Method document

Sample size for a stratified survey

How many people a survey needs when its population is divided into strata, such as regions, and every stratum is sampled: for one combined estimate or for every stratum on its own, with or without cluster sampling inside the strata, and for finite populations of people or of clusters.

Engine version 1.0.0 · Reviewed 8 October 2026

Open in calculator

https://statwisely.com/methods/precision-stratified/

When to use it

A national or regional survey of a prevalence, a mean, a median, a paired change or a set of category proportions that samples every region, district type or urban/rural area.
A survey that must also report each stratum: small strata are then sampled above their share, and the cost of doing so is worth seeing before fieldwork.

Not for

Comparing strata with each other. That is a comparison of groups, planned for a difference or for power.
Stratified randomisation in a trial, which balances arms and barely changes the sample size.

Notation

$H$: Number of strata
$W_h$: Population share of stratum h
$a_h$: Sample share of stratum h, set by the allocation
$n_{\mathrm{SRS}}$: Size a simple random sample would need for the margin
$n_h$: Sample in stratum h
$S_h^{2}$: Variance per person in stratum h: p_h(1 − p_h) for a proportion, σ² for a mean
$\mathrm{DE}_w$: Design effect of the weighting
$\mathrm{DE}_c$: Design effect of clustering within strata
$\rho$: Intracluster correlation
$m$: People sampled per cluster
$c_h$: Clusters sampled in stratum h
$N,\ C$: People, or clusters, in the population; N_h = W_h N and C_h = W_h C
$z_{1-\alpha/2}$: Standard normal quantile for confidence 1 − α

Formulas

The cost of the allocation

\mathrm{DE}_w = \sum_{h=1}^{H} \frac{W_h^{2}}{a_h}, \qquad n = \left\lceil n_{\mathrm{SRS}}\,\mathrm{DE}_w \right\rceil, \qquad n_h = \left\lceil a_h\, n \right\rceil

Sampling every stratum adds no design effect, but an allocation away from the population shares weights respondents unequally, and the combined estimate pays Kish’s weighting effect [1, 2]. It is 1 for proportional allocation and grows as the shares are bent. Without each stratum’s own proportion the strata are taken as equally variable, the cautious side; given them, the variance is Σ W_h² p_h(1 − p_h)/n_h, which credits the gain from strata that differ.

Allocations

a_h \propto W_h \;\;(\text{proportional}), \quad 1 \;\;(\text{equal}), \quad \sqrt{W_h} \;\;(\text{square root}), \quad W_h S_h \;\;(\text{Neyman})

The square root is Bankier’s compromise, giving small strata usable samples at a modest cost [5]; Neyman’s allocation samples where the variance is [4, 3]. A custom allocation takes any shares.

Clustering inside the strata

n = \left\lceil n_{\mathrm{SRS}}\,\mathrm{DE}_w\,\mathrm{DE}_c \right\rceil, \qquad \mathrm{DE}_c = 1 + (m-1)\rho, \qquad c_h = \max\!\left(2, \left\lceil \frac{n_h}{m} \right\rceil\right)

The two inflations multiply [1]. A cluster belongs to a stratum, so the cluster size is given and each stratum’s cluster count follows from its sample, never fewer than two.

Every stratum on its own

n_h = \left\lceil \frac{z_{1-\alpha/2}^{2}\, S_h^{2}}{E^{2}}\,\mathrm{DE}_c \right\rceil, \qquad n = \sum_h n_h

When each stratum is to be reported, each is its own survey, sized from its own variance. The combined estimate then comes out more precise than the margin.

Few clusters

n \;\leftarrow\; n \left(\frac{t_{\sum_h c_h - H,\,1-\alpha/2}}{z_{1-\alpha/2}}\right)^{2}

Optional. A stratified, clustered estimate has the clusters minus the strata as its degrees of freedom, the convention of survey software [6]; it is iterated because the clusters depend on n.

Finite strata of people

\operatorname{Var} = \sum_h W_h^{2} S_h^{2} \left(\frac{1}{n_h} - \frac{1}{N_h}\right), \qquad N_h = W_h N

Each stratum carries its own correction [3]. With proportional allocation this gives the familiar n/(1 + n/N). A stratum asked for more than it holds is taken whole, and the others are solved again.

Finite clusters

\mathrm{DE}_{c,h} = 1 + (m-1)\rho - \frac{c_h}{C_h}\, m\,\rho, \qquad C_h = W_h C

When the clusters sampled are a sizeable share of those there are, the between-cluster part of the variance shrinks by 1 − c/C. People within a cluster are taken as a small share of it, the cautious side. A stratum asked for more clusters than it holds takes all of them.

Non-response

n_{h,\text{approach}} = \left\lceil \frac{n_h}{1 - q} \right\rceil

Applied last, in every stratum.

The margin of a sample already in hand

E = z_{1-\alpha/2}\sqrt{\mathrm{DE}_c \sum_h W_h^{2}\, \frac{S_h^{2}}{n_h}\left(1 - \frac{n_h}{N_h}\right)}, \qquad E_h = z_{1-\alpha/2}\sqrt{\mathrm{DE}_c\, \frac{S_h^{2}}{n_h}\left(1 - \frac{n_h}{N_h}\right)}

The same variance read the other way: when the respondents in each stratum, n_h, are already fixed, by finished fieldwork or a budget, the margin follows instead of the sample. The last factor applies only when the population is given. Choose “The margin from a sample I already have” in the stratified section. Planning a sample for a margin and then entering that sample returns the planned margin.

Parameters

Confidence level $1-\alpha$

Values: 90%, 95% or 99%

95% is the convention in health research. A higher level widens the interval, so the same margin needs a larger sample.

Margin of error $E$

Values: Above 0; on the proportion scale for a proportion, in the outcome’s units for a mean

For the combined estimate, or for each stratum when every stratum is sized on its own.

Expected prevalence (or, for a mean, the standard deviation) $p$

Values: Between 0 and 1

The value used for every stratum unless each stratum’s own is given.

Population share of each stratum $W_h$

Values: Positive, adding to 100% (or to 1)

From a census or sampling frame; at least two strata.

Stratum names $\text{names}$ Optional

Values: One per stratum

Only labels the result.

Allocation $a_h$ Optional

Values: Proportional, square root, equal, Neyman or custom

Proportional is cheapest for the combined estimate; move towards equal when small strata must be reported.

Expected proportion in each stratum $p_h$ Optional

Values: Between 0 and 1, one per stratum

Needed for Neyman allocation. When given, the overall proportion becomes their population-weighted average, and the combined estimate credits the gain from strata that differ; with every stratum reported, each is sized from its own.

What the margin is for $\text{goal}$ Optional

Values: The combined estimate, or every stratum

Every stratum usually costs far more people.

Intracluster correlation $\rho$ Optional

Values: From 0 to 1

Within the strata; usually a little smaller than the overall ICC. A design effect can be given instead.

People per cluster $m$ Optional

Values: A whole number of at least 1

Within strata the cluster size is given, and each stratum’s cluster count follows.

Correction for few clusters $t$ Optional

Values: On or off

Worth applying with fewer than about 30 clusters in all.

People in the population $N$ Optional

Values: A whole number; without clustering

Matters only when a stratum’s sample is a sizeable share of that stratum.

Clusters in the population $C$ Optional

Values: A whole number; with clustering

Shared across strata by population share; matters only when a sizeable share of a stratum’s clusters is sampled.

Expected non-response rate $q$ Optional

Values: 0% up to, but not including, 100%

Applied in every stratum.

Assumptions

Every stratum is sampled independently, by simple random sampling or by clusters within it.
The analysis weights respondents back to the population shares and accounts for the strata and any clusters, for example with survey-design standard errors [6].
Non-response is unrelated to the outcome.

Worked examples

Proportional allocation

Four regions holding 50%, 30%, 15% and 5% of the population, a prevalence of about 30% and a margin of ±3 points: a simple random sample needs 897, and proportional allocation costs nothing, so North 449, South 270, East 135 and West 45.

North: 449 · South: 270 · East: 135 · West: 45 · Total: 899

Open this example in the calculator

Equal allocation

Giving every region 328 people lets West be reported, but the weights cost the combined estimate: 1312 in all.

328 per stratum · Total: 1312

Open this example in the calculator

Square-root compromise

West gets 118 at a total of 985: between the two.

North: 373 · South: 289 · East: 205 · West: 118 · Total: 985

Open this example in the calculator

Neyman allocation

With regional prevalences of 20%, 30%, 40% and 50%, the more variable regions get more: West 47, 814 in all.

North: 373 · South: 257 · East: 137 · West: 47 · Total: 814

Open this example in the calculator

Clusters of 20 inside the regions

With equal allocation, an ICC of 0.05 and 20 people per cluster, the weighting and clustering multiply: 639 per region in 128 clusters, 2556 in all.

639 per stratum · Total: 2556

Open this example in the calculator

Larger clusters, corrected for few of them

With 40 per cluster and the t correction on clusters minus regions: 991 per region in 100 clusters.

991 per stratum · Total: 3964

Open this example in the calculator

Every region on its own, ±5 points

Sized from each region’s own prevalence: North 246, South 323, East 369, West 385; 1323 in all.

North: 246 · South: 323 · East: 369 · West: 385 · Total: 1323

Open this example in the calculator

A population of 4,000

Equal allocation asks West for more people than its 200, so West is taken whole (200) and the other regions carry 268 each.

North: 268 · South: 268 · East: 268 · West: 200 · Total: 1004

Open this example in the calculator

80 clusters in the population

East and West hold only 12 and 4 clusters and take all of them; sampling so large a share lowers the design effect: 1202 in 62 clusters.

North: 441 · South: 441 · East: 240 · West: 80 · Total: 1202

Open this example in the calculator

Allowing for 10% non-response

Proportional allocation, with each region’s sample divided by 0.9: 999 to approach.

North: 449 · South: 270 · East: 135 · West: 45 · Total: 899 · To approach: 999 in total

Open this example in the calculator

Validation

Every worked example is recomputed by the calculation engine on every build.
Simulated 2,000 times with the R survey package from stratified, clustered populations, the plans gave intervals of the planned width with about 95% coverage, including the every-stratum goal and the correction for few clusters [6].
Simulated 4,000 times from finite populations, the corrections for finite strata and finite clusters delivered their target margins.

How to cite

StatWisely. Sample size for a stratified survey. Method document, engine version 1.0.0, reviewed 2026-10-08. https://statwisely.com/methods/precision-stratified/

Cite the original references for the formulas your design uses as well.

References

1.Kish L. Survey Sampling. New York: John Wiley & Sons; 1965.
2.Kish L. Weighting for unequal Pi. Journal of Official Statistics. 1992;8(2):183–200.
3.Cochran WG. Sampling Techniques. 3rd ed. New York: John Wiley & Sons; 1977.
4.Neyman J. On the two different aspects of the representative method: the method of stratified sampling and the method of purposive selection. Journal of the Royal Statistical Society. 1934;97(4):558–625. doi:10.2307/2342192
5.Bankier MD. Power allocations: determining sample sizes for subnational areas. The American Statistician. 1988;42(3):174–177. doi:10.1080/00031305.1988.10475559
6.Lumley T. Analysis of complex survey samples. Journal of Statistical Software. 2004;9(8):1–19. doi:10.18637/jss.v009.i08

When to use it

Not for

Notation

Formulas

The cost of the allocation

Allocations

Clustering inside the strata

Every stratum on its own

Few clusters

Finite strata of people

Finite clusters

Non-response

The margin of a sample already in hand

Parameters

Confidence level1−α1-\alpha1−α

Margin of errorEEE

Expected prevalence (or, for a mean, the standard deviation)ppp

Population share of each stratumWhW_hWh​

Stratum namesnames\text{names}namesOptional

Allocationaha_hah​Optional

Expected proportion in each stratumphp_hph​Optional

What the margin is forgoal\text{goal}goalOptional

Intracluster correlationρ\rhoρOptional

People per clustermmmOptional

Correction for few clusterstttOptional

People in the populationNNNOptional

Clusters in the populationCCCOptional

Expected non-response rateqqqOptional

Assumptions

Worked examples

Proportional allocation

Equal allocation

Square-root compromise

Neyman allocation

Clusters of 20 inside the regions

Larger clusters, corrected for few of them

Every region on its own, ±5 points

A population of 4,000

80 clusters in the population

Allowing for 10% non-response

Validation

How to cite

References

Confidence level $1-\alpha$

Margin of error $E$

Expected prevalence (or, for a mean, the standard deviation) $p$

Population share of each stratum $W_h$

Stratum names $\text{names}$ Optional

Allocation $a_h$ Optional

Expected proportion in each stratum $p_h$ Optional

What the margin is for $\text{goal}$ Optional

Intracluster correlation $\rho$ Optional

People per cluster $m$ Optional

Correction for few clusters $t$ Optional

People in the population $N$ Optional

Clusters in the population $C$ Optional

Expected non-response rate $q$ Optional