Risk ratio and odds ratio from one table, with the gap between them.
Risk ratio and odds ratio, from one table
RR 0.6000, OR 0.4286. Read as a risk ratio, the odds ratio makes the reduction look 43% larger than it is.
Exposed 30.0000% · unexposed 50.0000%
Relative risk = 0.600000
The odds ratio on the same table is 0.428571 — 0.71429 times the risk ratio. They are different quantities and only coincide when the outcome is rare.
Relative risk
0.600000
p₁ ÷ p₂
Odds ratio
0.428571
always further from 1
Risk difference
-20.0000pp
p₁ − p₂
p-value
0.000000
z = -9.1287
Baseline risk
50.0000%
the unexposed arm
Exposed risk
30.0000%
the group of interest
Difference CI
-24.145 to -15.752pp
Newcombe, 95%
OR ÷ RR
0.71429
1.000 only when rare
The odds ratio is 0.42857 and the relative risk is 0.60000. With a baseline risk of 50.0000%, the outcome is not rare, and the two diverge. The odds ratio is always further from 1 than the risk ratio — so reporting one as the other always overstates the effect, never understates it. A case-control study can only give you the odds ratio; a cohort study or trial gives you both, and the risk ratio is the one a reader can act on.
How far apart they get, at a fixed relative risk of 0.6000
The flat line at 1 is where the two agree. They part company as the baseline risk rises, which is why the rare-disease approximation is a statement about the outcome’s frequency and not about the study design. At a baseline near zero the gap vanishes; at 40% it is large enough to change how a result reads.
A ratio without its baseline cannot be acted on. A relative risk of 2.0 means the risk went from 0.001% to 0.002% or from 20% to 40%, and those are not the same finding. The risk difference of -20.0000 percentage points above is the part that says how many people are affected; the number needed to treat turns it into a count.
Neither ratio is a causal claim. Both are computed identically whether the exposure was randomised or merely observed, and the arithmetic cannot tell the difference. In an observational study the ratio also carries every confounder that was not measured — Simpson’s paradox is what it looks like when one of them is large enough to reverse the sign.
What this tool shows
At a 1% baseline the risk ratio is 0.5000 and the odds ratio 0.4975. At an 80% baseline the odds ratio is exactly half the risk ratio. They are different quantities that coincide only when the outcome is rare, and the odds ratio is always further from 1 — so quoting one as the other always overstates the effect. The tool computes both and plots the divergence across the whole baseline range.
Relative risk, odds ratio and the absolute risk difference from one table
The ratio between RR and OR, which is 1.000 only when the outcome is rare
A Newcombe confidence interval for the difference and a two-proportion p-value
The divergence curve: how far apart they get as the baseline risk rises
Whether the rare-disease approximation holds on your numbers
A link through to the number needed to treat, which the ratio cannot give
RR and OR Divergence shown CI included Rare-disease check
Updated 12 September 2026 · Works in any browser, no installation
Relative risk is the ratio of two probabilities: the risk in the exposed group divided by the risk in the unexposed one. An RR of 2 means the exposed group’s risk is twice as high — which says nothing about whether that is 0.002% against 0.001% or 40% against 20%. The absolute difference is the part that says how many people are affected.
At a glance
Formula shown
RR = p₁/p₂ where p₁ is the risk in the exposed group and p₂ in the unexposed. The odds ratio is OR = [p₁/(1−p₁)] ÷ [p₂/(1−p₂)] — a ratio of odds rather than of risks. Because odds exceed risks by the factor 1/(1−p), the OR is always further from 1 than the RR, and the two converge as p₂ → 0. The risk difference is p₁ − p₂, and its reciprocal is the number needed to treat.
Scenario support
Cohort studies and randomised trials, exposure-outcome comparisons, reading an epidemiological abstract, translating a published odds ratio into something interpretable, and checking whether the rare-disease approximation applies.
Educational estimate
Planning support from the values you enter — not professional advice.
The odds ratio is always further from 1
They are reported interchangeably and they are not interchangeable. The direction of the error is always the same.
Odds exceed risks by the factor 1/(1 − p). A risk of 0.5 is odds of 1; a risk of 0.8 is odds of 4. Taking a ratio of odds therefore stretches whatever the ratio of risks was, away from 1 in whichever direction it already sat.
So an odds ratio read as a risk ratio always overstates. Never understates. With a 30% treatment rate against a 50% control rate the RR is 0.600 and the OR 0.4286 — a 40% reduction presented as a 57% one, which is 43% larger than the truth.
At an 80% baseline the effect is stark. 600 of 1000 against 800 of 1000 gives an RR of 0.750 and an OR of 0.375: exactly half. A 25% reduction reads as a 62.5% one.
They converge only as the outcome becomes rare. At a 1% baseline, 5 of 1000 against 10 of 1000 gives an RR of 0.5000 and an OR of 0.4975 — agreement to within half a percent, which is where the habit of swapping them came from and why it survives.
The rule of thumb is a statement about the OUTCOME, not the study design. Above a baseline of roughly 10% the approximation stops being safe, whatever kind of study produced the numbers. The tool plots the divergence so you can see where your own case sits rather than applying the rule from memory.
When you are stuck with the odds ratio
If the risk ratio is the more interpretable measure, why does anyone report an odds ratio? Two reasons, and both are good ones.
A case-control study cannot give you a risk ratio. Cases and controls are sampled separately, so the proportion with the outcome in the study is chosen by the investigator rather than observed. Risks are not estimable; the odds ratio is, and it equals the population odds ratio regardless of the sampling fractions. That invariance is the whole reason the design works.
Logistic regression produces odds ratios by construction. It models the log odds, so its coefficients exponentiate to odds ratios. Any adjusted analysis from a logistic model reports them, which is most adjusted analyses in medicine.
The honest move is to convert, using a baseline you state. RR = OR ÷ (1 − p₂ + p₂·OR). That requires knowing the unexposed group’s risk, which a case-control study does not supply — so it has to come from elsewhere and be named.
What is not honest is quoting an odds ratio in words meant for a risk ratio. “Twice as likely” is a risk-ratio sentence. An odds ratio of 2 means twice the odds, and on a common outcome that is considerably less than twice the risk.
A ratio without its baseline cannot be acted on
This is the objection that applies to both measures equally, and it is the more important one.
A relative risk of 2.0 describes a move from 0.001% to 0.002% and a move from 20% to 40%. Identical ratio, and the two findings have nothing in common in terms of how many people are affected.
The risk difference is the part that counts people. In the first case it is 0.001 percentage points; in the second, 20. The tool prints it beside the ratios rather than below the fold.
And its reciprocal is the number needed to treat, which is the most directly actionable summary a trial produces — see the NNT calculator for what happens when people invert the relative figure instead.
Relative figures are not dishonest, they are incomplete. They travel better between populations, which is exactly why they get published — and it is the same property that makes them uninterpretable on their own.
Report both, always. “Risk fell from 30% to 15% (RR 0.50, absolute reduction 15 percentage points)” cannot be misread. Either half alone can be.
Which group is the reference changes the number, not the finding
A ratio has a numerator and a denominator, and swapping them inverts it. That sounds trivial and causes real confusion in reading.
An RR of 0.5 and an RR of 2.0 can describe the same data. One takes the exposed group as the numerator, the other the unexposed. Both are correct; only one matches the sentence being written.
The convention is that the reference group goes in the denominator — the unexposed, the control, the standard treatment. The tool follows it and labels the fields accordingly, so the ratio it prints is exposed-over-unexposed.
A reduction below 1 and an increase above 1 are not symmetric on a linear scale. An RR of 0.5 halves the risk; its inverse, 2.0, doubles it. They are symmetric on a LOG scale, which is why forest plots and meta-analyses are drawn logarithmically — and why a confidence interval for a ratio is asymmetric around the estimate when drawn linearly.
The tool reports the interval for the DIFFERENCE rather than for the ratio because the difference interval is symmetric, easier to check, and answers the question “is zero plausible” directly — which is the same question as “is the ratio 1 plausible”.
Neither ratio is a causal claim
The arithmetic is identical whether the exposure was randomised or merely observed, and nothing in the output distinguishes the two.
In a randomised trial the comparison is fair by design. Randomisation balances everything, measured and unmeasured, in expectation — so the difference between arms is attributable to the intervention.
In an observational study the ratio carries every confounder you did not measure. People who chose the exposure differ from those who did not, in ways that also affect the outcome, and adjustment can only address the differences that were recorded.
A confounder large enough can reverse the sign. That is Simpson’s paradox: every subgroup shows one direction and the pooled table shows the other, with no arithmetic error anywhere.
Reverse causation is invisible here too. A 2×2 table has no time in it, so an exposure that followed the outcome rather than preceding it produces the same ratio as one that caused it.
So the ratio is a description of an association. That is genuinely useful and it is not causation, and the difference cannot be recovered from the numbers.
Reporting a risk comparison
Five habits, each of which closes a specific way of being misread.
Say which measure it is. “Risk ratio” and “odds ratio” are different words for different quantities, and “relative risk” should never be used for an odds ratio however close they happen to be.
Give both event rates. The ratio plus one rate is enough to reconstruct everything; the ratio alone is not.
Give the absolute difference in the same sentence. A relative figure with no absolute one beside it is the single most common way a modest result is made to sound large.
Give a confidence interval. A ratio from small counts is extremely unstable — a study with 3 events against 6 gives an RR of 0.5 with an interval that spans almost everything.
And name the design. A ratio from a randomised trial and the same ratio from an observational cohort support very different claims, and the reader cannot tell them apart from the number.
Sources and methodology
References for risk ratios, odds ratios and their relationship.
Method. Both ratios are computed from the same 2×2 table with the unexposed arm as the reference, and the confidence interval is for the DIFFERENCE rather than the ratio, using Newcombe’s method built from two Wilson intervals. The suite verifies the direction of the divergence as a property rather than by example: the odds ratio is further from 1 than the risk ratio at every baseline above zero, the two agree to within half a percent at a 1% baseline, and the odds ratio is exactly half the risk ratio at the 80%-baseline case shipped as a preset. It also confirms the relationship between the two-proportion z-test and the 2×2 chi-square statistic — z squared equals chi-square to better than 1e-8 across 400 random tables — and that a relative risk is near-invariant when both event counts are scaled while the absolute difference changes in every case. That engine is verified on every change against 75 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Number Needed to TreatNNT from the absolute risk reduction, with the relative figure beside it — two trials reporting the identical “50% reduction” have NNTs of 7 and 1,000, and the common shortcut says 2 for both.
Odds RatioOdds ratio, relative risk, risk difference and number needed to treat from one 2x2 table — because an odds ratio of 6.00 can describe a relative risk of 1.50.
Two-Proportion Z-TestCompares two rates with the pooled z-test and a Newcombe interval for the difference, and shows that z² equals the 2×2 chi-square statistic exactly — so the “z-test or chi-square” question has no content.
Sensitivity and SpecificitySensitivity, specificity, PPV, NPV, likelihood ratios and MCC from a 2×2 table, with predictive values recomputed across the prevalence range — a 99%/99% test has a PPV of 50% at 1% prevalence and 9% at 0.1%.
Simpson's ParadoxDetects when a pooled comparison contradicts every subgroup, then repairs it by standardising — which takes the real Berkeley admissions gap from +14.16 points to −4.26 and changes its sign.
Confidence IntervalIntervals for a mean or a proportion using t at every sample size and Wilson rather than the textbook Wald formula — with both methods shown, because Wald returns [0,0] at zero successes.
An educational tool, not clinical or epidemiological advice. A risk ratio computed from observational data carries every unmeasured confounder with it, and no figure in this output can distinguish an association from a causal effect.
Published the risk ratio and odds ratio from one table with the divergence between them plotted against baseline risk. At a 1% baseline they agree to within half a percent (0.5000 against 0.4975); at an 80% baseline the odds ratio is exactly half the risk ratio.
The direction of the error is always the same and is verified as a property: the odds ratio is further from 1 than the risk ratio at every baseline above zero, so quoting one as the other always overstates the effect and never understates it.
The 50%-baseline case makes the cost concrete: a risk ratio of 0.600 against an odds ratio of 0.4286, which reads as a reduction 43% larger than it is.
The confidence interval is for the DIFFERENCE rather than the ratio, using Newcombe's method, because the difference interval is symmetric and answers 'is zero plausible' directly.
Verified that a relative risk is near-invariant when both event counts are scaled while the absolute reduction changes in every case - which is why both belong in a report.
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