Binomial Test
What Is the Binomial Test?
The binomial test is a simple but powerful statistical method for deciding whether the proportion of “successes” you observe in a series of trials matches a proportion you expected. Each trial has only two possible outcomes—success or failure, yes or no, heads or tails—which is why the test is called “binomial” (meaning “two names” or “two categories”).
Imagine you flip a coin 100 times and get 62 heads. You expected 50 heads if the coin were fair. Is 62 far enough from 50 to conclude that the coin is biased, or could this difference easily happen by luck? The binomial test answers exactly this kind of question.
Why Do We Need It?
Many real-world situations involve binary outcomes. A marketing team wants to know if the click-through rate on a new advertisement exceeds 10%. A psychologist asks whether more than half of participants choose option A over option B. A quality control inspector checks whether the defect rate in a factory has changed from its historical level. In each case, we are comparing an observed proportion of successes against a specific expected probability, and the binomial test is the natural tool for the job.
How Does It Work?
The test relies on the binomial distribution, which describes the probability of getting each possible number of successes in a fixed number of independent trials, given a known probability of success on any single trial. To run the test, you need three pieces of information: the number of trials (n), the number of observed successes (k), and the expected probability of success on each trial (p).
The test then calculates the probability of observing your result—or something more extreme—assuming the expected probability is correct. This probability is your p-value.
One-Tailed vs. Two-Tailed Tests
An important decision you need to make before running a binomial test is whether to use a one-tailed or two-tailed version. A two-tailed test asks: “Is the observed proportion different from the expected proportion in either direction?” For the coin example, this means asking whether the coin is biased toward heads or toward tails.
A one-tailed test asks a more specific question: “Is the observed proportion greater than (or less than) the expected proportion?” If a researcher hypothesizes ahead of time that a new training program will increase the pass rate above 70%, a one-tailed test focused on “greater than” is appropriate. The choice between one-tailed and two-tailed should always be made before you look at the data, based on your research question.
Interpreting the Result
The binomial test gives you a p-value. If this p-value falls below your chosen significance threshold (usually 0.05), you conclude that the observed proportion is significantly different from the expected one.
For example, suppose a survey asks 80 people whether they support a new policy, and 52 say yes. You want to know if support is significantly above 50%. A one-tailed binomial test with n = 80, k = 52, and p = 0.50 produces a p-value of about 0.0073. Since this is well below 0.05, you would conclude that support is significantly greater than 50%.
When to Use the Binomial Test
- Each trial has exactly two possible outcomes (binary data).
- The trials are independent of one another.
- You want to compare your observed proportion of successes to a specific expected probability.
- The probability of success is the same on every trial.
Typical applications include testing whether a coin is fair, evaluating whether a survey response rate differs from a benchmark, checking whether a treatment success rate meets a target, and assessing whether consumer preferences differ from an assumed split.
Key Assumptions
The binomial test requires the following conditions to be met:
- There are a fixed number of trials decided in advance.
- Each trial results in one of exactly two outcomes.
- The probability of success is constant across all trials.
- The trials are independent, meaning the outcome of one trial does not affect another.
When these conditions hold, the binomial test is an exact test—it computes precise probabilities rather than relying on approximations—making it especially trustworthy for small to moderate sample sizes.