Documentation: Census 1960 Tracts Only Set
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Publisher: U.S. Census Bureau
Document: Migration between State Economic Areas (Volume II, Part II - Subject Reports)
citation:
U.S. Bureau of the Census. U.S. Census of Population: 1960. Subject Reports, Migration between State Economic Areas. Final Report PC(2)-2E. U.S. Government Printing Office, Washington, D.C. 1967.
Migration between State Economic Areas (Volume II, Part II - Subject Reports)
Sample Design and Sampling Variability
Sample Design
For persons in housing units at the time of the 1960 Census, the sampling unit was the housing unit and all its occupants; for persons in group quarters, it was the person.

To provide a sampling frame for the 25-percent sample, the enumerators were instructed (on the first visit to an address) to assign a key letter (A, B, C, or D) to each
housing unit sequentially in the order in which he first visited the units, whether or not he completed an interview. The enumerators were given a random key letter to start their assignments, and the order of canvassing was indicated in advance. The instructions allowed some latitude in the order of visiting addresses. Each housing unit to which the key letter "A" was assigned was designated as a sample unit, and all persons enumerated in the unit were included in the sample. In group quarters, the sample consisted of every fourth person in the order listed.

Although the subsampling procedure did not automatically insure an exact 25-percent sample of persons, the sample design was unbiased. Generally, for large areas, the deviation from the estimated sample size was found to be quite small. Biases may have arisen, however, when the enumerator failed to follow his listing and sampling instructions exactly during the designation of the 25-percent sample.
Ratio Estimation
The statistics based on the 25-percent sample of the 1960 Census returns are estimates that have been developed through the use of a ratio estimation procedure. This procedure was carried out for each of the following 44 groups of persons in each of the sample weighting areas: 4

Group Sex, color, and age Relationship and tenure
  Male white:  
1 Under 5  
2 5 to 13  
3 14 to 24 Head of owner household
4 14 to 24 Head of renter household
5 14 to 24 Not head of household
6-8 25 to 44 Same groups as age group 14 to 24
9-11 45 and over Same groups as age group 14 to 24
  Male nonwhite:  
12-22 Same groups as male white  
  Female white:  
23-33 Same groups as male white  
  Female nonwhite:  
34-44 Same groups as male white  

For each of the 44 groups, the ratio of the complete count to the sample count of the population in the group was determined. Each specific sample person in the group was assigned an integral weight so that the sum of the weights would equal the complete count for the group. For example, if the ratio for a group was 4.2, one-fifth of the persons (selected at random) within the group were assigned a weight of 5, and the remaining four-fifths a weight of 4. The use of such a combination of integral weights rather than a single fractional weight was adopted to avoid the complications involved in rounding in the final tables. In order to control a potential bias in the estimates, where there were fewer than 50 persons in the complete count in a group, or where the resulting weight was over 16, groups were combined in a specific order until both of these conditions were satisfied.

These ratio estimates reduce the component of sampling error arising from the variation in the size of household and achieve some of the gains of stratification in the selection of the sample, with the strata being the groups for which separate ratio estimates are computed. The net effect is a reduction in the sampling error of most statistics below what would be obtained by weighting the results of the 25-percent sample by a uniform factor of four. The reduction in sampling error will be trivial for some items and substantial for others. A by-product of this estimation procedure, in general, is that estimates for this sample are generally consistent with the complete count with respect to the total population and for the subdivisions used as groups in the estimation procedure.


Footnote:
4Estimates of characteristics from the sample for a given area are produced using the formula:


vhere x is the estimate of the characteristic for the area obtained through the use of the ratio estimation procedure,
xi is the count of sample persons with the characteristic for the area in one (i) of the 44 groups,
yi is the count of all sample persons for the area in the same one of the 44 groups, and
Yi is the count of persons in the complete count for the area in the same one of the 44 groups.

Sampling Variability
The figures from the 25-percent sample tabulations are subject to sampling variability, which can be estimated roughly from the standard errors shown in tables A and B.

These tables do not reflect all the effect of response variance, or bias arising in the collection, processing, and estimation steps. Estimates of the magnitude of some of these factors in the total error are being evaluated and are being published in reports in Series ER 60, Evaluation and Research Program of the U.S. Censuses of Population and Housing: 1960. The chances are about two out of three that the difference due to sampling variability between an estimate and the figure that would have been obtained from a complete count of the population is less than the standard error. The chances are about 19 out of 20 that the difference is less than twice the standard error and about 99 out of 100 that it is less than 2 ½ times the standard error. The amount by which the estimated standard error must be multiplied to obtain other odds deemed more appropriate can be found in most statistical textbooks.

Table A. Rough Approximation to Standard Error of Estimated Number
(Range of 2 chances out of 3)
Estimated number Standard error
50 25
100 35
250 50
500 65
1,000 80
2,500 130
5,000 180
10,000 260
15,000 305
25,000 400
50,000 560


Table B. Rough Approximation to Standard Error of Estimated Percentage
(Range of 2 chances out of 3)

Estimated percentage Base of percentage
500 1,000 2,500 10,000 25,000 100,000
2 or 98 2.1 1.4 0.8 0.5 0.2 0.2
5 or 95 3.2 2.2 1.4 0.6 0.3 0.2
10 or 90 4.5 3.2 1.9 1.0 0.5 0.3
25 or 75 6.1 4.3 2.4 1.1 0.6 0.3
50 7.0 5.0 2.6 1.3 0.8 0.5


Table A shows rough standard errors of estimated numbers up to 50,000. The relative sampling errors of larger estimated numbers are somewhat smaller than for 50,000. For estimated numbers above 50,000, however, the nonsampling errors, e.g., response errors and processing errors, may have an increasingly important effect on the total error. Table B contains rough standard errors of data in the form of percentages, and although percentages are not published in this text, table B may be useful. Linear interpolation in tables A and B will provide approximate results that are satisfactory for most purposes.

The standard errors estimated from tables A and B are not directly applicable to differences between two sample estimates. For a difference between two estimates, the standard error is approximately the square root of the sum of the squares of the standard error of each estimate considered separately. The estimated differences shown in table 5 are based on the sample estimates in table 4. The approximate standard error of the differences in table 5 can be determined from the above formula using the standard error of the corresponding estimates from table 4. (see illustration).

Illustration
Table 4- shows that there were an estimated 6,398 persons whose residence in 1960 was in economic subregion 7, hut whose residence in 1955 was in economic subregion 1. Table 4 shows that for an estimated 6,398 persons, the approximate standarderror is 202. Table 4 also shows that there were an estimated 4,543 persons whose residence in 1960 was in economic subregion 1, but whose residence in 1955 was in economic subregion 7. Table A shows that for an estimated 4,543 persons, the approximate standard error is 171. Table 5 shows that there were an estimated 1,855 persons in net migration between economic subregions 1 and7. Therefore, the approximate standard error for this estimated 1,855 persons in net migration is

, or 265. This means that the chances are approximately two out of three that the results of a complete count would not differ by as much as 265 from a sample estimate. It also follows that there is only about one chance in 100 that the results of a complete count would differ by as much as 663, that is, by about 2 ½ times the standard error.