Statistical Power Calculator
The formula
What the statistical power tells you
The probability a test detects an effect that is genuinely there — a function of effect size, sample, and the significance bar.
Calculation: Power = Φ(d·√(n∕2) − zα) for a two-group one-sided test at α. Enter your inputs and every substitution is shown below the result.
Reading the result
The 80% convention exists because under-powered studies are the quiet disease of published literature: they over-report effects that survive.
Rules of thumb: Convention: 80% power at α=0.05; clinical trials 90%+
Limitations and best practice
Power calculated after the fact (post-hoc power from observed effects) is circular — it re-expresses the p-value and proves nothing.
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
Why does power matter for nulls?
What drives required n most?
Source & methodology
Cohen (1988) Statistical Power Analysis for the Behavioral Sciences.
Statistical Power: what this tool answers
You are here because you need a specific number, and you want to see how it was produced before you rely on it. Statistical Power 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 Statistical power in real time, and see every substitution printed below.
In plain terms, Statistical Power answers the question "Statistical power?". The measurement matters because it converts a situation that feels complicated — several inputs pulling in different directions — into a single number you can act on. Within Science & Research, Statistics & Data Science is the context where this figure is quoted most often, and professionals treat it as a first-pass screen rather than a verdict.
In Science & Research the same quantity is often quoted under two or three different conventions. The formula printed above fixes one convention and applies it consistently, which is what makes results from this page comparable with each other.
Every figure in the prose is generated by the engine rather than written beside it — the worked example resolves to statistical power = 88.5 % on the default inputs, and the sensitivity section ranks the inputs that move the result — Effect size first, at ≈8.3 % 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 Statistical Power formula, explained
Here is the complete model this tool runs — Power = Φ(d·√(n∕2) − zα) for a two-group one-sided test at α — with each input explained below. If a competitor’s page gives you a different answer from the same inputs, this is the text that settles which of you is right.
Effect size (Cohen's d) (example value 0) is a numeric input whose unit is fixed by the label; a wrong unit is a wrong answer wearing a plausible disguise. Keep every length field in the same unit; the formula assumes a consistency it cannot check for you.
Sample per group (example value 125) is a plain measured quantity — the unit shown in the label is the unit the formula expects. This input feeds the formula directly; a defensible figure here is worth more than false precision anywhere else.
Significance level (example value 0) is a measured number in the unit named by the label; the formula does not convert for you. Replace the example with the figure from your own records — the tool recomputes instantly and the printed steps follow your numbers.
Statistical Power on the example inputs: the complete substitution
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: Effect size (Cohen's d), Sample per group, Significance level. The endpoint of that chain is 88.5 % of statistical power, sitting in the “Adequately powered — the study will find this effect 8 times in 10 if it is real” band.
- z(α/2) = 1.96; effect signal = 0.40 × √(125/2) = 3.16
- Power = Φ(3.16 − 1.96) = 88.5%
Type II error risk (β) — 11.5 %. If your inputs change, this is usually the first companion figure to move.
n per group for 80% power — 99. Read next to the headline, this is the figure that shows the shape of the trade-off.
n per group for 90% power — 132. This is the number experienced users quote when the headline alone would mislead.
Non-centrality (d√(n/2)) — 3.16. 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.
Scenario table: Statistical Power at ±10 % and ±20 % of the key input
The honest version of any estimate is a corridor, and this one is cheap to produce: Statistical Power was recomputed at 5 points along the dominant input, with every other field left untouched.
| Effect size setting | Statistical power |
|---|---|
| 80 % of the example value | 71.6 % |
| 90 % of the example value | 81.2 % |
| example inputs | 88.5 % |
| 110 % of the example value | 93.6 % |
| 120 % of the example value | 96.7 % |
Across 5 settings of Effect size, the result spans 71.6 % to 96.7 %. A decision that only works at one row of this table is a decision waiting for an input to move. Reproduce it in the form above by editing Effect size alone; the substitution steps will follow your figures exactly.
The levers behind Statistical Power, ranked by effect
This list was computed for this specific engine just now: each numeric input was nudged ±10 % while every other field held its example value, and the statistical power was recalculated from scratch. For Statistical Power the ranking is measured rather than guessed, and it belongs to these defaults: shift the starting point and the order can shift with it.
- Effect size +10 % → Statistical Power = 93.6 % · −10 % → 81.2 % (the statistical power shifts ≈8.3 %)
- Sample per group +10 % → Statistical Power = 91.3 % · −10 % → 85.1 % (the statistical power shifts ≈3.9 %)
- Nudge Significance level by ±10 % and Statistical Power lands between 87.6 % and 89.3 % (sensitivity ≈1.0 %)
Effect size is the dominant lever here: ten percent either way moves the result by about 8.3 %, from 81.2 % to 93.6 %. That is where a careless figure does the most damage, so check it against a source, and treat Sample per group as the second priority (≈3.9 % per ten percent). The remaining fields can sit at safe defaults without materially changing the answer — knowing which is which is what separates a careful calculation from an anxious one.
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 “Adequately powered — the study will find this effect 8 times in 10 if it is real”; your own figures place you on a different rung of the same ladder. Before treating the band as fact, note its width: ten percent either way on Effect size already spans 81.2 % to 93.6 %. 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.
The comparison worth making is not with somebody else’s Statistical Power but with the version of your own situation measured last quarter — same formula, same conventions, different inputs.
The errors that quietly break a Statistical Power figure
Power calculated after the fact (post-hoc power from observed effects) is circular — it re-expresses the p-value and proves nothing. 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 Sample per group while Effect size (Cohen's d) is still a guess. The sensitivity ranking above shows exactly where the accuracy budget should go.
One mistake is specific to this engine: assuming every input matters equally. The computation above shows Effect size 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
- Verify each input against its original source document, not memory.
- Confirm the example values are plausible for your own situation before replacing them.
- Re-verify Effect size against a source document — ten percent of error there moves the headline by ≈8.3 %.
- Re-run the calculation with a ±10 % stress on the two most sensitive inputs.
- Check that every field the formula needs actually arrived from a document, not from recollection.
- Run the ±10 % stress on the top two levers and check that your conclusion survives both.
Five steps from raw inputs to a figure you can defend
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 — collect, then type. Everything this tool asks for is listed above (effect size (cohen's d), sample per group, significance level). Pull each figure from something you can point at — paperwork, a dashboard, a label — before touching a field. Guesswork at the input stage cannot be repaired at the interpretation stage. Re-running a number costs seconds; retracting a decision costs considerably more.
Step 2 — baseline before exploration. Get one defensible Statistical power on screen before you start nudging numbers. Without a baseline, every variation looks interesting and none of them means anything. Comparisons are cheap: run the identical calculation for the alternative and look at the gap.
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. Write the figure down before moving on; memory flatters whichever number you saw last.
Step 4 — ask what would have to be true. Take the result you want and work backwards through the inputs until you find the assumption that produces it; then decide honestly whether that assumption is defensible. The fastest sanity test is to guess the answer first and see how far the calculation disagrees.
Step 5 — write the one-line memo. “Statistical power = 88.5 % 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. If the verdict changes with a ten-percent nudge, report the nudge alongside the number.
Where Statistical Power matters in practice
Science & Research arguments end at arithmetic or they do not end. Statistical Power is the arithmetic: 88.5 % on the example inputs, reproducible by anyone who types the same numbers. A result worth acting on survives having its two shakiest inputs halved.
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. One decimal place more than your data supports is one decimal place of fiction.
There is a personal-administration use people overlook: periodic self-checking. Statistical Power 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. The input you reach for first is usually the one you measured longest ago.
Statistical Power — frequently asked questions
What affects Statistical Power the most?
The short answer: Effect size. The longer answer is the sensitivity table on this page, where every numeric input was probed by ten percent and the statistical power recalculated — Effect size moved it furthest, by roughly 8.3 % (81.2 % to 93.6 %).
Which matters more: sample per group or effect size?
By the numbers, Effect size: a ten-percent change in it moves the result ≈8.3 %, versus ≈3.9 % for Sample per group. Spend your measuring effort in that order — precision on the weaker lever buys almost nothing.
What is a typical Statistical Power?
On the worked example above Statistical Power comes out at 88.5 %. Treat that as one point on a distribution rather than a norm: the bands, the companion outputs and the sensitivity ranking tell you how wide the plausible range really is.
How do I calculate Statistical Power by hand?
Use the stated model — Power = Φ(d·√(n∕2) − zα) for a two-group one-sided test at α — and substitute in the order the tool prints (it starts from z(α/2) = 1.96; effect signal = 0.40 × √(125/2) = 3.16). On the example inputs the hand route lands on 88.5 %, 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 Statistical Power if effect 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 81.2 % to 93.6 % across the ten-percent probe, i.e. ≈8.3 % sensitivity on the statistical power.
Is the Statistical Power result of 88.5 % good or bad?
Neither, on its own — the label “Adequately powered — the study will find this effect 8 times in 10 if it is real” 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 Effect size alone moves it by ≈8.3 %.
Is Statistical Power accurate?
The arithmetic is exact: Power = Φ(d·√(n∕2) − zα) for a two-group one-sided test at α is evaluated with full floating-point precision and every substitution is printed in the worked example. What limits accuracy is input quality and the model’s assumptions, so treat the 88.5 % result as the correct answer to the question you typed — and check the assumptions section before acting on it.
How often should I check Statistical Power?
Whenever an input changes by more than a few percent, and otherwise on the rhythm of your decisions — before a purchase, a review, a renewal. Statistical Power is a snapshot, not a subscription; stale snapshots are the most common silent error in Science & Research planning.
Why does power matter for nulls?
A non-significant result from a 30%-powered study is a shrug, not a finding — absence of evidence with weak evidence of absence.
What drives required n most?
Small effects need quadratically more people: halving d quadruples n. Pilot-based honest d estimates save entire budgets.
Can this tool see the numbers I type?
It cannot. There is no submission step: the calculation happens in your browser and the result disappears when you close the page. Treat the fields above as a scratchpad rather than a form.
Key terms on this page
- Type II error risk (β)
- 11.5 % in the default Statistical Power calculation — one of the secondary figures an experienced reader checks before trusting the primary number.
- n per group for 80% power
- 99 on the example inputs — a companion figure printed beside the headline of Statistical Power so that no result is read in isolation.
- n per group for 90% power
- 132 on the example inputs — a companion figure printed beside the headline of Statistical Power so that no result is read in isolation.
- Statistical power
- The primary output of this tool: 88.5 % for the default inputs, computed exactly from Power = Φ(d·√(n∕2) − zα) for a two-group one-sided test at α.
- Worked example
- Every substitution in order, from the example inputs to 88.5 % — the audit trail for the headline figure.
- Verdict band
- The plain-language label — here “Adequately powered — the study will find this effect 8 times in 10 if it is real” — mapping the computed figure onto experience-based ranges for Science & Research.
- 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.
The provenance of this Statistical Power calculation
Everything above runs on one stated formula, applied without hidden coefficients: Power = Φ(d·√(n∕2) − zα) for a two-group one-sided test at α. Source of record: Cohen (1988) Statistical Power Analysis for the Behavioral Sciences. 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.
This page is an estimation aid, not professional advice. The design rule is simple: two people entering identical figures must get identical answers. The design constraint was set by usefulness rather than by comfort: an estimate you can interrogate beats an answer you have to trust.
Turning this Statistical Power into a decision
Save the number together with its inputs and today’s date; a Statistical Power figure without context is unusable the moment an input changes.
Whoever you are (people who need one trustworthy number before their next decision), the next step is the same: replace the example inputs with your own, note the date, and re-run whenever a figure moves. In Science & Research, a dated figure is a trend, and an undated one is a rumour.