Standard errors of estimated percentages. The reliability of an estimated percentage, computed using sample data for both numerator and denominator, depends on both the size of the percentage and the size of the total upon which the percentage is based. Estimated percentages are relatively more reliable than the corresponding estimates of the numerators of the percentages, particularly if the percentages are 50 percent or more. When the numerator and denominator of the percentage are in different categories, use the factor or parameters indicated by the numerator. The approximate standard error, (x,p) of an estimated percentage, p, on a base of size x can be obtained by use of the formula

Using the 60,000 estimate of standard error, the 68percent confidence interval as shown by the data is from 1,717,000 to 1,837,000. Therefore, a conclusion that the average estimate derived from all possible samples lies within a range computed in this way would be correct for roughly 68 percent of all possible samples. Similarly, we could conclude with 95 percent confidence that the number of Spanish-origin women 18 to 34 years old reporting on birth expectations in 1981 lies within the interval from 1,657,000 to 1,897,000 (using twice the standard error).

Table 2 shows that of the 3,289,000 Black women 18 to 34 years old reporting on birth expectations, 1,089,000 or 33.1 percent were currently married. Table C-5 shows the b parameter for CPS data for Black women to be 1,698;

using formula (4) the standard error, (x,p), of a 33.1 percent estimate is approximately 1.1 percent.3. Consequently, the 68-percent confidence interval of the percentage of currently married Black women 18 to 34 years old reporting on birth expectations is from 32.0 to 34.2 percent and the 95-percent confidence interval is from 30.9 to 35.3 percent.

Illustration of the computation of the standard error of a difference between percentages. Table 9 of this report shows that in 1981, 11.2 percent of the 22,358,000 White women 18 to 34 years old reporting on birth expectations expected no lifetime births whereas 9.0 percent of the 3,289,000 Black women in the corresponding age group expected no lifetime births. Thus, the apparent difference in the percentages between these two groups of women is 2.2. Using table C-2 and the appropriate factors from table C4, the standard error of the estimated percentages of women expecting no lifetime births is 0.2 for White women and 0.7 for Black women. Using formula (5) the standard error of the estimated difference of 22 percent is about

√(0.2)2 + (0.7)2 = 0.7 percent

*Formula (2) for this example with a ≈ 0.000.

* Formula (3) and the factor of 10 in table C4 for this example

xn'

and

2,000... 2,500. 3,000. 3,500.. 4,000.

...

1.05

1.14

1.18

1.22

1.26

1.30

1.34

1.38

It should be noted that for data involving only one event per woman, e.g., one additional birth expected, table C-2, the

Illustration of the computation of the standard error of a fertility ratio. Table 2 of this report shows that in 1981 Spanish origin women 18 to 34 years old expected 865 future births per 1,000 women. Table 2 also shows that there were 1,777,000 Spanish-origin women 18 to 34 years old. Table C-3 shows the standard error of 865 future births with a base of 1,777,000 women to be approximately 32. Multiplying the standard error, 32, by the factor for Spanishorigin women found in table C-4 (i.e., 1.14), the standard error becomes 36 per 1,000 women. Consequently, the 68-percent confidence interval for the fertility ratio is from 829 to 901 births per 1,000 Spanish-origin women and the 95-percent confidence interval is from 793 to 937 future births per 1,000 Spanish-origin women 18 to 34 years old. The standard errors of estimated numbers as calculated for this report are not applicable to estimates of total women by age or race. As estimates by age, sex and race are independently derived totals, they are not subject to any sampling error. Because of the use of ratio estimation, any published number which is a large subset of a given age-sex-race group (i.e., 50 percent or more) will have a sampling error smaller than that shown in table C-1. For such items, a closer approximation to the standard errors can be obtained by using the population of the age-sex-race group as the base of the percentage and coverting the standard errors in table C-2 from percentages to numbers by multiplying them by the bases.

For example, table 2 of this report shows that in 1981 there were 15,348,000 currently married women 18 to 34 years old. If this figure was a subset of a considerably larger independent age-race group, the standard error on this estimate would be taken from table C-1. However, since figure of 15,348,000 currently married women re