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Design-based inference evaluates an estimator by the random mechanism selecting a sample from a fixed finite population. Population values $y_1,\dots ,y_N$ are held fixed; the selected subset $S$ is random under a known sampling design $p(s)$. ThusThe subscript $p$ identifies the source of randomness. An unbiased estimator of a finite population total has ${\mathbb{E}}_p\hat{T}=T$ for each eligible fixed population. The unequal-probability total developed by D. G. Horvitz and D. J. Thompson is a central example.^[ref]

For an original example, let the fixed population values be $(1,1,10)$ and sample one label uniformly. Every inclusion probability is $1/3$, so the Horvitz-Thompson estimator is $3y_i$. Its three equally probable values are $3,3,30$, while the actual total is $12$. Consequently,The estimator is unbiased even though none of its possible estimates equals the target. Unbiasedness describes averaging across the sampling design, rather than guaranteed accuracy in an individual sample.

A confidence interval with design-based coverage must likewise succeed at its stated rate across repeated selections under the specified design. That property requires a justified construction; unbiasedness of a point estimate alone supplies no coverage guarantee. Repeating a convenient calculation on the same selected units does not reproduce the selection process. Assumptions about measurement error or missing responses must also be stated separately, since knowledge of the original selection probabilities alone does not repair those later stages.

D. G. Horvitz coauthored the 1952 paper that generalized finite-population sampling without replacement to unequal selection probabilities. The joint work with D. J. Thompson provides the foundation for the Horvitz-Thompson estimator.^[ref]

The central idea is to compensate for unequal chances of inclusion. For fixed population values $y_i$ and positive inclusion probabilities ${\pi }_i$, write a total estimate asUsing an inclusion indicator $I_i$ gives $\mathbb{E}(I_i/{\pi }_i)=1$, so $\mathbb{E}\hat{Y}={\sum }_i{y}_i$. Unbiasedness follows from the design probabilities rather than from assuming that the sampled values are an independent realization of a population model.^[ref]

For an illustrative population $(4,6)$, select exactly one unit with probabilities $(1/4,3/4)$. The total estimate is $16$ or $8$, respectively, with expected value $(1/4)16+(3/4)8=10$, the true total. The less frequently selected unit receives the larger multiplier. This example explains how a sampling design can deliberately favor some units without forcing an unweighted analysis to be correct.

D. J. Thompson coauthored the 1952 general treatment of sampling without replacement from a finite population with unequal selection probabilities. The paper with D. G. Horvitz is the source of the Horvitz-Thompson estimator.^[ref]

The contribution separates unbiased total estimation from assumptions of independent selection. If $I_i$ is an inclusion indicator, then $\mathbb{E}{I}_i={\pi }_i$, while pair products satisfy $\mathbb{E}(I_i{I}_j)={\pi }_{ij}$. Consequently, for fixed population values,This identity makes joint inclusion probabilities essential to a variance calculation, even though the expectation of the total estimate only needs the individual inclusion probabilities.

For an illustrative population $(1,1)$, sample one unit uniformly. Each selected value is multiplied by $2$, so the total estimate is always $2$ and its variance is zero. The individual random contributions are nevertheless variable: their covariance cancels their variances because they cannot both be selected. Ignoring that sampling dependence would give the wrong precision. The example shows why design-based estimation concerns the whole selection mechanism, not just a set of multipliers.

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.

The Horvitz-Thompson estimator estimates a finite population total by expanding each observed unit's contribution by the reciprocal of its inclusion probability. D. G. Horvitz and D. J. Thompson introduced the general unequal-probability construction in 1952.^[ref]

Let $y_1,\dots ,y_N$ be fixed values attached to labelled population units. A sampling design selects a random subset $S$, with known inclusion probabilities ${\pi }_i=\mathbb{P}(i\in S)>0$. Thenwhere $I_i$ is the inclusion indicator. This is inverse probability weighting for a finite population.^[ref] Its expected value is immediatelyThe subscript $p$ emphasizes design-based inference: the fixed values are not assumed to follow a probability model; the sample-selection mechanism supplies the randomness. The inclusion indicators need not be independent. Dividing by known $N$ gives an unbiased estimator of the finite population mean.

For an original exact example, use values $(y_1,y_2,y_3)=(2,4,6)$ and choose samples $\{1,2\}$, $\{1,3\}$, $\{2,3\}$ with probabilities $1/2,1/3,1/6$, respectively. The inclusion probabilities are $(5/6,2/3,1/2)$. The corresponding estimates of the total are $8.4,14.4,18$, andSimply multiplying each sample mean by three instead gives totals $9,12,15$, with expected value $11$. Equal treatment of the observed values ignores the unequal chances of observing them.

The sampling variance depends on pair information. Put ${\pi }_{ij}=\mathbb{P}(i,j\in S)$, including ${\pi }_{ii}={\pi }_i$. Bilinearity of covariance givesFor the three-unit design it is $14.4$, found directly from $\frac{1}{2}(8.4-12{)}^2+\frac{1}{3}(14.4-12{)}^2+\frac{1}{6}(18-12{)}^2$. Thus an unbiased estimator need not give a nearly correct total in each sample.

With pairwise inclusion positivity, the same expansion device applied to pairs constructs the Horvitz-Thompson variance estimator:Its expected value equals the preceding population sum because ${\mathbb{E}}_p(I_i{I}_j/{\pi }_{ij})=1$. In the worked design its three possible values are $9.76,58.56,-60$, whose probability-weighted average is $14.4$. This particular unbiased estimator of variance can therefore be negative for an individual sample. Its expected value identity is a statement about repeated selection.

If ${\pi }_i=0$, an arbitrary unobserved $y_i$ cannot contribute to an unbiased total estimate. If a distinct pair has ${\pi }_{ij}=0$, its product cannot be recovered by the pair-expansion argument. Finally, dividing the weighted sum by the random sum of its inverse-probability weights changes the estimator into a ratio; the indicator proof above no longer establishes that this ratio is an unbiased estimator. The known population size and the actual sampling design are part of the result.

A joint inclusion probability is the probability that two specified population units both belong to a random sample $S$:For the same unit twice, ${\pi }_{ii}={\pi }_i$, where ${\pi }_i$ is its inclusion probability. For distinct units, ${\pi }_{ij}$ is generally not ${\pi }_i{\pi }_j$: selection can make their inclusion indicators dependent. These probabilities describe the sampling procedure, rather than a relationship between the measured values of the two units.^[ref]

For simple random sampling of $n$ distinct labels from $N>1$ labels, counting subsets containing both $i$ and $j$ giveswith value zero if $n<2$. Since ${\pi }_i=n/N$,For $N=4,n=2$, each unit has inclusion probability $1/2$, but a particular pair has joint inclusion probability $1/6$. Conditional on including one member, the other is included with probability $(1/6)/(1/2)=1/3$, not $1/2$.

Equal individual inclusion probabilities do not determine these pair probabilities. A different procedure that chooses only $\{1,2\}$ or $\{3,4\}$, each with probability $1/2$, still includes each unit with probability $1/2$. Yet ${\pi }_{12}=1/2$ and ${\pi }_{13}=0$. Thus two procedures can agree on individual representation while producing different dependence and different precision for a population summary. The distinction motivates keeping pair information when calculating the variance of weighted survey estimators; the classical general treatment is due to D. G. Horvitz and D. J. Thompson.^[ref]

A sampling design specifies a probability distribution over the possible samples from a defined population:For sampling distinct labelled units, $s$ is a subset of the population. The design includes the selection mechanism, not merely the final sample size. A procedure with repeated draws can instead require a distribution over ordered sequences; the resulting inclusion probabilities concern whether a label appears at least once.

The design determines inclusion probabilities and joint inclusion probabilities throughThese connect the physical sampling procedure to the Horvitz-Thompson estimator and its sampling variance.^[ref]

An original comparison uses four fixed values $(0,0,10,10)$ and selects two units. Under simple random sampling, every one of the six pairs has probability $1/6$. Each ${\pi }_i=1/2$, so the Horvitz-Thompson estimator is twice the selected sum. It takes values $0,20,40$ with probabilities $1/6,4/6,1/6$. Its expected value is the true finite population total $20$, and its variance isA second design chooses only $\{1,2\}$ or $\{3,4\}$, each with probability $1/2$. It has the same sample size and the same individual inclusion probabilities, but the estimator is now $0$ or $40$, giving variance $400$. Pair selection matters even when individual representation agrees.

A third design chooses uniformly among the four pairs containing one label from each of those two groups. Its total estimate is always $20$ for this particular population. This illustrates the potential gain from stratified sampling when groups correspond to relevant variation, not a claim that the third design dominates for every possible population. A reproducible design records the eligible labels and actual random selection rule before interpreting the observed sample.