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Suppose
$D_i=$ Observed number of deaths in the age group $(x_i, x_{i+1})$,
$P_i=$ Mid-year population for age group $(x_i, x_{i+1})$,
$N_i=$ Number of people alive at $x_i$ among whom $d_i$ deaths occur.
I can't distinguish between $P_i$ and $N_i$. What is the difference between $P_i$ and $N_i$?
The number of people alive $(N_i)$ is the number at the start of the time period so the number at risk of dying during the following year. The mid-year population $(P_i)$ is just that, the number alive at the middle of the time period under study. This can be used, for instance, if you want to estimate the number of person-years that population were at risk of something like an illness or accident. You cannot use $N_i$ to calculate person-years at risk because, especially at advanced ages, many will die part way through the year.
