ANOVA Calculator
The formula
What the anova design power tells you
Planning an analysis of variance: given groups, per-group n and an expected η², whether the omnibus F test will actually see the effect.
Calculation: f² = η²/(1−η²); F-df₁ = k−1, df₂ = N−k; power from non-centrality λ = f²N. Enter your inputs and every substitution is shown below the result.
Reading the result
Four groups × 25 with η²=7% lands near 80% — the textbook balance where most real experiments should be aiming.
Rules of thumb: 80% power at α=.05 on the planned η²; balanced cells when you can
Limitations and best practice
Expected η² from optimistic pilots is the leading cause of under-powered ANOVA; deflate pilot effects by a third and re-plan.
This calculator gives an estimate based on standard formulas; it is not professional financial, medical, legal or engineering advice. Verify anything you plan to act on with a qualified professional and your own records.
Frequently asked questions
Fixed or random groups?
Does unbalanced n break ANOVA?
Source & methodology
Cohen (1988); standard power-planning for ANOVA.
How ANOVA works — and what your result really means
You are here because you need a specific number, and you want to see how it was produced before you rely on it. ANOVA is the kind of number people argue about from memory. This page ends that argument with arithmetic you can watch happen: enter your figures, get Design power (η²=8%) in real time, and see every substitution printed below.
It helps to separate the mechanics from the interpretation. The mechanics are fixed: f² = η²/(1−η²); F-df₁ = k−1, df₂ = N−k; power from non-centrality λ = f²N. Anyone, anywhere, entering the same inputs will get the same 24.6 %. The interpretation is where experience counts: whether 24.6 % is good, normal or alarming depends on comparisons covered below, and that is the part most articles about ANOVA never explain.
A calculation like this one is worth roughly as much as a careful sentence: it forces several vague considerations into one comparable quantity. Once ANOVA is on the table, the argument can stop being about impressions.
What makes a calculation page trustworthy is that its copy is derived from its own arithmetic: here design power (η²=8%) = 24.6 %, recomputed every time a field changes, and the sensitivity section ranks the inputs that move the result — Per group size first, at ≈13.9 % of the headline for a ten-percent error either way in that input. Work down the page and by the end you could re-derive the result in a spreadsheet from memory.
The ANOVA formula, explained
The engine implements a single stated model: f² = η²/(1−η²); F-df₁ = k−1, df₂ = N−k; power from non-centrality λ = f²N. There is no hidden coefficient and no login-gated precision — the formula printed is the formula executed, which is also what makes the result citable.
Number of groups (example value 5) is a quantity rather than a category — precision here is cheap and worth having. Quantities scale everything downstream — a one-unit error here is multiplied across the whole result.
Per group size (example value 27) is a plain measured quantity — the unit shown in the label is the unit the formula expects. Measure twice and use the unit the label asks for — length terms often scale the result geometrically.
Expected η² (%) (example value 8) is a numeric input whose unit is fixed by the label; a wrong unit is a wrong answer wearing a plausible disguise. Check the convention first: if the source says “6.5 % per year”, type 6.5 and let the formula do the rest.
ANOVA on the example inputs: the complete substitution
Abstract formulas are hard to trust, so here is the complete substitution using the default inputs, written the way an examiner would want it. Starting figures: Number of groups, Per group size, Expected η² (%). Run to the end, the design power (η²=8%) resolves to 24.6 % — Under-powered — add groups’ n or expect false nulls on the bands this tool uses.
- f² = 0.080 ÷ 0.920 = 0.087; λ = f²N = 11.7
- F(df₁=4, df₂=130); power from λ against the critical ratio ≈ 24.6%
Cohen’s f — 0.295. If your inputs change, this is usually the first companion figure to move.
Degrees of freedom — 4 vs 130. Compare it against the main result of 24.6 % before drawing conclusions.
Non-centrality λ — 11.7. Compare it against the main result of 24.6 % before drawing conclusions.
n per group for 80% power — 37. Compare it against the main result of 24.6 % before drawing conclusions.
Overwrite any input above and every line of this example updates with it — the substitution is live, not a screenshot of somebody else’s numbers.
The ANOVA corridor: your result across a realistic band
Read this table as the practical error bar on the headline figure. It answers the question people actually have after seeing a single number: how wrong could this be if the main input is off by ten or twenty percent?
| Per group size setting | Design power (η²=8%) |
|---|---|
| 80 % of the example value | 17.6 % |
| 90 % of the example value | 21.2 % |
| example inputs | 24.6 % |
| 110 % of the example value | 27.9 % |
| 120 % of the example value | 30.9 % |
The corridor runs from 17.6 % to 30.9 % across that band — which is the range worth planning inside rather than the single 24.6 % point estimate. Reproduce it in the form above by editing Per group size alone; the substitution steps will follow your figures exactly.
The levers behind ANOVA, ranked by effect
Generic advice says “it depends on your inputs”; this section says exactly how much. Each field was perturbed by ±10 % in isolation and the design power (η²=8%) recomputed, so the ranking below is measured rather than asserted. Read it as a map of where ANOVA is fragile — drawn for the example inputs, which is the map you want when deciding what to measure carefully.
- Per group size +10 % → ANOVA = 27.9 % · −10 % → 21.2 % (the design power (η²=8%) shifts ≈13.9 %)
- Nudge Expected η² by ±10 % and ANOVA lands between 22.2 % and 27.1 % (sensitivity ≈10.3 %)
- Number of groups +10 % → ANOVA = 25.2 % · −10 % → 23.9 % (the design power (η²=8%) shifts ≈3.0 %)
Per group size is the dominant lever here: ten percent either way moves the result by about 13.9 %, from 21.2 % to 27.9 %. That is where a careless figure does the most damage, so check it against a source, and treat Expected η² as the second priority (≈10.3 % per ten percent). The rest of the inputs earn only a rounding-error of attention, and that asymmetry is the practical payoff of measuring rather than asserting.
Interpreting the ANOVA verdict
A number on its own is trivia; a number against a range is intelligence. The verdict attached to the default calculation — “Under-powered — add groups’ n or expect false nulls” — is one band on a ladder your own figure may sit somewhere else on. Before treating the band as fact, note its width: ten percent either way on Per group size already spans 21.2 % to 27.9 %. The practical reading rules: read the band as a starting hypothesis, then test it: change the two dominant inputs by ten percent and see whether the verdict survives. Verdicts that survive stress are the ones worth acting on.
In Science & Research, ANOVA is one instrument in an orchestra: on its own it is a number, next to Cohen’s f and Degrees of freedom it becomes a diagnosis. The tools linked above exist so the headline never has to carry that weight alone.
Why ANOVA results go wrong — and how to prevent it
Expected η² from optimistic pilots is the leading cause of under-powered ANOVA; deflate pilot effects by a third and re-plan. This calculator gives an estimate based on standard formulas; it is not professional financial, medical, legal or engineering advice. Verify anything you plan to act on with a qualified professional and your own records.
Chasing precision in Per group size while Number of groups is still a guess. The sensitivity ranking above shows exactly where the accuracy budget should go.
Attention is also misallocated between fields. Having measured this engine, we can say where it belongs: Per group size first, everything else second.
Another is comparing across contexts: an ANOVA computed with one set of conventions should never be lined up against a figure computed with another. Check the inputs behind both before ranking them.
Six things to confirm before acting on ANOVA
- Recompute once with the dominant input deliberately set low, and see whether the decision changes.
- Compare the result with one independent method or published benchmark.
- Check Per group size before anything else: it is the measured dominant input, ≈13.9 % of ANOVA per ten percent.
- Verify each input against its original source document, not memory.
- Check that every field the formula needs actually arrived from a document, not from recollection.
- Repeat the ±10 % test on the inputs that matter and see whether the decision — not just the number — still holds.
How to use the ANOVA tool properly
Using a calculator well sounds trivial, but the difference between a decision-grade figure and a throwaway one lives entirely in the routine. The routine for ANOVA takes a minute to learn and applies to every input above.
Step 1 — start from evidence. The inputs behind ANOVA are number of groups, per group size, expected η² (%). Write them down with their source next to each one; if you cannot name where a figure came from, that is the input most likely to be wrong. A figure nobody can reproduce is a figure nobody will act on twice.
Step 2 — start from the case you can defend. Enter pessimistic-but-honest figures first; the optimistic run is a ceiling, not a plan. A conclusion that holds on conservative inputs holds everywhere. Estimates improve most when the worst case is written down in ink.
Step 3 — change one input at a time. Nudge a single field by ten percent, watch the headline and its companions move, then undo it. Two or three fields will dominate — the sensitivity table above already ranks them for this engine. Note the assumption you are least sure about; that is where the real risk lives.
Step 4 — run the scenario trio. Expected, pessimistic, optimistic. The gap between pessimistic and expected is the number to plan against; the optimistic result is useful only as a ceiling. Say the answer out loud before you save it; if it sounds wrong, the model did not change, your understanding did.
Step 5 — write the one-line memo. “Design power (η²=8%) = 24.6 % on this date, from these inputs.” That line is the entire deliverable; everything else on this page is how you earned the right to write it. The input you reach for first is usually the one you measured longest ago.
From calculation to decision: where ANOVA earns its keep
In Statistics & Data Science the figure computed here is usually the first one somebody asks to see. Having it — with the formula behind it — changes your position in the conversation from participant to reference. If a result feels obviously right, that is the moment to check the inputs hardest.
The companion outputs — Cohen’s f and Degrees of freedom — exist for the second stage of that screen: the headline sizes the prize, while the companions describe the trade-off you are accepting to get it. The point of transparency is that a reader can disagree with a specific line rather than with you.
There is a personal-administration use people overlook: periodic self-checking. ANOVA quietly drifts as life changes — rates move, bodies change, costs compound — and re-running it a few times a year catches drift while correction is still cheap. When two sources disagree, it is almost never the arithmetic that differs.
ANOVA — frequently asked questions
What affects ANOVA the most?
The short answer: Per group size. The longer answer is the sensitivity table on this page, where every numeric input was probed by ten percent and the design power (η²=8%) recalculated — Per group size moved it furthest, by roughly 13.9 % (21.2 % to 27.9 %).
Which matters more: expected η² or per group size?
Per group size, by a factor of about 1.4. Concretely: ten percent on Per group size is worth ≈13.9 % of ANOVA, the same nudge on Expected η² only ≈10.3 %. That ranking is re-computed for each page from that page’s own defaults.
What is a typical ANOVA?
With the example inputs on this page the result is 24.6 %, labelled “Under-powered — add groups’ n or expect false nulls”. “Typical” varies with situation, which is why every result here ships with its verdict band and its sensitivity list: they turn one figure into a range you can locate yourself inside.
How do I calculate ANOVA by hand?
Use the stated model — f² = η²/(1−η²); F-df₁ = k−1, df₂ = N−k; power from non-centrality λ = f²N — and substitute in the order the tool prints (it starts from f² = 0.080 ÷ 0.920 = 0.087; λ = f²N = 11.7). On the example inputs the hand route lands on 24.6 %, exactly what the calculator shows. A spreadsheet reproduces it in two or three cells; doing it once by hand is what makes the automated answer trustworthy.
What happens to ANOVA if per group size changes by 10 %?
Try it in the form above and the printed substitution will follow you; the pre-computed answer for the defaults is 21.2 % to 27.9 % across the ten-percent probe, i.e. ≈13.9 % sensitivity on the design power (η²=8%).
Is the ANOVA result of 24.6 % good or bad?
Neither, on its own — the label “Under-powered — add groups’ n or expect false nulls” is the band this figure falls in for the example inputs, and your own inputs will place you elsewhere on the same ladder. Judge the number against your history and your alternatives, and remember that a ten-percent error in Per group size alone moves it by ≈13.9 %.
How many significant figures are reliable in ANOVA?
About as many as your least precise input. The tool computes at full precision and displays rounded output; report 24.6 % to a decimal place your inputs can actually support.
What affects ANOVA the most?
Sensitivity depends on where you start, but the inputs Number of groups and Per group size typically move ANOVA further than the rest because they enter the formula multiplicatively. If you can only measure two things well, measure those and hold the others at conservative defaults.
Fixed or random groups?
Fixed-effects when the levels are the ones you care about; random when sampling a population of conditions — the F denominators change.
Does unbalanced n break ANOVA?
It bends it: F remains valid but loses power and variance-equality sensitivity — Type II/III sums matter then.
Is my data private on this page?
Yes — computation happens locally in your browser, with no account, no tracking of your inputs and nothing sent to a server. The printed formula is the whole story of what happens to your figures.
What the labels on this page mean
- Cohen’s f
- 0.295 on the example inputs — a companion figure printed beside the headline of ANOVA so that no result is read in isolation.
- Degrees of freedom
- 4 vs 130 on the example inputs — a companion figure printed beside the headline of ANOVA so that no result is read in isolation.
- Non-centrality λ
- 11.7 in the default ANOVA calculation — one of the secondary figures an experienced reader checks before trusting the primary number.
- Design power (η²=8%)
- The primary output of this tool: 24.6 % for the default inputs, computed exactly from f² = η²/(1−η²); F-df₁ = k−1, df₂ = N−k; power from non-centrality λ = f²N.
- Worked example
- The calculation written out in full so that nothing on this page has to be taken on trust.
- Verdict band
- The interpretation attached to a result (“Under-powered — add groups’ n or expect false nulls” on these inputs), which turns a bare number into a position on a scale.
- Sensitivity (±10 %)
- How far the ANOVA moves when a single input is nudged ten percent either way; the ranked list on this page is computed live from this engine.
Methodology, sources and limits
Everything above runs on one stated formula, applied without hidden coefficients: f² = η²/(1−η²); F-df₁ = k−1, df₂ = N−k; power from non-centrality λ = f²N. Source of record: Cohen (1988); standard power-planning for ANOVA. Specialist software layers extra coefficients on top of the same algebra; none are hidden here, and that transparency is what makes a disagreement traceable to a specific input instead of a black box.
Nothing here replaces a qualified professional, and nothing here is approximate either — the arithmetic is exact, and the honesty is in the inputs you supply. What that buys you is reproducibility — two people, two browsers, the same arithmetic, and a printed trail from inputs to result that either of them can audit.
Turning this ANOVA into a decision
Save the number together with its inputs and today’s date; an ANOVA figure without context is unusable the moment an input changes.
The audience here is people who need one trustworthy number before their next decision. Everything you need to defend the result — formula, substitution, sensitivity, source — is printed above, so the only remaining variable is the honesty of your inputs.