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 $N_i$?