Odds ratio and relative risk together — because they are not the same number.
Odds ratio and risk ratio, never one without the other
60% against 40%. The odds ratio is 50% larger than the risk ratio.
Risk 60.000% in the exposed group against 40.000%
RR 1.50000 · OR 2.25000
The relative risk says the event is 1.5000× as likely. The odds ratio says the odds are 2.2500× as high. These are different statements about the same table.
Relative risk
1.500000
Odds ratio
2.250000
Risk difference
20.0000 pp
absolute, not relative
Number needed
5.00000
to see one extra event
The odds ratio overstates the risk ratio by a factor of 1.5000 here. The two only agree when the outcome is rare — the “rare disease assumption” — and this outcome is not: 60.0% and 40.0%. Odds and probability diverge as probability rises: at 50% the odds are 1, at 90% they are 9. So “2.250times the odds” reported as “2.250 times as likely” overstates the finding by 50%. Case-control studies can only produce an odds ratio, so this is not a mistake the researchers made — it is one made in the reporting.
In absolute terms the risk changes by 20.0000 percentage points. You would need 5.00000 people exposed to see one additional event. A relative measure with no baseline is the other half of this page’s problem: “doubles your risk” describes a move from 40% to 80% and a move from 0.001% to 0.002% equally well, and those are not the same news. Whenever a relative figure is quoted, the absolute one is the missing context.
What this tool shows
An odds ratio of 6.00 can describe a relative risk of 1.50. They agree when the outcome is rare and diverge sharply when it is not — so “six times the odds” reported as “six times as likely” overstates the finding fourfold. This shows both, plus the absolute difference.
Odds ratio and relative risk from one 2×2 table
How much the odds ratio overstates the risk ratio
Absolute risk difference in percentage points
Number needed to treat or harm
A warning when the outcome is too common for the approximation
Why a relative figure needs its baseline
Both measures shown Overstatement quantified Absolute risk too NNT included
“Doubles your risk” describes 40%→80% and 0.001%→0.002% equally well.
Updated 8 September 2026 · Works in any browser, no installation
An odds ratio compares odds; a relative risk compares probabilities. Odds and probability are the same thing below about 10% and diverge above it — at 50% the odds are 1, at 90% they are 9. So the two ratios agree for rare outcomes and come apart for common ones, and only one of them is what most readers think they are being told.
At a glance
Formula shown
From a 2×2 table with a = exposed with the event, b = exposed without, c = unexposed with, d = unexposed without: relative risk = [a/(a+b)] ÷ [c/(c+d)]; odds ratio = (a/b) ÷ (c/d). Risk difference = a/(a+b) − c/(c+d), and the number needed to treat is its reciprocal.
Scenario support
Reading a medical study; comparing exposure and outcome in a 2×2 table; case-control studies, which can only produce an odds ratio; interpreting “X raises your risk by Y%” claims; logistic regression output, whose coefficients are log odds ratios.
Educational estimate
Planning support from the values you enter — not professional advice.
Odds are not probability, and the gap grows
The two measures are computed from the same four numbers and answer different questions. The confusion is not that people cannot tell them apart — it is that odds and probability feel like synonyms in ordinary English and are not.
Probability is events divided by everyone. Odds are events divided by non-events. A 50% probability is odds of 1. A 90% probability is odds of 9. As probability rises the odds accelerate away from it, and above about 10% the divergence is material.
So the ratio of odds and the ratio of probabilities separate too. Three tables from the tool’s presets:
2% against 1%: relative risk 2.00, odds ratio 2.02. Practically identical — this is the rare disease assumption doing its work.
60% against 40%: relative risk 1.50, odds ratio 2.25. The odds ratio is 50% larger.
90% against 60%: relative risk 1.50, odds ratio 6.00. Four times the risk ratio, from a table describing exactly the same 1.5× increase in probability.
Neither number is wrong. The odds ratio genuinely is 6.00 and the study is entitled to report it. The error is in translation: “six times the odds” becoming “six times as likely” in a summary, a headline or a conversation. The tool prints the overstatement factor so it is a number rather than a caveat.
Why studies report the odds ratio anyway
Given all that, it is fair to ask why researchers do not simply report the relative risk. There are three real reasons, and only one of them is a matter of preference.
Case-control studies cannot produce a relative risk. They start from the outcome — recruit people who have the disease and people who do not — and look backwards at exposure. The proportion with the disease is fixed by the researchers, so it carries no information about incidence, and a risk ratio computed from it would be meaningless. The odds ratio, remarkably, is unaffected by this and estimates the population odds ratio correctly. For rare diseases that makes it a good estimate of the relative risk, which is precisely why case-control studies work at all.
Logistic regression produces odds ratios by construction. Its coefficients are log odds ratios, so exponentiating gives an odds ratio and nothing else. Since logistic regression is the standard method for binary outcomes with covariates, most adjusted results in the literature are odds ratios whether or not that was the intent.
The odds ratio is symmetric in a way the risk ratio is not. The odds ratio for the event and the odds ratio for the non-event are exact reciprocals. Risk ratios are not, so the two directions give inconsistent answers — an awkwardness that matters in meta-analysis.
What follows practically: if the study is a cohort or a trial, ask for the relative risk, because it exists. If it is case-control or logistic regression, the odds ratio is what there is — and it should be read as an odds ratio, with the outcome’s baseline stated so a reader can judge how far it sits from the risk ratio.
A relative figure with no baseline says nothing
This is the second failure, and it is independent of the first: even a correctly computed relative risk is uninterpretable on its own.
“Doubles your risk” describes a move from 40% to 80% and a move from 0.001% to 0.002% equally well. Both are a relative risk of 2.00. One is a life-changing finding and the other is nothing, and the relative figure cannot distinguish them.
The absolute risk difference is the number that carries the weight. 40 percentage points in the first case; 0.001 in the second. It is the quantity a decision actually depends on, and it is routinely omitted — because a relative figure sounds larger, which is convenient for a press release.
The number needed to treat turns it into people. It is the reciprocal of the risk difference: how many must be treated for one additional person to benefit. A treatment cutting risk from 2% to 1% has an NNT of 100 — 99 people treated for no benefit, one helped. That is a defensible trade for a cheap safe drug and a poor one for an expensive risky procedure, and the relative risk reduction of 50% conveys none of it.
The tool reports all four together — relative risk, odds ratio, absolute difference and NNT — because any one of them alone permits a misleading summary, and no two of them together do.
Reading a 2×2 result
Four checks before acting on one, beyond the arithmetic.
Ask what the baseline is. If the report gives only a ratio, the finding is incomplete. This is the fastest way to tell a careful summary from a promotional one.
Ask which measure it is. “Odds ratio”, “risk ratio”, “hazard ratio” and “rate ratio” are four different quantities and are used interchangeably in summaries. A hazard ratio in particular is a ratio of instantaneous rates over time and is not a probability ratio at all.
Look at the confidence interval, not the point estimate. An odds ratio of 2.5 with an interval of 0.9 to 7.1 is compatible with no effect. The interval on a ratio is asymmetric — it is symmetric on the log scale — so a point estimate sitting near the middle of a quoted range is already a hint that something was reported carelessly. The confidence interval calculator covers the general case.
Remember it is observational unless it was randomised. A 2×2 table shows association. Confounding, reverse causation and selection can all produce one, and the correlation page sets out the mechanisms. An odds ratio adjusted for covariates has controlled for the confounders someone thought of, which is not the same as controlling for confounding.
Watch for zero cells. An odds ratio is infinite when any cell is zero, and the usual fix — adding 0.5 to every cell — is a convention rather than a derivation. With small counts the estimate is unstable regardless, and an exact method is a better route than a corrected ratio.
The interval on a ratio is not symmetric
A ratio lives on a multiplicative scale, and that changes what its uncertainty looks like in a way the usual "estimate plus or minus" phrasing hides.
The confidence interval is computed on the log scale and exponentiated back. Log odds are roughly normal; odds ratios are not. So a symmetric interval on the log becomes an asymmetric one on the ratio — an odds ratio of 2.0 might run from 1.2 to 3.3, with the point estimate nearer the lower end than the upper.
That asymmetry is a quick authenticity check. If a reported ratio sits exactly in the middle of its interval, the interval was probably built on the wrong scale. The point estimate should be the geometric centre: the ratio of the upper bound to the estimate should match the ratio of the estimate to the lower bound.
The null value is 1, not 0. An interval that contains 1 is compatible with no effect, which is the ratio equivalent of a difference interval containing zero. An odds ratio of 2.5 with an interval of 0.9 to 7.1 is not evidence of anything, however large the point estimate looks.
Small counts widen it dramatically, and zero cells break it. The standard error depends on the reciprocals of all four cell counts, so a single cell of 2 or 3 dominates it. With a zero cell the odds ratio is infinite and the usual repair — adding 0.5 to every cell — is a convention rather than a derivation. At those counts an exact method is the honest route, and the beta posterior handles the zero case without a fudge.
Sources and methodology
References for the measures and the reporting problem.
Method. Both measures are computed from the raw 2×2 counts rather than from rounded percentages, so the overstatement factor is exact rather than an artefact of the display. The tool reports the ratio of the odds ratio to the relative risk explicitly and warns above a 10% divergence, which is roughly where the rare-disease approximation stops holding. The absolute risk difference and the number needed to treat are shown alongside unconditionally — not as an option — because a relative measure without a baseline permits a misleading summary and the two together do not. The suite asserts the three preset tables to twelve decimals (RR 2.00 with OR 2.0204 at 2% against 1%; RR 1.50 with OR 2.25 at 60% against 40%; RR 1.50 with OR 6.00 at 90% against 60%), that the odds ratio always exceeds the relative risk when the risk difference is positive, and that the odds ratio for the event and for the non-event are exact reciprocals while risk ratios are not. That engine is verified on every change against 49 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
ProbabilityTwo events, repeated trials and Bayes, with the three usual errors handled — the dropped overlap in P(A or B), n×p instead of the complement, and the base rate that makes a 99% test 17% right.
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.
Chi-SquareGoodness of fit and tests of independence with every expected count and per-cell contribution shown — because the validity condition is about expected counts, not observed ones, and most calculators hide them.
p-valueA p-value from a t or z statistic, one- or two-tailed — with a panel that holds an effect fixed and grows the sample, so you can watch significance appear from nothing but n.
Beta DistributionTakes raw successes and failures and runs the conjugate update, so 5 out of 5 returns 6/7 rather than the 100% a plain proportion claims — with a credible interval that stays honest at zero.
Effect SizeCohen d, Hedges g and the overlap between groups, with a sample-size control that moves the p-value while leaving the effect size fixed — the same d gives t = 1.29 at n=30 and 23.57 at n=10,000.
An educational tool, not medical advice. A 2×2 table shows association rather than causation, and an odds ratio approximates a relative risk only when the outcome is rare — above roughly 10% incidence the two diverge materially.
Published an odds ratio calculator that reports the relative risk beside it unconditionally, because the two agree only when the outcome is rare. At 2% against 1% they differ by 1%; at 90% against 60% the odds ratio is 6.00 and the relative risk is 1.50.
Prints the overstatement factor explicitly and warns above a 10% divergence, which is roughly where the rare-disease approximation stops holding.
Shows the absolute risk difference and the number needed to treat alongside, since a relative figure with no baseline describes a move from 40% to 80% and one from 0.001% to 0.002% equally well.
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