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# FAMILY SIZE DISTRIBUTION

A family is often defined as a group of people who are related through marriage, blood, or adoption. The size of this unit depends on the criteria used for establishing membership. One common application of the term family size refers to women alone and counts only the number of children born to them. That is the usage that will be employed here.

Suppose that a person asked a woman how many children she had borne and that she replied, "three." Suppose that the person also asked each of those three children how many children his or her mother had borne. The answer should also be "three." For an individual family, there should be no difference between the "family size" of the mother and the "family size" of a child (that is, the number of children borne to the mother of a particular child).

When measurement is extended to a population, on the other hand, these two measures need not and typically will not have the same value. Consider a population in which one-half of the women have one child and the other half have seven. The mean family size among the women is four. But the mean family size among their children will not be four but some larger number. The reason for the discrepancy is that each woman with seven children leaves seven times as many children to testify about her family size as a woman with one child. In this particular example, the mean family size among the children will be [7(7) + 1(1)]/8 = 6.25.

It is clear that the relation between the family size of women and the family size of children depends on how much variability exists in the fertility performance of women. If all women had three children, then all children would derive from three-child families and the mean family size of women, three, would be the same as the mean family size of children. But if there is any spread at all in the distribution of women's family sizes, then women with higher fertility will be overrepresented in reports by children about their mother's childbearing.

The formula that relates the mean family size of children, C*, to the mean family size of women, W*, is:

C* = W* + V/W*,

where V is the variance in family sizes among women. If there is no variance in childbearing among women, then C* will equal W*. Any variance whatsoever will increase C* above W*.

This relationship would be a mere statistical oddity were it not for the huge variability in childbearing among women in most populations. Among women in the United States who had completed childbearing between 1890 and 1970, the mean family size of their children exceeded the mean family size of women by 1.8 to 3.1 children. The mean family size of children was never less than 4.4 during this period. The variance of family size among women often grows in the course of a fertility transition as subgroups of the population develop small family norms while others retain their previous behaviors. When this happens, family sizes among children decline more slowly than family sizes of women or may even rise.

One striking disparity between the two measures of family size occurred in the United States when the low fertility rates of the Great Depression were replaced by the high fertility rates of the baby boom. Women who bore the bulk of their children during the 1930s wound up with about 2.3 children, whereas those at the peak of their childbearing years in the 1950s bore an average of about 2.7. But the mean family size of the Depression-era children, 4.9, was actually higher than that of the baby boomers, 4.5. The reason for the discrepant trends is that the baby boom was accompanied by much lower variability in family sizes among women. Facile attributions of baby boomers' characteristics to their unusually large families were clearly based on a false premise.

There are several other useful implications of the disparity between the two measures. First, one should not try to infer directly the aggregate fertility levels of women in the past from the accounts of their children. Such histories provide a very biased view of fertility in the past unless they are corrected for variability in the distribution of family sizes. Such corrections are rarely undertaken, and the result is that people often gain an inflated impression of the past volume of childbearing from personal testimonials about the fertility of ancestors.

A second implication is that there must be a "revolt against childbearing" each generation simply to keep a population's fertility rate constant. Women must bear, on average, fewer children than their mothers, or a population's level of fertility would rise sharply every generation.

The relation between family sizes of women and family sizes of children is analogous to several other relations in demography. For example, the mean size of a household when households are the unit of analysis is always less than the mean size of households when individuals are the unit of analysis. The schemes employed for calculating mean household size by most statistical offices, which treat each household as one unit, underestimate the size of household as experienced by members of the population. This distortion extends to the classification of households by other criteria as well. For example, only 38 percent of households in the United States in 1980 contained a child under age 17, but 59 percent of the U.S. population lived in a household containing a child. The reason for the discrepancy is that households containing children were, on average, 52 percent larger than the mean for all household types.

## bibliography

Bongaarts, John. 2001. "Household Size and Composition in the Developing World in the 1990s." Population Studies 55: 263–279.

Burch, Thomas K. 1972. "Some Demographic Determinants of Average Household Size: An Analytic Approach." In Household and Family in Past Time, ed. Peter Laslett. New York: Cambridge University Press.

King, Miriam, and Samuel H. Preston. 1990. "Who Lives with Whom? Individual versus Household Measures." Journal of Family History 15: 117–132.

Preston, Samuel H. 1976. "Family Sizes of Children and Family Sizes of Women." Demography 13:105–114.

Samuel H. Preston