Relative risk:
formula and worked example

What relative risk tells you

Relative risk, sometimes called the risk ratio, compares two groups on a single outcome. One group has an exposure of interest (a drug, a habit, a machine setting, a marketing email) and the other does not. You take the proportion of each group that experienced the outcome during the same period, and divide one by the other.

The answer is a multiplier. A value of 1 means the two groups look alike. A value above 1 means the exposed group had the outcome more often, and a value below 1 means less often. Because it is a ratio, it travels well between studies of different sizes, which is why it is the headline figure in cohort studies, clinical trials and A/B tests alike.

What it does not carry is scale. Two studies can report the same relative risk while describing very different amounts of real-world harm or benefit, so it should always be read next to the underlying risks.

The formula and its two inputs

Each risk is a proportion between 0 and 1: the number of people (or units) who had the outcome divided by the number in that group at the start. A risk of 0.15 means 15 out of every 100.

• Risk exposed: events in the exposed group divided by the size of the exposed group.
• Risk unexposed: events in the comparison group divided by the size of that group. This is the denominator, so it must not be zero.

Worked example with 15% against 10%

Suppose 15 out of 100 people who used a product developed a skin reaction, while 10 out of 100 people who did not use it did. The figures are invented to show the arithmetic.

1. 1Convert each group to a proportion: 15 / 100 = 0.15 and 10 / 100 = 0.10.
2. 2Divide the exposed risk by the unexposed risk: 0.15 / 0.10.
3. 3The quotient is 1.5, so the exposed group had the reaction 1.5 times as often.
4. 4Cross-check with the gap: 0.15 - 0.10 = 0.05, which is 5 extra cases for every 100 people.

How to read a ratio of 1.5

Read the number in three layers. First the direction: above 1 points to more events among the exposed, below 1 points to fewer. Second the size: values close to 1, say 0.9 to 1.1, are often indistinguishable from noise unless the groups are very large. Third the base: convert back to counts per 100 or per 1,000 so a reader can see what is actually at stake.

There is also a ceiling worth knowing. Since a risk cannot exceed 1, the relative risk cannot exceed 1 divided by the unexposed risk. With a baseline of 0.10 the largest possible value is 10, and with a baseline of 0.60 it is only about 1.67. Common outcomes therefore produce modest ratios by construction, while rare outcomes can produce very large ones.

A single ratio from a sample also carries sampling uncertainty. Published work usually pairs it with a confidence interval; if that interval spans 1, the data are compatible with no difference at all.

What happens when the exposed risk moves

The ratio is linear in the numerator: raise the exposed risk by 10% and the relative risk rises by exactly 10%, from 1.5 to 1.65. It is inversely related to the baseline, so halving the unexposed risk would double the ratio even if nothing changed in the exposed group. That is why a quiet improvement in the comparison group can make an unchanged exposure look worse.

Mistakes that distort relative risk

• Using odds instead of risk. The odds ratio from a 2x2 table is larger than the risk ratio whenever the outcome is common, and reading it as a risk ratio overstates the effect.
• Comparing groups followed for different lengths of time, so one risk has had longer to accumulate than the other.
• Assuming cause from an observational comparison. Age, income or health status may differ between groups and drive both exposure and outcome.
• Quoting the ratio with no baseline, which leaves the reader unable to tell a trivial change from a major one.
• Dividing the wrong way round, which returns the reciprocal and flips the story.

Using the number in a decision

Before acting on a relative risk, write down the absolute risks, the follow-up period, how participants were assigned to groups and the number of events behind each proportion. A ratio built on four events is fragile; one built on four hundred is far steadier.

A quick scenario shows why. Two clinics each report that a complication is 1.5 times as common after a procedure. In the first the baseline is 2 in 100, so the procedure adds 1 case per 100. In the second the baseline is 20 in 100, so it adds 10 per 100. The ratio is identical, yet the second clinic faces ten times the extra burden and may rightly weigh the decision differently.