Birth order distribution
A birth order distribution for a set of births is the breakdown of those births by birth order, i.e., it specifies what proportion of those births was at a particular birth order.
The more the birth order distribution is skewed toward lower birth orders, the lower the fertility is likely to be (the detailed mathematical relation will be described below).
Types of birth sets for which we are looking for a birth order distribution
Births within a given time period
We might consider all the births that occur within a given year and consider the birth order distribution for these. Note that there is no direct or immediate numerical relationship between the birth order distribution for a single year and the total fertility rate for that year. We can think of the birth order distribution for a particular year as an attempt to use information about a single year to read more long-term trends, but the nature of the slice is different from the nature of the slice that is used when computing the total fertility rate. Ideally, we would find ways of using both pieces of information together.
Births to a given cohort (such as a birth cohort) over their childbearing years
Note that this differs from the previous type in that we are now looking at all births where the mother was born in a certain year (or year range). This birth order distribution, along with information on the total population of the birth cohort of mothers, is equivalent, information-wise, to the completed fertility distribution of the population.