Given a set of populations \(P_i\) for age intervals \([x_{i-1},x_{i})\) the cumulative population at \(x_i\) is $$ N_i=\left\{ \begin{array}{ll} 0,& i=0\\ \sum_{k\le i } P_i,&\mathrm{otherwise} \end{array}\right. $$ The cumulative population \(N(x)\) at any age \(x\) is calculated by cubic spline interpolation of \(N_i\).