 Finite population total

A finite population total adds a specified numerical value over every labelled unit in a fixed, finite population $U=\{1,\dots ,N\}$:The target must specify both the population and the measured variable. It may be a total inventory mass, the total number of employees across establishments, or the sum of a binary property. If $y_i\in \{0,1\}$, the total counts units having the property; it is not automatically the population size $N$. A finite population mean is $\overline{Y}=T/N$, when $N$ is known and positive.

An original inventory has four containers holding $3,7,10,20\mathrm{kg}$ of a material. The total is $40\mathrm{kg}$ and the finite population mean is $10\mathrm{kg}$ per container. Giving every container the same mass fraction of an analyte is a separate assumption: the total analyte mass instead requires summing each container's mass multiplied by its own mass fraction. The target follows the physical quantity being added.

When only a sample $S$ is observed, the unweighted sample sum is generally not an unbiased estimator of $T$. Using inclusion indicators $I_i$ and inclusion probabilities ${\pi }_i$ givesIf every ${\pi }_i>0$, the Horvitz-Thompson estimator instead satisfiesThis is a consequence of the sampling design, without requiring the fixed population values to follow a statistical distribution.^[ref] A full census removes sample-selection variation, but imperfect measurement can still make the recorded sum differ from the intended physical total.