Binomial test is an exact test of the statistical significance of deviations from a theoretically expected distribution of observations into two categories using sample data.
A binomial test is a statistical hypothesis test used to determine whether the proportion of successes in a sample differs from an expected proportion in a binomial distribution.
It is useful for situations when there are two possible outcomes (e.g., success/failure, yes/no, heads/tails), i.e., where repeated experiments produce binary data.
The formula of the binomial distribution gives the probability of those
-value for a two-tailed test is slightly more complicated, since a binomial distribution isn't symmetric if
-value is calculated as, One common use of the binomial test is the case where the null hypothesizes that two categories occur with equal frequency (
However, as the example below shows, the binomial test is not restricted to this case.
[1] Most common measures of effect size for Binomial tests are Cohen's h or Cohen's g. For large samples such as the example below, the binomial distribution is well approximated by convenient continuous distributions, and these are used as the basis for alternative tests that are much quicker to compute, such as Pearson's chi-squared test and the G-test.
However, for small samples these approximations break down, and there is no alternative to the binomial test.
The most usual (and easiest) approximation is through the standard normal distribution, in which a z-test is performed of the test statistic
is the number of successes observed in a sample of size
An improvement on this approximation is possible by introducing a continuity correction: For very large
, this continuity correction will be unimportant, but for intermediate values, where the exact binomial test doesn't work, it will yield a substantially more accurate result.
In notation in terms of a measured sample proportion
in both numerator and denominator, which is a form that may be more familiar to some readers.
We have now observed that the number of 6s is higher than what we would expect on average by pure chance had the die been a fair one.
But, is the number significantly high enough for us to conclude anything about the fairness of the die?
To find an answer to this question using the binomial test, we use the binomial distribution As we have observed a value greater than the expected value, we could consider the probability of observing 51 6s or higher under the null, which would constitute a one-tailed test (here we are basically testing whether this die is biased towards generating more 6s than expected).
In order to calculate the probability of 51 or more 6s in a sample of 235 under the null hypothesis we add up the probabilities of getting exactly 51 6s, exactly 52 6s, and so on up to probability of getting exactly 235 6s: If we have a significance level of 5%, then this result (0.02654 < 5%) indicates that we have evidence that is significant enough to reject the null hypothesis that the die is fair.
Normally, when we are testing for fairness of a die, we are also interested if the die is biased towards generating fewer 6s than expected, and not only more 6s as we considered in the one-tailed test above.
In order to consider both the biases, we use a two-tailed test.
Note that to do this we cannot simply double the one-tailed p-value unless the probability of the event is 1/2.
This is because the binomial distribution becomes asymmetric as that probability deviates from 1/2.
One method is to sum the probability that the total deviation in numbers of events in either direction from the expected value is either more than or less than the expected value.
The second method involves computing the probability that the deviation from the expected value is as unlikely or more unlikely than the observed value, i.e. from a comparison of the probability density functions.
This can create a subtle difference, but in this example yields the same probability of 0.0437.
In both cases, the two-tailed test reveals significance at the 5% level, indicating that the number of 6s observed was significantly different for this die than the expected number at the 5% level.
Binomial tests are available in most software used for statistical purposes.