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Demography measures the size, composition and change of human populations. It combines population censuses, registrations of births and deaths, and statistical sampling with mathematical models of population dynamics. A population census refers its counts to a specified time and territory; collecting questionnaires on different days should not silently change the reference date.^[ref]

For a fixed territory, a demographic accounting identity iswhere $B,D,I,E$ count births, deaths, immigration and emigration during the intervening period. For example, $N(t_0)=10000$, $B=140$, $D=90$, $I=80$ and $E=110$ give $N(t_1)=10020$. This demographic balancing equation distinguishes natural increase from net human migration; it does not assume an exponential growth law.

Counts also need denominators. A birth rate compares births with the relevant population exposure, rather than comparing raw birth totals between differently sized human populations. Changes in age composition can alter an aggregate rate even when the underlying age-specific rates remain fixed, motivating analysis by demographic cohorts.

For demography, human migration is movement that changes a person's place of usual residence across a specified territorial boundary. Counts depend on the boundary, the residence criterion and the observation interval. A commuter crossing daily and a person changing residence therefore need not be counted alike.

In the demographic balancing equation, immigration $I$ adds residents and emigration $E$ removes them, so net migration is $I-E$. If a district gains 800 and loses 500 residents by migration during a year, its net migration is 300, while the total number of moves is 1300. Net migration alone can conceal substantial turnover. Movement between two districts contributes to each district's account but cancels in the national total if both districts are inside the same national boundary. This accounting viewpoint is the quantitative aspect used here.

A human population for quantitative demography is the collection of people satisfying a stated territorial, residence and time definition. Its size $N(t)$ is a count; its age or other composition describes how that count is distributed between subgroups. A resident population and the people physically present on a particular night can have different sizes.

For example, a town with 10000 residents and 200 births during one year has an approximate crude birth rate of 20 births per 1000 residents per year if its midyear population is 10000. A town with 20000 residents and the same birth count has approximately half that rate. Raw event counts therefore cannot replace denominators. Population censuses and the demographic balancing equation connect successive measurements, while demographic cohorts allow changes in composition to be tracked.

Thomas Robert Malthus made the comparison between population growth and the growth of food supply explicit in his 1798 An Essay on the Principle of Population by Thomas Robert Malthus. In Chapters I–II, he assumed that unchecked numbers could follow a geometric progression, taking doubling every 25 years as his rule, while food supply gained a fixed amount in each such period, an arithmetic progression. These were different hypotheses about two quantities, rather than a calculation from a single universal growth law.^[ref]

With equal initial scales, his comparison can be written as $N_k=N_0{2}^k$ and $F_k=F_0(1+k)$ after $k$ periods of 25 years. Then $N_k/F_k=(N_0/F_0)2^k/(1+k)$ grows without bound. The corresponding continuous exponential growth rate for numbers would be $\log (2)/25$ per year. The mathematics describes the consequence of the assumed rates; it does not establish that actual birth, death, human migration and food-production rates stay fixed. Demography can test population assumptions against counts, age composition and measured rates.