Proportional and Equal Allocation in Stratified Samples

Why Allocation Matters

In a stratified sample, the population is divided into strata, and units are sampled from each stratum. Once the total sample size is chosen, a second decision remains: how many units should come from each stratum? That decision is the allocation. It affects how closely the sample’s composition follows the population and how much information is available about each subgroup.

Two common choices are proportional allocation, which assigns sample counts in proportion to stratum sizes, and equal allocation, which assigns the same count to every stratum. Neither is automatically best for every question. Proportional allocation naturally represents population composition when the within-stratum selection procedures are comparable. Equal allocation gives smaller strata more sample members relative to their population size, which can be useful when the goal includes comparing subgroups.

Definition: Suppose a population has \(H\) strata. Let \(N_h\) be the number of population units in stratum \(h\), \(N=\sum_{h=1}^{H}N_h\) the total population size, and \(n\) the desired total sample size. Let \(n_h\) be the number sampled from stratum \(h\), so that \(\sum_{h=1}^{H}n_h=n\). An allocation specifies the values of the \(n_h\)'s.

Proportional Allocation

Proportional allocation gives each stratum a sample count matching its share of the population. If stratum \(h\) contains a fraction \(N_h/N\) of the population, it is assigned that same fraction of the total sample:

n_h=n\frac{N_h}{N}.

This formula can produce a non-integer value, although an actual sample count must be a whole number. When that happens, the counts must be rounded and adjusted so they add to \(n\). The rounded allocation will usually be close to proportional but not exactly proportional.

Worked Example: Calculating Proportional Allocation

A hypothetical population has 1,200 members in Stratum A, 800 in Stratum B, and 400 in Stratum C. A total sample of \(n=240\) is planned. Find the proportional allocation.

The total population size is \(N=1{,}200+800+400=2{,}400\). Apply the allocation formula separately to each stratum.

For Stratum A: \(n_A=240(1{,}200/2{,}400)=240(0.5)=120\).

For Stratum B: \(n_B=240(800/2{,}400)=240(1/3)=80\).

For Stratum C: \(n_C=240(400/2{,}400)=240(1/6)=40\). The counts sum to \(120+80+40=240\), as required. Each stratum contributes 10% of its members: \(120/1{,}200=0.10\), \(80/800=0.10\), and \(40/400=0.10\).

Equal Allocation

Equal allocation assigns the same sample count to every stratum. If all strata are to receive an equal share of the sample, the count is:

n_h=\frac{n}{H}\quad\text{for each }h.

This requires \(n/H\) to be a whole number for an exactly equal allocation. If it is not, equal whole-number counts are impossible while also using exactly \(n\) units; the counts must differ by at least one, or the planned total sample size must change. Feasibility also requires \(n_h\leq N_h\) for every stratum. In particular, a small stratum cannot supply more distinct units than it contains when sampling without replacement.

Worked Example: Calculating Equal Allocation

A hypothetical population consists of two strata: 900 rural residents and 300 urban residents. The planned total sample size is \(n=120\), with an equal number sampled from each stratum.

Here \(H=2\), so each stratum receives \(n/H=120/2=60\) sample members. The allocation is \(n_{\text{rural}}=60\) and \(n_{\text{urban}}=60\), which sums to 120. Both counts are feasible because neither stratum has fewer than 60 members.

The sampling fractions are different: \(60/900=0.0667\), or about 6.67%, in the rural stratum, and \(60/300=0.20\), or 20%, in the urban stratum. Thus equal allocation does not mean that the same percentage of each stratum is sampled. It means the same number of units comes from each stratum.

Rounding Proportional Counts

A practical proportional allocation often requires rounding. A sound procedure is to first calculate all unrounded counts, take their integer parts, and then distribute any remaining sample places to the strata with the largest fractional remainders. This makes the final counts add to the planned total while keeping them as close as possible to the proportional targets under this procedure.

Worked Example: Rounding a Proportional Allocation

A hypothetical population has 510 units in Stratum A, 290 in Stratum B, and 200 in Stratum C. The total population is \(N=1{,}000\), and the desired sample size is \(n=97\). Calculate a whole-number allocation using largest remainders.

The unrounded proportional counts are: \(97(510/1{,}000)=49.47\) for A, \(97(290/1{,}000)=28.13\) for B, and \(97(200/1{,}000)=19.40\) for C.

Taking the integer parts gives 49, 28, and 19, which total 96. One sample place remains. The fractional remainders are 0.47, 0.13, and 0.40; the largest is 0.47 for Stratum A. Assign the remaining place to A. The final allocation is therefore \(n_A=50\), \(n_B=28\), and \(n_C=19\), and \(50+28+19=97\).

The resulting sampling fractions are \(50/510=0.0980\), \(28/290=0.0966\), and \(19/200=0.0950\). They are close to, but not exactly, the intended common fraction \(97/1{,}000=0.0970\), because whole-number sample counts were required.

What the Allocation Implies About Selection Chances

Allocation by itself does not determine each unit’s chance of selection. The following probability statements require an additional assumption: within each stratum, units are selected with equal probability, for example by simple random sampling without replacement. Under that assumption, each member of stratum \(h\) has inclusion probability \(n_h/N_h\). A different probability design within a stratum could give different members different inclusion probabilities, even if the stratum’s sample count is proportional.

For exact proportional allocation, substitute \(n_h=nN_h/N\) into the inclusion probability:

\frac{n_h}{N_h} = \frac{nN_h/N}{N_h} = \frac{n}{N}.

Thus, with exact proportional counts and equal-probability selection within each stratum, every population member has the same inclusion probability \(n/N\). This is a result of both the allocation and the within-stratum selection assumption, not of proportional allocation alone. With rounded proportional counts, the fractions \(n_h/N_h\) can differ slightly across strata.

Under equal allocation, the inclusion probability in stratum \(h\), again assuming equal-probability selection within that stratum, is \(n/(HN_h)\). It is generally higher in smaller strata and lower in larger strata. For the rural–urban example, these probabilities were 6.67% and 20%, respectively. Equal allocation therefore deliberately gives the smaller urban stratum a larger sampling fraction.

Estimating an Overall Population Mean

When a quantitative variable is measured, let \(\bar{x}_h\) be the sample mean in stratum \(h\), and let \(\mu_h\) be that stratum’s population mean. If each stratum is sampled by simple random sampling without replacement, \(\bar{x}_h\) estimates \(\mu_h\). The overall population mean gives each stratum weight according to its share of the population:

\mu=\sum_{h=1}^{H}\frac{N_h}{N}\mu_h.

Replacing each stratum mean with its sample mean gives the corresponding population-weighted estimate:

\hat{\mu}_{\mathrm{str}}=\sum_{h=1}^{H}\frac{N_h}{N}\bar{x}_h.

The within-stratum simple random sampling assumption matters here: it makes the unweighted mean within each stratum appropriate for estimating that stratum’s mean. If a stratum instead uses unequal inclusion probabilities, an unweighted \(\bar{x}_h\) need not estimate \(\mu_h\) without bias; a design-weighted stratum estimate is needed.

Under proportional allocation, the sample’s stratum shares approximately match the population’s shares, so the ordinary combined sample mean will be close to the population-weighted form. Under equal allocation, the combined sample mean gives equal weight to the stratum sample means, which generally differs from the population-weighted estimate. For the rural–urban example, the appropriate population weights are \(900/1{,}200=0.75\) and \(300/1{,}200=0.25\), not one-half each.

Choosing Between the Two Allocations

The choice depends on what the sample is meant to support. Proportional allocation is a natural choice when the main aim is to describe the population overall and the cost and variability of sampling are similar across strata. It gives larger strata more sample members and, under the stated within-stratum assumption, gives population members equal inclusion probabilities when allocation is exact.

Equal allocation can be useful when separate descriptions or comparisons of the strata are important. If one stratum is much smaller than another, proportional allocation may leave it with very few observations. Equal allocation increases its sample count relative to its population size, while reducing the count taken from the larger stratum compared with proportional allocation. The benefit is more observations available for studying the small group; the trade-off is that an overall population estimate must account for the different stratum sizes.

Neither method guarantees that a sample is representative in every respect. Allocation addresses how the sample is divided across known strata. The selection process within each stratum still matters, and the sample must be drawn as planned. The allocation formulas alone do not correct for unequal inclusion chances within a stratum or for other sources of error.

Common Mistakes

Confusing equal counts with equal percentages. Equal allocation gives the same number to each stratum, not the same sampling fraction. The fraction is \(n_h/N_h\), so it differs when stratum sizes differ.
Assuming proportional allocation always gives equal inclusion probabilities. That conclusion requires exact proportional counts and equal-probability selection within each stratum. Allocation alone does not guarantee it.
Using the ordinary combined sample mean after equal allocation. If strata have different population sizes, an unweighted combined mean gives the strata weights based on sample counts, not their population shares. Use the population-weighted stratum means under equal-probability sampling within strata.
Rounding each count independently without checking the total. Whole-number counts must sum to \(n\). Calculate the unrounded targets first, then adjust the rounded counts so the planned total is met.

Key takeaway: Proportional allocation assigns \(nN_h/N\) units to stratum \(h\); equal allocation assigns \(n/H\). Check that the integer counts sum to the planned sample size, and state the within-stratum selection assumption before interpreting inclusion probabilities or calculating an overall estimate.

Check Your Understanding

Use the definitions and assumptions above to answer each question.

A population has strata of sizes 600 and 400, and the planned sample size is 150. What are the proportional sample counts?
There are four strata and a planned sample size of 200. What is the equal allocation, and what feasibility check should be made?
In a population with strata of sizes 800 and 200, equal allocation gives 50 units to each. Assuming simple random sampling within each stratum, calculate each stratum’s inclusion probability.
Why does exact proportional allocation not, by itself, prove that every member of the population has the same chance of selection?
For a population with two strata of sizes 900 and 300, explain why an overall mean based on equal sample counts should generally weight the stratum means by 0.75 and 0.25 rather than one-half each.