Illustration. Table 4 of this report shows that in 1990, 23.1 percent of the 4,840,000 Hispanic families were maintained by female householders. Table 4 also shows that 16.0 percent of all non-Hispanic families (61,250,000) were maintained by female householders. The apparent difference between the percentage of Hispanic and non-Hispanic families maintained by female householders in 1989 is 7.1 percent. Using formula (4) with b = 1,539 from table B-5, the approximate standard error, Sx, for Hispanic female householders is 0.8. The standard error, sy, for non-Hispanic female householders is 0.2 (b = 1,703). Using formula (5), the standard error of the estimated difference of 7.1 percent is about

Sx-y =√(0.8)2+(0.2)2 =0.8

This means that the 90-percent confidence interval around the difference is from 5.8 to 8.4, i.e., 7.1 ± 1.6(0.8). Because this interval does not contain zero, we can conclude with 90 percent confidence that the percentage of families maintained by female householders is larger for Hispanics than for non-Hispanics. Standard Error of a Mean for Grouped Data. The formula used to estimate the standard error of a mean for grouped data is

c is the number of groups; i indicates a specific group, thus taking on values 1 through c.

P, is the estimated proportion of households, families or persons whose values, for the characteristic (x-values) being considered, fall in group i.

x, is (Z1.1 + Z) /2 where Z., and Z, are the lower and upper interval boundaries, respectively, for group i.

x; is assumed to be the most representative value for the characteristic for households, families, and unrelated individuals or persons in group i. Group c is open-ended, i.e., no upper interval boundary exists. For this group the approximate average value is

The standard error of the numerator, s,, and that of the denominator, s,, may be calculated using formula (2). Alternatively, use formula (1) and tables B-1, B-2, and B-5. In formula (10), r represents the correlation between the numerator and the denominator of the estimate.

For one type of ratio, the denominator is a count of families or households and the numerator is a count of persons in those families or households with a certain characteristic. If there is at least one person with the characteristic in every family or household, use 0.7 as an estimate of r. An example of this type is the mean number of children per family with children.

For all other types of ratios, r is assumed to be zero. If r is actually positive (negative), then this procedure will

Standard Error of a Median. The sampling variability of an estimated median depends on the form of the distribution and the size of the base. One can approximate the reliability of an estimated median by determining a confidence interval about it. (See the section on sampling variability for a general discussion of confidence intervals.)

Estimate the 68-percent confidence limits of a median based on sample data using the following procedure. 1. Determine, using formula (4), the standard error of the estimate of 50 percent from the distribution.

2. Add to and subtract from 50 percent the standard error determined in step 1.

3. Using the distribution of the characteristic, determine upper and lower limits of the 68-percent confidence interval by calculating values corresponding to the two points estab lished in step 2.

Use Pareto interpolation for any point in an income interval greater than $2,500 in width, and linear interpolation otherwise. The formulas for interpolation are:

A mathematically equivalent result is obtained by using common logarithms (base 10) and antilogarithms.

4. Divide the difference between the two points determined in step 3 by two to obtain the standard error of the median.

The new, more detailed income intervals used in this report have $2,500 increments up to $40,000 for households and families and up to $20,000 for persons, and Pareto interpolation is needed only when a median income falls in an interval of width larger than $2,500 (beginning with March 1980 CPS). Therefore, this type of interpolation will seldom be needed (i.e., only in cases where the estimated median income exceeds $40,000 for households and families and $20,000 for persons). For this reason, illustration of the use of Pareto interpolation in computing a confidence interval for a median has been omitted. An illustration of this procedure can be found in the source and reliability section of Current Population Reports, Series P-60, No. 123.

Use of the above procedure could result in standard errors which differ from those given in the detailed tables. The reasons for this discrepancy are the use of a more detailed distribution than that given in the tables in determining the published standard errors, and the rounding of the numbers to thousands in the published tables. Linear interpolation was almost always used to compute the published medians and standard errors. Occasionally, a median may lie in an open-ended interval. To calculate its standard error the user must call Housing and Household Economic Statistics Division of the Census Bureau to obtain the methodology.

Illustration. Table 1 shows that the median age of the Mexican population in 1990 in the United States was 24.1. Table 1 also shows that the base of the distribution from which this median was determined was 13,305,000. 1. Using formula (4) and b = 2,622 from table B-5, the standard error of 50 percent on a base of 13,305,000 is about 0.7 percentage points.

2. Adding to and subtracting from 50 percent the standard error found in step 1 to obtain a 68-percent confidence interval on the estimated median yields limits of 49.3 percent and 50.7 percent.

3. From table 1, 58.1 percent (7,730,000) of the Mexican population was 20 years of age or older and 48.3 percent (6,426,000) was 25 years of age or older. Thus, the entire 68-percent confidence interval falls in the age interval 20 to 25. The upper and lower limits of the confidence interval for the median age of the Mexican population can be calculated using linear interpolation. Using formula (12), the lower limit on the estimate is about