An intraclass correlation of 0.02 sounds negligible and is not. With fifty people in each of twenty clusters the design effect is 1.98, so a thousand enrolled subjects carry the information of 505.05 — 494.9 observations bought and discarded. The multiplier is 1 + (m − 1)ρ, and the cluster size is doing the work: at ρ = 0.02 it takes only fifty-one people per cluster to double the required sample. Effective observations per cluster are 25.253 against a ceiling of 50, so this design has already spent half of what the correlation permits.
Design effect 1.9800 — 1000 subjects carry the information of 505.05
Effective observations per cluster are 25.253 against a ceiling of 50.00, so 50.5% of what the correlation allows is already spent. 494.9 observations are lost to within-cluster similarity.
Design effect
1.9800
1 + (m − 1)ρ
Effective sample
505.05
of 1000 enrolled
Observations wasted
494.9
49.5% of the study
Per-cluster ceiling
50.00
reached 50.5% at 50 per cluster
The ceiling, and the approach to it
Design effect and effective observations at a range of cluster sizes
Per cluster
Design effect
Effective per cluster
Share of the ceiling
Gain from the last one
2
1.020
1.961
3.9%
0.9608
5
1.080
4.630
9.3%
0.8560
10
1.180
8.475
16.9%
0.7160
20
1.380
14.493
29.0%
0.5222
30
1.580
18.987
38.0%
0.3976
50
1.980
25.253
50.5%
0.2525
100
2.980
33.557
67.1%
0.1111
200
4.980
40.161
80.3%
0.0397
500
10.980
45.537
91.1%
0.0081
The last column is the argument. Effective observations per cluster are m/(1 + (m − 1)ρ), which converges to 1/ρ however large the cluster grows — so each extra person contributes less than the one before, and beyond a point the contribution rounds to nothing. That ceiling is a property of the correlation alone and no amount of enrolment moves it.
The design effect is exactly 1 when either the correlation is zero or there is one observation per cluster, which the verification suite asserts along with the formula itself on 150 generated designs. It also asserts that effective observations per cluster never exceed 1/ρ, which is the ceiling this page is built around.
An intraclass correlation is itself an estimate, usually from a previous study, and a noisy one — a random-intercept model on thirty clusters gives an unbiased estimate with a wide spread. Since the design effect is linear in it, a correlation underestimated by half halves the inflation you plan for.
Effective sample size The 1/ρ ceiling Marginal gain per person The ICC is an estimate
What this tool shows
At an intraclass correlation of 0.05, no cluster size ever delivers more than twenty effective observations — however many people are enrolled. Two hundred per cluster gets 18.265 of that twenty, and going from a hundred to two hundred bought 1.4581 effective observations across the whole cluster. A correlation of 0.02 sounds negligible and costs half the study at fifty per cluster: a thousand subjects carry the information of 505.05.
The design effect 1 + (m − 1)ρ and the effective sample size it implies
The ceiling of 1/ρ effective observations per cluster, and how much of it a given cluster size uses
The marginal gain in effective observations from each extra person, across nine cluster sizes
How many observations a design is buying and discarding, as a count and a share
A preset where the correlation genuinely is small enough to ignore, for contrast
Why a correlation estimated from a previous study carries its own uncertainty into the plan
Effective sample size The 1/ρ ceiling Marginal gain per person The ICC is an estimate
Updated 13 September 2026 · Works in any browser, no installation
When observations come in clusters that resemble each other, the study carries less information than its size suggests, and the design effect 1 + (m − 1)ρ is how much less. Divide the enrolled sample by it to get the sample an independent study would have needed. The multiplier is driven by the cluster size rather than the correlation, which is why a correlation of 0.02 — a number people round to nothing — doubles the required sample at fifty per cluster.
At a glance
Formula shown
DEFF = 1 + (m − 1)ρ, where m is the observations per cluster and ρ the intraclass correlation. Effective sample size is the total divided by that. Effective observations per cluster are m/(1 + (m − 1)ρ), which as m grows converges to 1/ρ — a hard ceiling that no amount of enrolment passes. DEFF is exactly 1 when either ρ = 0 or m = 1.
Scenario support
Planning a cluster-randomised trial, adjusting a survey’s margin of error for its sampling design, deciding whether to add more clusters or more people per cluster, and reading a published clustered study that reported its raw sample size.
Educational estimate
Planning support from the values you enter — not professional advice.
Effective observations per cluster have a hard ceiling
The design effect is usually presented as a penalty that grows with the cluster. The more useful way to see it is as a limit that the cluster approaches and never passes.
Effective observations per cluster are m/(1 + (m − 1)ρ), which converges to 1/ρ as m grows.
At ρ = 0.05 that ceiling is exactly 20. Two hundred people per cluster deliver 18.265 of it — 91.3%.
Going from a hundred to two hundred bought 1.4581. A hundred additional people, across the whole cluster, worth less than one and a half observations.
Which reframes the design question entirely. Past a certain size the only lever is more clusters, and the page prints the share of the ceiling already used so that point is visible rather than discovered late.
A correlation of 0.02 is not small
The intuition that a two per cent correlation can be ignored is the most expensive mistake this page exists to prevent.
At ρ = 0.02 with fifty per cluster the design effect is 1.98. A thousand subjects carry the information of 505.05.
Fifty-one people per cluster doubles the requirement exactly. 1 + 50 × 0.02 = 2.
The cluster size does the work, not the correlation. The same 0.02 with five per cluster gives a design effect of 1.08, which really can be ignored.
So “the ICC is small” is never a complete sentence. The fourth preset shows what genuinely negligible looks like: ρ = 0.001 at thirty per cluster, a design effect of 1.029 and under three per cent lost.
More clusters or bigger ones
Once the ceiling is understood the design trade-off becomes concrete, and it usually points the opposite way from what is convenient.
Each additional cluster adds its full effective contribution. Independent of every other cluster, and therefore worth the same as the first.
Each additional person adds less than the one before. At ρ = 0.05 the second person in a cluster adds 0.905 effective observations and the five-hundredth adds 0.001.
Recruiting clusters is almost always harder, which is why studies drift towards fewer, larger ones — and why they end up underpowered.
The arithmetic is unambiguous even when the logistics are not, and the sample-size page turns the trade-off into a table of clusters needed at each cluster size.
Where clustering comes from
The design effect applies far more widely than cluster-randomised trials, and it is often unaccounted for because nobody called the data clustered.
Repeated measures on the same subject are a cluster. Five readings from one person are not five independent observations.
Multi-site studies cluster by site. Patients at the same hospital share protocols, staff and case mix.
Household surveys cluster by household, which is why survey margins of error are routinely wider than the raw sample implies.
And matched or paired designs are clusters of two, where the correlation is usually the whole point — there the design effect is below 1, because the pairing helps rather than hurts.
The correlation is a number you had to guess
Everything here is exact arithmetic given ρ, and ρ itself almost never is.
It usually comes from a previous study, where it was estimated from a limited number of clusters.
The estimator is unbiased and noisy. A random-intercept model on thirty clusters recovers the truth on average and any single estimate can be well off.
The design effect is linear in it, so an underestimate carries straight through: plan with 0.01 when the truth is 0.02 and you have planned half the inflation you needed.
Which argues for planning at the upper end of a plausible range, and for reporting the assumed correlation so anyone reading the study can see what the plan rested on.
The design effect is for planning, not analysis
This calculation belongs in a protocol. At analysis time there are better tools, and substituting one for the other is a common shortcut.
Dividing the sample by the design effect gets the planning arithmetic right. It is the standard and correct use.
It is not a valid way to analyse clustered data. Inflating a standard error by √DEFF after the fact assumes equal cluster sizes and a known correlation, and neither holds.
A mixed model or a cluster-robust variance estimator uses the actual sizes and the actual correlation, and gives different answers when the clusters are unbalanced.
The design effect stays useful as a summary, which is why a random-intercept fit reports one — a single number saying how much the clustering cost, computed from the data rather than assumed.
Reporting a clustered design
Four items, and the second is the one whose absence makes a sample size unverifiable.
Give the effective sample size, not just the enrolled one. A thousand subjects in twenty clusters is not a thousand observations.
Give the assumed correlation and where it came from. It is the one input nobody can check without being told.
Give the number of clusters separately from the cluster size. They are not interchangeable, and the first is what most of the power rests on.
And say what was done at analysis. A design effect used for planning and a mixed model used for analysis is the normal pairing; a design effect used for both is a shortcut worth declaring.
Sources and methodology
References for the design effect and effective sample size.
Method. The design effect is 1 + (m − 1)ρ exactly, and the effective sample size the total divided by it, so everything here is arithmetic rather than estimation — which means the verification suite can check it against hand-worked values rather than tolerances. It asserts the formula on 150 generated designs, that the effect is exactly 1 when either the correlation or the cluster size term vanishes, and that effective observations per cluster never exceed 1/ρ on the same 150. The marginal-gain column is the difference between consecutive cluster sizes rather than a derivative, which is what makes it directly readable as “what the last person added”. That engine is verified on every change against 128 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Cluster Sample SizeClusters needed for a target power, with the individual-randomisation comparison and the whole trade-off between more clusters and bigger ones.
Mixed ModelA random-intercept model from variance components, with the intraclass correlation, the design effect it implies, both standard errors for the overall mean, and shrunk group estimates.
Intraclass CorrelationAll six ICC forms from one subject-by-rater matrix, with the rater means that drive them apart: one 8x3 matrix gives ICC(1,1) = 0.1277 and ICC(3,1) = 0.9852.
Sample SizeResponses needed for a target margin of error, with the finite-population correction and a table of the whole cost curve — because n scales with 1/margin², so the last point of precision costs more than the first ten.
Margin of ErrorMargin of error for a percentage or an average, shown across seven sample sizes so the square-root law is visible — every doubling buys exactly 29.3%, never more.
Survival Sample SizeSchoenfeld's event requirement converted to an enrolment target, with the sensitivity to the hazard ratio and the power achieved at every event count.
An educational tool. The design effect is a planning device and not a method of analysis — inflating a standard error by its square root after the fact assumes equal cluster sizes and a known correlation, and a mixed model or cluster-robust estimator is the right tool at analysis time. The intraclass correlation it depends on is itself an estimate, and the design effect is linear in it.