Last updated: 2026-06-23

Relative Risk

Relative risk, also called the risk ratio, measures how much more or less likely an outcome is in one group compared with another. It is one of the most widely reported effect measures in epidemiology and clinical research, used whenever two groups are followed forward in time to see who develops a disease, side effect, or other event. This calculator takes the four counts of a 2×2 table and returns the relative risk together with a 95% confidence interval, so you can judge both the size of the effect and how precisely it has been estimated.

Risk and the 2×2 table

The risk of an outcome in a group is simply the proportion of that group who experience it. To compare two groups — typically an exposed (or treated) group and an unexposed (or control) group — the data are arranged in a two-by-two table:

Here $a$ is the number of exposed subjects who experienced the outcome, $b$ the exposed subjects who did not, and $c$ and $d$ the corresponding counts for the unexposed group. The risk in the exposed group is $a/(a+b)$ and the risk in the unexposed group is $c/(c+d)$.

The relative risk formula

Relative risk is the ratio of those two risks:

$RR = \frac{a / (a + b)}{c / (c + d)}$

The intuition is direct: divide the exposed group's risk by the unexposed group's risk to express one as a multiple of the other. Because it is a ratio of two proportions, it is dimensionless and easy to communicate.

Interpreting the result

• $RR \gt 1$ — the outcome is more likely in the exposed group. An $RR$ of 3 means three times the risk.
• $RR \approx 1$ — the risk is the same in both groups; exposure makes no difference.
• $RR \lt 1$ — the outcome is less likely in the exposed group. An $RR$ of 0.5 means half the risk, often interpreted as a protective effect.

A ratio alone hides the baseline rate, so relative risk should be read alongside the absolute risks. Doubling a risk from 1 in a million to 2 in a million is a relative risk of 2 but a trivial absolute change, whereas doubling a risk from 20% to 40% is the same relative risk but a substantial real-world difference. Report both the relative and the absolute risk whenever the baseline matters.

The 95% confidence interval

Because the relative risk is estimated from a sample, it carries uncertainty. The standard error is computed on the natural-log scale, where the sampling distribution of the log risk ratio is approximately normal:

$SE(\ln RR) = \sqrt{\frac{1}{a} - \frac{1}{a+b} + \frac{1}{c} - \frac{1}{c+d}}$

The 95% interval is then formed in log space and exponentiated back to the ratio scale:

$CI_{95\%} = \exp\!\left(\ln RR \pm 1.96 \cdot SE(\ln RR)\right)$

If the interval excludes 1, the difference is statistically significant at the 5% level; if it includes 1, the data are consistent with no difference. The formula requires the outcome counts $a$ and $c$ to be positive, which is why both are validated before the interval is computed.

Worked example

Suppose a cohort study follows 100 exposed and 100 unexposed subjects. Among the exposed, 30 develop the outcome and 70 do not; among the unexposed, 10 develop it and 90 do not. These four counts are the 2×2 cells a, b, c, and d shown below.

The exposed group is three times as likely to experience the outcome. Because the confidence interval (1.55 to 5.80) lies entirely above 1, the elevated risk is statistically significant.

Relative risk versus the odds ratio

Relative risk can only be estimated when the underlying risk is known for each group, which requires a cohort study or randomized trial that follows subjects forward in time. In a case-control study, subjects are selected by their outcome status, so the risks cannot be measured and the Odds Ratio Calculator is reported instead. When the outcome is rare, the odds ratio closely approximates the relative risk; when the outcome is common, the odds ratio exaggerates the effect. For clinical decisions, the relative risk is often paired with the Number Needed to Treat (NNT) Calculator, which translates a risk difference into the number of patients who must be treated for one to benefit.