Below is text from one section of Douglas Galbi’s work, “Sense in Communication.” This work includes text and images. Some images may be missing (due to use restrictions) or improperly formatted below. The full work in pdf format, as well as other text sections, are available at www.galbithink.org
Adjusting Name Popularity Statistics for Family Size
For comparing and interpreting the popularity of highly popular names, family size can be important. Consider, for example, the popularity of the name Mary. Calculating the share of females named Mary among all females ignores that some females might not have been at risk, i.e. eligible, to be called Mary. In particular, giving living sisters the same name could cause confusion in identifying them. If parents as a rule do not repeat names among living siblings, then the number of females who were eligible to be named Mary depends on the family size distribution and the distribution of the name Mary relative to birth order. In a naming equilibrium, both dynastic and popularity considerations favor Mary having a higher probability of being represented among older daughters. When Mary is a highly popular name, large average family size can significantly affect the share of females eligible to be named Mary.
A rough adjustment for family size can be calculated using these parameters:
f = the share (fraction of females) named Mary
s = the probability of a female child surviving to adulthood
z= share of births to unmarried women
b = average female births per married woman
n = the share of females named Mary who are the first-born daughter (the share of females named Mary who are the second-born daughter will be assumed to be 1-n).
A simple calculation gives the proportion of women, with at least one daughter, who have a daughter name Mary. In a steady-state name-share equilibrium, the share of women named Mary is f. Now consider the share of women who were not eligible to be named Mary. Assume that all females born to unmarried women have no sisters. Let N be the number of women. A woman has on average (1-z)s(b-1) sisters. Then the nfN first-born women named Mary have nfN(1-z)s(b-1) sisters who were ineligible to be called Mary. The (1-n)fN second-born females named Mary have (1-n)fN(1-z)s(b-2) sisters who were ineligible to be called Mary. The total number of adult females who were ineligible to be named Mary is thus i = fN(1-z)s(n(b-1)+(1-n)(b-2)). The share of women named Mary, among those women who were eligible to be named Mary, is v=f/(1-i). In a population naming equilibrium with no repeated names for living daughters, this statistic v estimates the proportion of women, with at least one daughter, who have a daughter named Mary.
The parameterization and result for mid-eighteenth century England are:
f = 0.24 [see Appendix A, infra.]
s = 0.58 [Livi-Bacci (2000) Table 5.1, p. 94-5]
b = 3.3 [calc. from id.]
z = 0.03 [Laslett, Costerveen, and Smith (1980) p. 14]
n = 0.5 [conjecture]
These figures imply v = 0.32.