Effect Size (r) Calculator
The formula
What the correlation effect size tells you
What a correlation coefficient means as an effect: r² of shared variance, its t-test, and the Fisher-z confidence interval around it.
Calculation: r² shared variance; t = r√(n−2)∕√(1−r²); Fisher z for intervals. Enter your inputs and every substitution is shown below the result.
Reading the result
r = 0.35 is a serious-sounding correlation that still leaves 88% of variance unexplained — the r² reflex is the honesty habit.
Rules of thumb: 0.1 / 0.3 / 0.5 = small / medium / large correlation conventions
Limitations and best practice
Correlation is scale-fragile: restriction of range, outliers and nonlinearity each bend r without any change in the underlying relationship.
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
Is r=0.5 strong?
Why Fisher z instead of a direct CI?
Source & methodology
Cohen (1988); Fisher (1921) z-transformation.
How Effect Size (r) works — and what your result really means
This page was written for one job: turning your own inputs into a defensible figure, with the working shown. Searchers land on an Effect Size (r) 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.
Like most useful measures, Effect Size (r) is a summary, and summaries hide as much as they reveal. The companion outputs and the input table are here to reopen what the headline compressed.
A calculation like this one is worth roughly as much as a careful sentence: it forces several vague considerations into one comparable quantity. Once Effect Size (r) 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 shared variance (r²) = 12.2 %, recomputed every time a field changes, and the sensitivity section ranks the inputs that move the result — Correlation coefficient r first, at ≈21.0 % 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 Effect Size (r) formula, explained
One formula, published in full: r² shared variance; t = r√(n−2)∕√(1−r²); Fisher z for intervals. 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.
Correlation coefficient r (example value 0) is a quantity rather than a category — precision here is cheap and worth having. The example value is there to demonstrate the mechanics, not to describe your situation — overwrite it.
Sample size (example value 170) is a quantity rather than a category — precision here is cheap and worth having. Keep every length field in the same unit; the formula assumes a consistency it cannot check for you.
A fully worked Effect Size (r) example
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: Correlation coefficient r, Sample size. The computed shared variance (r²) comes out at 12.2 %, which the interpretation band labels “r = 0.35 explains 12.2% of variance — medium on Cohen’s correlation ladder; t(168) ≈ 4.84 clears the 5% bar”.
- r² = 0.122 → 12.2% of variance in common
- Fisher z = 0.365; CI back-transformed = [0.21, 0.48]
t-statistic — 4.84. This is the number experienced users quote when the headline alone would mislead.
95% CI via Fisher z — 0.35. Read next to the headline, this is the figure that shows the shape of the trade-off.
Effect ladder position — 0.35. This is the number experienced users quote when the headline alone would mislead.
n for r to reach p<.05 (two-tailed) — 30. This is the number experienced users quote when the headline alone would mislead.
Re-running with your own figures is instant; re-running by hand once is the best way to know the tool is telling you the truth.
How far Effect Size (r) can reasonably move
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?
| Correlation coefficient r setting | Shared variance (r²) |
|---|---|
| 80 % of the example value | 7.8 % |
| 90 % of the example value | 9.9 % |
| example inputs | 12.2 % |
| 110 % of the example value | 14.8 % |
| 120 % of the example value | 17.6 % |
Across 5 settings of Correlation coefficient r, the result spans 7.8 % to 17.6 %. 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 Correlation coefficient r alone; the substitution steps will follow your figures exactly.
Sensitivity: which inputs change Effect Size (r) the most
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 shared variance (r²) was recalculated from scratch. Read it as a map of where Effect Size (r) is fragile — drawn for the example inputs, which is the map you want when deciding what to measure carefully.
- Nudge Correlation coefficient r by ±10 % and Effect Size (r) lands between 9.9 % and 14.8 % (sensitivity ≈21.0 %)
Correlation coefficient r is the dominant lever here: ten percent either way moves the result by about 21.0 %, from 9.9 % to 14.8 %. Measure that input from a document rather than from memory. Everything below that can stay approximate, which is a licence to stop chasing data that would not change the decision.
Is your Effect Size (r) good — reading the result bands
Nobody acts on a bare figure — they act on a figure with a comparison attached. The default case lands in the “r = 0.35 explains 12.2% of variance — medium on Cohen’s correlation ladder; t(168) ≈ 4.84 clears the 5% bar” band, and bands are deliberately wide because real situations overlap. The sensitivity list shows how far a modest input error moves you along that ladder: for Correlation coefficient r, ±10 % covers 9.9 % to 14.8 %. 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.
Placing this figure in context: calculators of this kind differ less in arithmetic than in what they choose to ignore. This one keeps Correlation coefficient r explicit so that the omissions are visible rather than hidden in a preset.
The errors that quietly break an Effect Size (r) figure
Correlation is scale-fragile: restriction of range, outliers and nonlinearity each bend r without any change in the underlying relationship. 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 size while Correlation coefficient r 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: Correlation coefficient r first, everything else second.
The subtlest error is answering the wrong question precisely. Confirm that shared variance (r²) 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 Effect Size (r)
- Record the date and the inputs alongside the number so future-you knows what was assumed.
- Compare the result with one independent method or published benchmark.
- Put Correlation coefficient r under a magnifier; the sensitivity table above is the reason why.
- Verify each input against its original source document, not memory.
- Re-run the calculation with a ±10 % stress on the two most sensitive inputs.
- Repeat the ±10 % test on the inputs that matter and see whether the decision — not just the number — still holds.
Step by step: getting a trustworthy Effect Size (r) out of this tool
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 Effect Size (r) takes a minute to learn and applies to every input above.
Step 1 — start from evidence. The inputs behind Effect Size (r) are correlation coefficient r, sample size. 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. Suspiciously tidy answers deserve the same scrutiny as untidy ones.
Step 2 — compute the version you would not be embarrassed by. If a stranger audited your inputs, which set would you hand over? Enter that set, read the result, and treat everything else as upside. The fastest sanity test is to guess the answer first and see how far the calculation disagrees.
Step 3 — play with the inputs on purpose. Ten percent up, ten percent down, one field at a time. Within a few minutes you will know this calculation better than most people who quote its results. If the answer changes your behaviour, it deserves a second run with different assumptions.
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 — save the story with the figure. A number without its inputs and its date is unusable within a quarter. Record 12.2 %, the assumptions behind it, and today’s date; your future self will be auditing a stranger’s arithmetic. The test of a good estimate is not that it feels right but that it changes what you do.
From calculation to decision: where Effect Size (r) earns its keep
Numbers earn their keep in conversations, and Effect Size (r) is no exception. In Science & Research discussions around Statistics & Data Science, this figure decides whether an option stays on the table: it converts an opinionated debate into an arithmetic one, and arithmetic is the only referee both sides accept. Half of a good calculation is knowing which input you would hate to be wrong about.
The companion outputs — t-statistic and 95% CI via Fisher z — 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. Numbers are cheap to produce and expensive to retract; the extra minute is the cheapest insurance available.
Treat Effect Size (r) like a smoke alarm rather than a painting — its value is in being tested periodically, not in being admired once. A calculation you cannot explain to somebody else is a calculation you cannot defend.
The questions that come up most about Effect Size (r)
What affects Effect Size (r) the most?
The short answer: Correlation coefficient r. The longer answer is the sensitivity table on this page, where every numeric input was probed by ten percent and the shared variance (r²) recalculated — Correlation coefficient r moved it furthest, by roughly 21.0 % (9.9 % to 14.8 %).
What is a typical Effect Size (r)?
On the worked example above Effect Size (r) comes out at 12.2 %. 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 Effect Size (r) by hand?
Three steps: write the formula (r² shared variance; t = r√(n−2)∕√(1−r²); Fisher z for intervals), 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 Effect Size (r) than any number of calculator runs.
What happens to Effect Size (r) if correlation coefficient r changes by 10 %?
On these inputs a ±10 % change in correlation coefficient r takes Effect Size (r) to 14.8 % against the 12.2 % example — around 21.0 % of the headline. The same test on every other field is in the sensitivity section.
Is the Effect Size (r) result of 12.2 % good or bad?
Neither, on its own — the label “r = 0.35 explains 12.2% of variance — medium on Cohen’s correlation ladder; t(168) ≈ 4.84 clears the 5% bar” 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 Correlation coefficient r alone moves it by ≈21.0 %.
What affects Effect Size (r) the most?
Sensitivity depends on where you start, but the inputs Correlation coefficient r and Sample size typically move Effect Size (r) 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.
How often should I check Effect Size (r)?
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. Effect Size (r) is a snapshot, not a subscription; stale snapshots are the most common silent error in Science & Research planning.
Is r=0.5 strong?
In personality and epidemiology it is large; in psychophysics with tight measurement it is weak. Context decides.
Why Fisher z instead of a direct CI?
r’s sampling distribution is skewed near ±1; z-transforms to near-normal, builds the interval, then back-transforms.
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.
Glossary for Effect Size (r)
- t-statistic
- 4.84 on the example inputs — a companion figure printed beside the headline of Effect Size (r) so that no result is read in isolation.
- 95% CI via Fisher z
- 0.35 in the default Effect Size (r) calculation — one of the secondary figures an experienced reader checks before trusting the primary number.
- Effect ladder position
- 0.35 for the Effect Size (r) worked case; it sits next to the main result because it carries context the headline leaves out.
- Shared variance (r²)
- The primary output of this tool: 12.2 % for the default inputs, computed exactly from r² shared variance; t = r√(n−2)∕√(1−r²); Fisher z for intervals.
- Worked example
- Every substitution in order, from the example inputs to 12.2 % — the audit trail for the headline figure.
- Verdict band
- A named zone on the range of plausible results; the example inputs land in “r = 0.35 explains 12.2% of variance — medium on Cohen’s correlation ladder; t(168) ≈ 4.84 clears the 5% bar”, and your inputs may not.
- Sensitivity (±10 %)
- How far the Effect Size (r) moves when a single input is nudged ten percent either way; the ranked list on this page is computed live from this engine.
Method, sources and what this tool cannot tell you
Everything above runs on one stated formula, applied without hidden coefficients: r² shared variance; t = r√(n−2)∕√(1−r²); Fisher z for intervals. Source of record: Cohen (1988); Fisher (1921) z-transformation. Where a professional version of this calculation differs, it differs in inputs and jurisdiction-specific coefficients, not in the algebra — which is why re-checking your inputs beats hunting for a second calculator.
Treat the output as a precise answer to the question you typed: exact arithmetic, transparent formula, published source, no hidden rounding. The design constraint was set by usefulness rather than by comfort: an estimate you can interrogate beats an answer you have to trust.
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.