Method document

Sample size to estimate several category proportions at once

How many participants are needed to estimate the share of every category of a variable, such as blood groups or levels of a scale, so that all the intervals are within a chosen margin together. Covers finite populations, cluster sampling, stratified sampling and non-response.

Engine version 1.0.0 · Reviewed 8 October 2026

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

When to use it

Describing the whole distribution of a categorical variable, when every category’s share matters.

Not for

A single category of interest, such as the prevalence of one condition. Use a single proportion, which needs fewer participants.
Testing whether distributions differ between groups, which is a test of independence. To estimate the shares within each group, set the number of groups.

Notation

$n$: Sample size under simple random sampling
$k$: Number of categories (and, under clustering, of clusters; the context makes clear which)
$p_i$: Expected proportion in category i
$E$: Margin of error for every category, on the proportion scale
$\chi^{2}_{1,\,1-\alpha/k}$: Chi-square quantile with 1 degree of freedom at 1 − α/k
$g$: Number of groups, each planned to the same precision
$z_{1-\alpha/2}$: Standard normal quantile for confidence 1 − α (1.960 for 95%)
$N$: Population size
$\mathrm{DE}$: Design effect
$\rho$: Intracluster correlation coefficient (ICC)
$m$: Usable observations per cluster
$\mathrm{CV}$: Coefficient of variation of cluster size
$C$: Clusters in the population
$W_h$: Population share of stratum h
$a_h$: Sample share of stratum h, set by the allocation
$q$: Non-response rate

Formulas

Simple random sampling

n = \left\lceil \frac{\chi^{2}_{1,\,1-\alpha/k}\;\max_i\,p_i(1-p_i)}{E^{2}} \right\rceil

Goodman’s simultaneous intervals: the confidence level is shared across the k categories, so that all k intervals hold together with confidence 1 − α [1]. Separate 95% intervals would cover all categories at once with well under 95% confidence. Without expected shares, the widest possible category, p = 0.5, is assumed.

Several groups

n_{\text{total}} = g\,n, \qquad \alpha \;\leftarrow\; \alpha / g \ \text{(simultaneous)}

Optional. When every category proportion is wanted in each of g independent groups, each group needs the one-group n. Read together, g intervals each at 95% all hold less than 95% of the time; choosing simultaneous intervals splits α across the groups (Bonferroni), which guarantees the joint level at the cost of a larger n in each group. Cluster sampling is not offered with several groups, and a population size applies to each group.

Finite population correction

n' = \left\lceil \frac{n}{1 + (n - 1)/N} \right\rceil

Applied when a population size is given [3]. It matters only when the sample is a sizeable share of the population, and is not offered with cluster sampling.

Design effect from an ICC

\mathrm{DE} = 1 + (m - 1)\,\rho

For clusters of equal size [4]. A design effect can also be entered directly, in which case no ICC is used.

Unequal cluster sizes

\mathrm{DE} = 1 + \left[(\mathrm{CV}^{2} + 1)\,m - 1\right]\rho, \qquad \mathrm{CV} = \frac{\operatorname{SD}(s_j)}{\bar{s}}

When each cluster is given its own share s_j of the sample, the CV of those shares enlarges the design effect [5]. With a design effect entered directly the shares only divide the sample, since a measured design effect already reflects its survey’s cluster sizes.

Number of clusters for a chosen cluster size

k = \max\!\left(2,\ \left\lceil \frac{n \cdot \mathrm{DE}}{m} \right\rceil\right), \qquad n_{\text{clustered}} = k\,m

The requirement is rounded up to whole clusters, so the total can exceed n × DE slightly. At least two clusters are needed to estimate between-cluster variation.

Cluster size for a fixed number of clusters

m = \left\lceil \frac{n\,(1-\rho)}{k - n\,\rho} \right\rceil, \qquad k > n\,\rho

Solving k·m = n·DE for m. Below that number of clusters no cluster size reaches the margin, because each extra person per cluster adds less and less information; the calculator reports the minimum number of clusters instead.

Sampling a share of a finite set of clusters

\mathrm{DE} = 1 + (m - 1)\,\rho - \frac{k}{C}\, m\,\rho, \qquad k = \left\lceil \frac{n\,[1 + (m-1)\rho]}{m\,(1 + n\rho/C)} \right\rceil

Optional. When the clusters sampled are a sizeable share of the C there are, the between-cluster part of the variance shrinks by 1 − k/C, and the number of clusters has the closed form shown [3]. People within a cluster are taken as a small share of it, the cautious side. The people-level correction does not apply to a cluster sample.

Stratified sampling

n_{\text{stratified}} = \left\lceil n\,\mathrm{DE}_w\,\mathrm{DE} \right\rceil, \qquad \mathrm{DE}_w = \sum_h \frac{W_h^{2}}{a_h}, \qquad n_h = \lceil a_h\, n_{\text{stratified}} \rceil

Optional. When every stratum (region, urban or rural area) is sampled, the strata add no design effect of their own; an allocation away from the population shares weights respondents unequally, which costs the combined estimate Kish’s weighting effect DE_w, 1 for proportional allocation [4]. It multiplies the cluster design effect when clusters are sampled within the strata. Every stratum can instead be sized for its own margin, and finite strata carry their own corrections: see “Sample size for a stratified survey” for the allocations, those options and worked examples.

Few clusters

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

Optional. With few clusters the interval is built on k − 1 degrees of freedom, so the normal critical value is replaced by t. Because k depends on n, this is iterated until it settles.

Non-response

n_{\text{approach}} = \left\lceil \frac{n}{1 - q} \right\rceil

The number to approach so that n respond. Applied last, after any clustering.

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 and below 1 on the proportion scale

The margin every category’s interval must meet.

Number of categories $k$

Values: A whole number of at least 2

Used when no expected shares are given. More categories split the confidence level further, so they cost more participants.

Expected proportions $p_1, \dots, p_k$ Optional

Values: One value per category, each above 0, adding to 1

Give them when known: only the category nearest 50% drives the requirement, so shares away from 50% reduce it.

Number of groups $g$ Optional

Values: A whole number of at least 1; 1 for one sample

The independent groups, such as study arms or exposure categories, in which the same estimate is wanted.

Simultaneous intervals across groups $\alpha / g$ Optional

Values: On or off

Turn on when the groups will be compared or read together, so that every interval holds at once.

Population size $N$ Optional

Values: A whole number of at least 2

Leave empty for a large population. It matters only when the sample would be more than about 5% of the population. Not available with cluster sampling.

Expected non-response rate $q$ Optional

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

The share of people approached who will not take part; use the response rate of a comparable study. Approaching more people restores the number of respondents, not their representativeness: if non-responders differ, the estimate can still be biased.

Intracluster correlation (ICC) $\rho$ Optional

Values: From 0 to 1

The share of the outcome’s variation that lies between clusters. Published ICCs for health outcomes in primary-care and community clusters are mostly small, often below 0.05 [7], but depend on the outcome and the kind of cluster; prefer an estimate from a comparable study.

Design effect $\mathrm{DE}$ Optional

Values: Above 0; usually between 1 and 3

An alternative to the ICC, when a previous study of the same design reports its design effect.

Observations per cluster $m$ Optional

Values: A whole number of at least 1

Usable responses per cluster. Larger clusters are cheaper to reach, but when the ICC is above zero each additional person in a cluster adds less information.

Number of clusters $k$ Optional

Values: A whole number of at least 2

Instead of a cluster size, when the clusters are fixed: the emirates of a country, the schools of a district. The calculator then finds the cluster size, or reports how many clusters the margin needs.

Share of the sample for each cluster $s_j$ Optional

Values: One positive value per cluster, on any scale

When clusters will contribute unequally, for example in proportion to their population. With an ICC, unequal shares enlarge the design effect [5].

Population share of each stratum $W_h$ Optional

Values: Positive, adding to 100%, for at least two strata

Tick “Stratified sampling” to use it. From a census or sampling frame.

Allocation across strata $a_h$ Optional

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

Proportional costs nothing; move towards equal when small strata must be reported on.

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

Values: The combined estimate, or every stratum on its own

Every stratum on its own usually costs far more people.

Clusters in the population $C$ Optional

Values: A whole number, at least the clusters sampled

Leave empty when the clusters sampled are a small share of all there are. Give it when they are not, say 20 of 40 districts; the design effect then falls.

Correction for few clusters $t_{k-1}$ Optional

Values: On or off

Uses t on k − 1 degrees of freedom instead of the normal distribution. Worth applying when there are few clusters, roughly fewer than 30.

Assumptions

Participants, or clusters, are selected at random, and respond independently apart from the clustering specified.
Every category is expected to hold enough people for the normal approximation to apply.
Without expected shares the result is conservative, because no real distribution has every category at 50%. Thompson’s method finds the least favourable distribution instead and gives a smaller sample [2].
With clustering, one ICC is assumed for the membership indicators of all categories, or the largest category-specific ICC as a conservative bound.
With cluster sampling: one ICC applies to every cluster; clusters are of equal size unless shares are given; the sampled clusters are a small fraction of all clusters; and the analysis accounts for the clustering, for example with survey-design standard errors [4, 6].
Non-response is unrelated to the outcome.

Worked examples

Four categories, shares unknown

4 categories, margin ±5 points: χ²₁ at 1 − 0.05/4 is 6.239, so n = 6.239 × 0.25 / 0.05² = 623.85, rounded up to 624.

Sample size: 624

Protocol statement
To estimate the proportions of 4 categories simultaneously with 95% confidence and a margin of error of ±5 percentage points, the required sample size is 624 participants. Calculated with StatWisely (statwisely.com) using: n = χ²_(1-α/k, 1) × max pᵢ(1-pᵢ) / E², Goodman simultaneous intervals over k categories. These are planning estimates; confirm the final design with a qualified statistician.

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Four categories with expected shares

Expected shares 40%, 30%, 20% and 10%: the widest is 40%, with 0.4 × 0.6 = 0.24, so n = 6.239 × 0.24 / 0.05² = 598.9, rounded up to 599.

Sample size: 599

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Cluster sampling with an ICC

ICC 0.02 and 20 per cluster: DE = 1 + 19 × 0.02 = 1.38. Then 624 × 1.38 = 861.1, so 44 clusters and 880 participants.

Sample size: 880

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Category proportions in each of two groups

The same four categories estimated in each of 2 groups, each group on its own: 624 per group, 1248 in all.

Group 1: 624 · Group 2: 624 · Total: 1248

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Validation

Every worked example on this page is recomputed by the StatWisely engine whenever the software is built. If a number or formula here stopped matching the calculator, the release would be stopped until this document was corrected.
An independent implementation in R recomputes this procedure, with and without cluster sampling, every night and before each production release, and the results are compared with the engine’s.
The basic result agrees with the closed form: 6.239 × 0.25 / 0.0025 = 623.85, rounded up to 624. In R, ceiling(qchisq(1 - 0.05/4, 1) * 0.25 / 0.05^2) returns 624.

How to cite

StatWisely. Sample size to estimate several category proportions at once. Method document, engine version 1.0.0, reviewed 2026-10-08. https://statwisely.com/methods/precision-multinomial/

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

References

1.Goodman LA. On simultaneous confidence intervals for multinomial proportions. Technometrics. 1965;7(2):247–254.
2.Thompson SK. Sample size for estimating multinomial proportions. The American Statistician. 1987;41(1):42–46.
3.Cochran WG. Sampling Techniques. 3rd ed. New York: John Wiley & Sons; 1977.
4.Kish L. Survey Sampling. New York: John Wiley & Sons; 1965.
5.Eldridge SM, Ashby D, Kerry S. Sample size for cluster randomized trials: effect of coefficient of variation of cluster size and analysis method. International Journal of Epidemiology. 2006;35(5):1292–1300. doi:10.1093/ije/dyl129
6.Donner A, Klar N. Design and Analysis of Cluster Randomization Trials in Health Research. London: Arnold; 2000.
7.Adams G, Gulliford MC, Ukoumunne OC, Eldridge S, Chinn S, Campbell MJ. Patterns of intra-cluster correlation from primary care research to inform study design and analysis. Journal of Clinical Epidemiology. 2004;57(8):785–794. doi:10.1016/j.jclinepi.2003.12.013

When to use it

Not for

Notation

Formulas

Simple random sampling

Several groups

Finite population correction

Design effect from an ICC

Unequal cluster sizes

Number of clusters for a chosen cluster size

Cluster size for a fixed number of clusters

Sampling a share of a finite set of clusters

Stratified sampling

Few clusters

Non-response

Parameters

Confidence level1−α1-\alpha1−α

Margin of errorEEE

Number of categorieskkk

Expected proportionsp1,…,pkp_1, \dots, p_kp1​,…,pk​Optional

Number of groupsgggOptional

Simultaneous intervals across groupsα/g\alpha / gα/gOptional

Population sizeNNNOptional

Expected non-response rateqqqOptional

Intracluster correlation (ICC)ρ\rhoρOptional

Design effectDE\mathrm{DE}DEOptional

Observations per clustermmmOptional

Number of clusterskkkOptional

Share of the sample for each clustersjs_jsj​Optional

Population share of each stratumWhW_hWh​Optional

Allocation across strataaha_hah​Optional

What the margin is forgoal\text{goal}goalOptional

Clusters in the populationCCCOptional

Correction for few clusterstk−1t_{k-1}tk−1​Optional

Assumptions

Worked examples

Four categories, shares unknown

Four categories with expected shares

Cluster sampling with an ICC

Category proportions in each of two groups

Validation

How to cite

References

Confidence level $1-\alpha$

Margin of error $E$

Number of categories $k$

Expected proportions $p_1, \dots, p_k$ Optional

Number of groups $g$ Optional

Simultaneous intervals across groups $\alpha / g$ Optional

Population size $N$ Optional

Expected non-response rate $q$ Optional

Intracluster correlation (ICC) $\rho$ Optional

Design effect $\mathrm{DE}$ Optional

Observations per cluster $m$ Optional

Number of clusters $k$ Optional

Share of the sample for each cluster $s_j$ Optional

Population share of each stratum $W_h$ Optional

Allocation across strata $a_h$ Optional

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

Clusters in the population $C$ Optional

Correction for few clusters $t_{k-1}$ Optional