APPENDIX A: Beers formulas for calculation of base population by single ages

Beers formulas (Beers, 1945) to separate the population by five-year age groups into single years of age. The 0-4 age group is split using the following formulas:

(A1)

(A2)

(A3)

(A4)

(A5)

where p₁, p₂, p₃, p₄, and p₅ are the population aged 0-4, 5-9, 10-14, 15-19, and 20-24 respectively and a₀, a₁, a₂, a₃, and a₄ are the populations at single ages 0, 1, 2, 3, and 4 respectively. Similarly, the 5-9 age group is split using the following formulas:

(A6)

(A7)

(A8)

(A9)

(A10)

The age groups from 10-14 to 70-74 are split using the following formulas:

(A11)

(A12)

(A13)

(A14)

(A15)

where a₁, a₂, a₃, a₄, and a₅ are the first, second, third, fourth, and fifth ages in the particular age group and p_a-2 is the population of the age group two groups younger than the reference group, p_a-1 is the population of the age group one group younger than the reference group, and so on. The 75-79 age group is split with the formulas:

(A16)

(A17)

(A18)

(A19)

(A20)

Ages 80 and above are aggregated, and by definition are not split into single years.