The vaccine efficacy calculator applies one published formula to four numbers you enter. It computes the attack rate in each arm of a trial, their ratio, the efficacy figure that ratio implies, and a confidence interval around it. It does nothing else. It holds no data about any vaccine, names no product, and attaches no judgement to any number it produces.
Arb Digital publishes it as an arithmetic and definitions tool alongside its other free epidemiology calculators. The reason to have it is narrow and specific: efficacy figures are quoted constantly, and very few of the people quoting them can say what the denominator was, how wide the interval was, or what outcome the figure was measured against. Those three things determine what the number means.
What Vaccine Efficacy Is
Vaccine efficacy is defined as the proportionate reduction in the rate of an outcome in a vaccinated group compared with an unvaccinated group under trial conditions. The World Health Organization sets out the concept in its feature story on vaccine efficacy, effectiveness and protection, describing efficacy as measured in controlled clinical trials and expressing it as the reduction in risk of the outcome relative to the comparison group.
It is a ratio of two rates observed in one study. It is not a probability that any individual will or will not get ill, it is not a property of a vaccine independent of the trial that measured it, and it is not comparable across trials that used different case definitions, populations or follow-up periods.
How to Use It
- Find the four counts in the trial report: cases and total participants in each arm. All four must come from the same analysis population and the same follow-up period.
- Check the case definition. A trial counting laboratory-confirmed symptomatic disease and one counting any infection are measuring different things.
- Enter the counts and choose a confidence level.
- Read the interval, not just the point estimate. The width tells you how much the counts constrain the answer.
- Compare with the figure the report published. A difference usually means the trial used person-time or an adjusted model rather than the simple attack-rate formula.
The Formula
The attack rate in a group is the number of cases divided by the number of participants. Efficacy is one minus the ratio of those two attack rates:
VE = 1 − (attack rate in the vaccinated arm ÷ attack rate in the unvaccinated arm) = 1 − RR
The interval is computed on the log scale, which is the standard approach because the sampling distribution of a ratio is skewed. The standard error of the log risk ratio is the square root of (1/a − 1/n₁ + 1/c − 1/n₂), where a and c are the case counts and n₁ and n₂ the group sizes. Multiply by the normal critical value for your confidence level, add and subtract from the log ratio, exponentiate, and subtract each end from one. Note that the ends swap: the lower bound of the risk ratio gives the upper bound of the efficacy.
Work the placeholder values through. With 20 cases among 10,000 vaccinated participants the attack rate is 0.002; with 80 cases among 10,000 controls it is 0.008. The risk ratio is 0.25 and the efficacy is 1 − 0.25 = 0.75, or 75%. The standard error of the log ratio is the square root of (1/20 − 1/10000 + 1/80 − 1/10000) = 0.2496, so the 95% interval on the risk ratio runs from 0.1533 to 0.4077, and the efficacy interval runs from 59.22% to 84.67%.
What a Wide Interval Does and Does Not Mean
The width of the interval is driven almost entirely by the number of cases, not the number of participants. A trial with 200,000 participants and six cases produces a very wide interval; a trial with 8,000 participants and 300 cases produces a narrow one. This is why trials are powered on expected case counts and why they run until enough cases accrue rather than until enough people are enrolled.
A wide interval means the data are consistent with a broad range of underlying values. That is a statement about how much the study constrains the estimate. It is not evidence that a vaccine works, and it is not evidence that it does not. Reading a wide interval as though it supported either conclusion is the single most common misuse of this arithmetic, and it is worth resisting in both directions. The same care applies to any ratio measure — our relative risk calculator and absolute risk reduction calculator make the same point from the general epidemiological side.
Efficacy Is Not Effectiveness
These two words name different quantities and the distinction is not pedantry. Efficacy is measured under trial conditions: a defined protocol, a selected population, monitored administration, active case-finding. Effectiveness is what is observed when a vaccine is used in an ordinary population, where age structure, health status, storage, dosing intervals, adherence, circulating variants and background exposure all differ from the trial.
Effectiveness studies are also observational rather than randomised, so they carry confounding that a randomised trial is designed to remove. The two figures can differ in either direction and both can be correct for what they measure. A page like this computes efficacy from trial counts; it cannot tell you anything about effectiveness, and a number produced here should never be described as one.
Efficacy Against What, Exactly
A single vaccine and a single trial can yield several different efficacy numbers depending on the outcome counted. Efficacy against infection, against symptomatic disease, against severe disease, against hospitalisation, against death and against onward transmission are distinct endpoints, and they are often measured with different precision because the rarer the outcome, the fewer the cases and the wider the interval.
The methodological literature makes this decomposition explicit. The American Journal of Epidemiology paper Estimating Influenza Vaccine Efficacy From Challenge and Community-based Study Data sets out a framework separating efficacy for susceptibility, for symptomatic illness given infection, and for infectiousness, and argues that a single combined figure obscures more than it reveals. The practical consequence for a reader is simple: a quoted percentage is meaningless without the endpoint attached to it, and two figures for the same product may be measuring entirely different things.
Why One Trial Is Not the Evidence Base
A point estimate from a single study is one observation of a quantity that varies by population, by period, by circulating strain and by how long after vaccination the outcome was measured. Efficacy typically changes with time since dosing, and a trial reporting a fixed follow-up window is reporting an average over that window rather than a permanent property.
Assessment of any vaccine is done by regulators and public health bodies using the full trial dataset, safety data, immunogenicity data, post-authorisation surveillance and systematic reviews across studies. Those bodies publish their assessments, and they are the appropriate source for any question about a specific product. An arithmetic tool that recomputes one ratio from four numbers is not a substitute for that process and should not be presented as one.
When the Formula Breaks Down
Three situations have no valid answer from this method, and the calculator says so in writing rather than printing a number. If there are no cases in the unvaccinated arm the ratio has a zero denominator and efficacy is undefined. If there are no cases in the vaccinated arm the point estimate is 100%, but the log-scale interval requires dividing by zero, so no interval can be computed this way — published trials in that position use exact or continuity-corrected methods instead. And if either group size is zero there is no attack rate to compute.
There is also a case the arithmetic handles but the interpretation should not gloss over: if the attack rate is higher in the vaccinated arm, efficacy comes out negative. That is a description of the counts entered, nothing more. Small case counts produce negative point estimates routinely by chance alone, which is precisely why the interval is reported alongside.
Arb Digital builds free calculators and explainers that compute what is published and say plainly what a number does not carry. Browse the library, or tell us what your audience keeps searching for.
Browse Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Quoting a point estimate without its interval, which hides how much or how little the underlying counts constrain it.
- Comparing efficacy figures across trials with different case definitions, populations, follow-up windows or circulating strains.
- Using the number of participants without the number of cases to judge precision — cases drive the interval width.
- Calling an efficacy figure an effectiveness figure, or the reverse. They are measured under different conditions by different study designs.
- Reading efficacy as an individual probability. It is a ratio of group rates in one study, not a personal risk.
Related Free Tools From Arb Digital
Work with the general two-by-two case in the relative risk calculator, see the absolute scale alongside the ratio in the absolute risk reduction calculator, handle person-time denominators with the incidence rate calculator, work with death rates in the mortality rate calculator, and explore the teaching model of transmission in the epidemic SIR model calculator. The free online tools hub lists every statistics tool we publish.
Frequently Asked Questions
As one minus the ratio of the attack rate in the vaccinated group to the attack rate in the unvaccinated group. Each attack rate is the number of cases divided by the number of participants in that group.
Efficacy is measured under controlled trial conditions with a defined protocol and population. Effectiveness is what is observed in ordinary use outside a trial, in observational studies that carry confounding a randomised trial is designed to remove.
Because the width is driven by the number of cases rather than the number of participants. Few cases mean the data are consistent with a broad range of underlying values, which is a statement about precision and not a conclusion either way.
No. It is a ratio of rates between two groups in one study, not an individual probability. Questions about your own situation belong with a qualified clinician and your national health authority.
That has to be stated. Infection, symptomatic disease, severe disease, hospitalisation and death are separate endpoints, and one trial can produce a different figure for each with different precision.
Published analyses often use person-time denominators, adjusted models, or a different analysis population than the simple attack-rate formula used here. The trial report states its method; this tool applies only the basic definition.
With no cases in the unvaccinated arm the ratio is undefined. With no cases in the vaccinated arm the point estimate is 100% but the log-scale interval cannot be computed, and other methods are required. The tool reports a written message in both situations.
This page is an arithmetic tool for general information only. It is not medical advice, it makes no claim about any vaccine, and no number it produces should be read as a conclusion about whether a vaccine works. For any question about vaccination, speak to a qualified clinician and follow the current guidance of your national health authority.
