Statistical Power Calculator
The formula
What the statistical power tells you
The chance a real effect of the size you care about is detected as significant — the complement of the false negative that under-powered tests drown in.
Calculation: Power = Φ(|Δ|/SE − z_conf) for two proportions. Enter your inputs and every substitution is shown below the result.
Reading the result
80% is the working convention; below that you run tests mostly to fail, and a win at 30% power is more likely a stopped-early peek than a discovery.
Rules of thumb: 0.8 target; consumer-experiment effects need it most where base rates are small
Limitations and best practice
Power at the planned sample is not power at the observed one: post-hoc power from results is circular — use it only before launch.
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
Power for the metric I chose, or the decision?
Can I just extend the test until significant?
Source & methodology
Standard two-proportion power computation.
Statistical Power explained: formula, worked example and honest limits
You are here because you need a specific number, and you want to see how it was produced before you rely on it. Searchers land on a Statistical Power page for three reasons — to plan, to compare options, and to check that somebody else’s arithmetic is honest. This tool serves all three, and everything after the calculator explains the model instead of hiding it.
The discipline Statistical Power enforces is naming your assumptions. Baseline rate (%) and Target rate to detect (%) are guesses dressed as inputs; once they are on paper, the argument stops being about vibes and starts being about whether the guesses are defensible — which is how good decisions get made.
People rarely disagree about arithmetic; they disagree about assumptions that were never written down. Printing the substitution and the sensitivity ranking removes that category of disagreement for Statistical Power.
Every figure in the prose is generated by the engine rather than written beside it — the worked example resolves to statistical power = 73.5 % on the default inputs, and the sensitivity section ranks the inputs that move the result — Target rate to detect first, at ≈67.5 % 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.
Every input in Statistical Power, and what it does to the answer
One formula, published in full: Power = Φ(|Δ|/SE − z_conf) for two proportions. Anything you can read, you can reproduce in a spreadsheet, and that reproducibility is the whole point of printing it rather than hiding it behind a button.
Baseline rate (%) (example value 3) is a measured number in the unit named by the label; the formula does not convert for you. Rates are the most mis-transcribed input on any page like this; copy it from the document rather than retyping from memory.
Target rate to detect (%) (example value 4) is a measured number in the unit named by the label; the formula does not convert for you. Check the convention first: if the source says “6.5 % per year”, type 6.5 and let the formula do the rest.
Visitors per arm (example value 9,167) is a numeric input whose unit is fixed by the label; a wrong unit is a wrong answer wearing a plausible disguise. Replace the example with the figure from your own records — the tool recomputes instantly and the printed steps follow your numbers.
Confidence level (%) (example value 92) is a numeric input whose unit is fixed by the label; a wrong unit is a wrong answer wearing a plausible disguise. Enter it exactly as your document states it — quoted annual, monthly and effective rates are three different numbers.
Statistical Power with real numbers, step by step
The tool prints its own working; this is the same arithmetic on the example values, reproduced so you can follow it by hand. Starting figures: Baseline rate (%), Target rate to detect (%), Visitors per arm, Confidence level (%). Run to the end, the statistical power resolves to 73.5 % — Under the 80% convention — borderline, will miss real effects often on the bands this tool uses.
- SE = sqrt(0.0300(0.9700)/9167 + ...) = 0.00264
- z = Δ/SE − 1.65 → power Φ(z) = 73.5%
Probability of a false negative — 26.5 %. Read next to the headline, this is the figure that shows the shape of the trade-off.
Effect detectable at 80% power here — 0.66 %. Compare it against the main result of 73.5 % before drawing conclusions.
n per arm for 80% power — 10,962.2. This is the number experienced users quote when the headline alone would mislead.
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.
How far Statistical Power can reasonably move
Single-point answers invite false confidence, so the same engine was run across a band around the input that matters most here. Each row is a full recalculation of Statistical Power, not an interpolation.
| Target rate to detect setting | Statistical power |
|---|---|
| 80 % of the example value | 12.2 % |
| 90 % of the example value | 23.9 % |
| example inputs | 73.5 % |
| 110 % of the example value | 97.1 % |
| 120 % of the example value | 99.9 % |
The corridor runs from 12.2 % to 99.9 % across that band — which is the range worth planning inside rather than the single 73.5 % point estimate. Reproduce it in the form above by editing Target rate to detect alone; the substitution steps will follow your figures exactly.
The levers behind Statistical Power, ranked by effect
Nobody should have to guess which input matters. The engine was re-run once per field below — ten percent up, ten percent down, everything else frozen — and the results ranked by how far they moved the statistical power. Read it as a map of where Statistical Power is fragile — drawn for the example inputs, which is the map you want when deciding what to measure carefully.
- Nudge Target rate to detect by ±10 % and Statistical Power lands between 23.9 % and 97.1 % (sensitivity ≈67.5 %)
- Baseline rate +10 % → Statistical Power = 29.7 % · −10 % → 96.7 % (the statistical power shifts ≈59.6 %)
- Nudge Confidence level by ±10 % and Statistical Power lands between 38.1 % and 73.5 % (sensitivity ≈48.1 %)
- Visitors per arm: ten percent up gives 77.0 %, ten percent down gives 69.6 % — ≈5.4 % of the statistical power either way
Target rate to detect is the dominant lever here: ten percent either way moves the result by about 67.5 %, from 23.9 % to 97.1 %. That is where a careless figure does the most damage, so check it against a source, and treat Baseline rate as the second priority (≈59.6 % per ten percent). Everything below that can stay approximate, which is a licence to stop chasing data that would not change the decision.
Interpreting the Statistical Power verdict
The value above only becomes useful the moment you know what it is being measured against. On the example inputs the band reads “Under the 80% convention — borderline, will miss real effects often”; your own figures place you on a different rung of the same ladder. How much of that band is luck? On Target rate to detect alone ten percent either way covers 23.9 % to 97.1 %. The practical reading rules: compare against your own history before comparing against other people; treat band boundaries as zones, not walls — one point on the wrong side of a threshold rarely changes a real outcome; and when a companion output contradicts the headline, believe the pair.
Related measures answer related questions, and the distinctions matter: Statistical Power is the quantity, Statistical power is this tool’s reading of it, and the decision that follows belongs to neither of them separately.
How people get Statistical Power wrong
Power at the planned sample is not power at the observed one: post-hoc power from results is circular — use it only before launch. 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.
Reporting the headline and quietly discarding Probability of a false negative. The companions are where the shape of the trade-off survives the rounding.
One mistake is specific to this engine: assuming every input matters equally. The computation above shows Target rate to detect dominating — a careful figure there beats false precision everywhere else.
The subtlest error is answering the wrong question precisely. Confirm that statistical power is actually the quantity your decision needs — the arithmetic is flawless either way, which is what makes the mistake invisible.
Six things to confirm before acting on Statistical Power
- Check that every field the formula needs actually arrived from a document, not from recollection.
- Re-run the calculation with a ±10 % stress on the two most sensitive inputs.
- Put Target rate to detect under a magnifier; the sensitivity table above is the reason why.
- Verify each input against its original source document, not memory.
- Recompute once with the dominant input deliberately set low, and see whether the decision changes.
- Run the ±10 % stress on the top two levers and check that your conclusion survives both.
How to use the Statistical Power tool properly
The tool does the arithmetic; the routine below is what makes the arithmetic trustworthy. Five steps, a minute total — and afterwards you will know not just what the number is, but which assumption produced it.
Step 1 — one document, one pass. Open the statement, label or record that holds baseline rate (%), target rate to detect (%), visitors per arm, confidence level (%) and transcribe straight into the form. Transcription from a single source removes the mismatch errors that appear when figures are assembled from three different places. The point of transparency is that a reader can disagree with a specific line rather than with you.
Step 2 — enter the conservative case first. Resist the flattering scenario. Type the inputs you would defend in front of a sceptical reader, read the Statistical power (73.5 % for the example values), and note the verdict band. A result worth acting on survives having its two shakiest inputs halved.
Step 3 — isolate your variables. Change one field, observe, revert. The live sensitivity list does the ranking for the default inputs; verify it against your own, because dominance shifts when the starting point shifts. A figure with a date beside it becomes evidence; without one it stays an anecdote.
Step 4 — stress before you commit. A decision that survives a ten-percent squeeze on its two biggest inputs needs no luck to be sound. One that does not survive is a decision about hope, not about arithmetic. Estimates improve most when the worst case is written down in ink.
Step 5 — save the story with the figure. A number without its inputs and its date is unusable within a quarter. Record 73.5 %, the assumptions behind it, and today’s date; your future self will be auditing a stranger’s arithmetic. A figure nobody can reproduce is a figure nobody will act on twice.
Where Statistical Power matters in practice
Business & Entrepreneurship arguments end at arithmetic or they do not end. Statistical Power is the arithmetic: 73.5 % on the example inputs, reproducible by anyone who types the same numbers. The test of a good estimate is not that it feels right but that it changes what you do.
Used properly this is a five-minute filter that saves weeks. Compute Statistical Power, reject what fails, and only then bring in the expensive analysis for the survivors. When two sources disagree, it is almost never the arithmetic that differs.
Treat Statistical Power like a smoke alarm rather than a painting — its value is in being tested periodically, not in being admired once. The test of a good estimate is not that it feels right but that it changes what you do.
People also ask about Statistical Power
What affects Statistical Power the most?
The short answer: Target rate to detect. The longer answer is the sensitivity table on this page, where every numeric input was probed by ten percent and the statistical power recalculated — Target rate to detect moved it furthest, by roughly 67.5 % (23.9 % to 97.1 %).
Which matters more: baseline rate or target rate to detect?
Target rate to detect, by a factor of about 1.1. Concretely: ten percent on Target rate to detect is worth ≈67.5 % of Statistical Power, the same nudge on Baseline rate only ≈59.6 %. That ranking is re-computed for each page from that page’s own defaults.
What is a typical Statistical Power?
There is no honest universal average for Statistical Power, but there is a reference point: the defaults on this page produce 73.5 % (Under the 80% convention — borderline, will miss real effects often). Replace the inputs with your own and compare — that comparison is the useful meaning of “typical”.
How do I calculate Statistical Power by hand?
Three steps: write the formula (Power = Φ(|Δ|/SE − z_conf) for two proportions), insert your values, and simplify. The worked example section shows exactly that chain for the default inputs; replicating it once on paper teaches you more about Statistical Power than any number of calculator runs.
What happens to Statistical Power if target rate to detect changes by 10 %?
On these inputs a ±10 % change in target rate to detect takes Statistical Power to 97.1 % against the 73.5 % example — around 67.5 % of the headline. The same test on every other field is in the sensitivity section.
Is the Statistical Power result of 73.5 % good or bad?
Neither, on its own — the label “Under the 80% convention — borderline, will miss real effects often” 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 Target rate to detect alone moves it by ≈67.5 %.
Can Statistical Power be negative or empty?
Yes, and that is informative rather than broken: a negative Statistical Power usually means the inputs encode an outflow or a shortfall, and the verdict band above labels exactly that situation instead of hiding it.
How do I calculate Statistical Power manually?
Write down the inputs, then follow the printed steps in order — Power = Φ(|Δ|/SE − z_conf) for two proportions. A spreadsheet replicates it with two or three cells. The manual route is worth doing once to build intuition; after that the calculator saves time and eliminates transcription slips.
Power for the metric I chose, or the decision?
The decision: if you would ship on a 5% lift, power the test at 5%, not at the 15% you secretly hope for.
Can I just extend the test until significant?
No — optional stopping breaks the p-value’s error control; extend only per a pre-registered interim-analysis plan.
Does this Statistical Power tool store my numbers?
No. The entire calculation runs in your browser session; nothing entered above is transmitted or saved, which is the correct behaviour for sensitive Business & Entrepreneurship figures and the reason you can re-run this page as often as you like.
Glossary for Statistical Power
- Probability of a false negative
- 26.5 % in the default Statistical Power calculation — one of the secondary figures an experienced reader checks before trusting the primary number.
- Effect detectable at 80% power here
- 0.66 % on the example inputs — a companion figure printed beside the headline of Statistical Power so that no result is read in isolation.
- n per arm for 80% power
- 10,962.2 for the Statistical Power worked case; it sits next to the main result because it carries context the headline leaves out.
- Statistical power
- The primary output of this tool: 73.5 % for the default inputs, computed exactly from Power = Φ(|Δ|/SE − z_conf) for two proportions.
- Worked example
- The calculation written out in full so that nothing on this page has to be taken on trust.
- Verdict band
- A named zone on the range of plausible results; the example inputs land in “Under the 80% convention — borderline, will miss real effects often”, and your inputs may not.
- Sensitivity (±10 %)
- How far the Statistical Power moves when a single input is nudged ten percent either way; the ranked list on this page is computed live from this engine.
Where the Statistical Power model comes from
Everything above runs on one stated formula, applied without hidden coefficients: Power = Φ(|Δ|/SE − z_conf) for two proportions. Source of record: Standard two-proportion power computation. Any other implementation of this model should return the same figure for the same inputs. If one does not, the difference is in its assumptions — which this page prints and that one probably does not.
Treat the output as a precise answer to the question you typed: exact arithmetic, transparent formula, published source, no hidden rounding. The promise to the reader is narrow and testable: same inputs, same output, no silent rounding, and no figure presented without the formula that produced it.
Next steps after the calculation
Re-run the calculation with your own figures, stress the two most sensitive inputs by ±10 %, and only then act on the result.
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.