 with a margin of error of 77,920. So, the estimated number of people who have never been married is 39,401,560 + 33,385,649 = 72,787,209. 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 60,325. The standard error for the number of females never married is calculated using the margin of error:
SE(33,385,649) = 77,920 / 1.645 = 47,368.
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 72,787,209 using the standard error, simply multiply 76,699 by 1.645, then add and subtract the product from 72,787,209. Thus the 90 percent confidence interval for this estimate is [72,787,209 - 1.645(76,699)] to [72,787,209 + 1.645(76,699)] or 72,661,039 to 72,913,379.
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 33,385,649 and the number of people who have never been married is 72,787,209. 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 47,368 and the standard error for the number of people never married calculated from example 2 as 76,699.
The estimate is (33,385,649 / 72,787,209) * 100% = 45.9%
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 45.9 using the standard error, simply multiply 0.05 by 1.645, then add and subtract the product from 45.9. Thus the 90 percent confidence interval for this estimate is
[45.9 - 1.645(0.04)] to [45.9 + 1.645(0.04)], or 45.8% to 46.0%.
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 39,401,560 with a standard error of 60,325, and the estimates for the number of unmarried women is 33,385,649 with a standard error of 47,368.
The estimate of the ratio is 39,401,560 / 33,385,649 = 1.180.
The standard error of this ratio is
The 90 percent margin of error for this estimate would be 0.00246 multiplied by 1.645, or about 0.004. The 90 percent lower and upper 90 percent confidence bounds would then be [1.180 - 0.004] to [1.180 + 0.004], or 1.176 and 1.184.
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 75,086,485 with a margin of error of 218,471 from subject table S2504 for 2006, and the percent of 1-unit detached owner-occupied housing units is 81.4% (0.814) with a margin of error of 0.1 (0.001). So the number of 1-unit detached owner-occupied housing units is 75,086,485 * 0.814 = 61,120,399. Calculating the standard error for the estimates using the margin of error we have:
SE(75,086,485) = 218,471 / 1.645 = 132,809 and SE(0.814) = 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 61,118,243 using the standard error, simply multiply 117,348 by 1.645, then add and subtract the product from 61,120,399. Thus the 90 percent confidence interval for this estimate is [61,120,399 - 1.645(117,348)] to [61,120,399 + 1.645(117,348)] or 60,927,362 to 61,313,436.