In statistics, the binomial test is an exact test of the statistical significance of deviations from a theoretically expected distribution of observations into two categories. This article is about the field of statistics. ... An exact (significance) test is a test where all assumptions that the derivation of the distribution of the test statistic is based on are met. ... In statistics, a result is significant if it is unlikely to have occurred by chance, given that a presumed null hypothesis is true. ...

For example, suppose we have a board game that depends on the roll of a die, and special importance attaches to rolling a 6. In a particular game, the die is rolled 235 times, and 6 comes up 51 times. If the die is fair, we would expect 6 to come up 235/6 = 39.17 times. Is the proportion of 6s significantly higher than would be expected by chance, on the null hypothesis of a fair die? A board game is a game played with counters or pieces that are placed on, removed from, or moved across a board (a premarked surface, usually specific to that game). ... Dice (the plural of die, from Old French de, from Latin datum something given or played [1]) are small polyhedral objects, usually cubical, used for generating random numbers or other symbols. ... In statistics, a null hypothesis is a hypothesis set up to be nullified or refuted in order to support an alternative hypothesis. ...

To find an answer to this question using the binomial test, we consult the binomial distribution B(235,1/6) to find out what the probability is of finding exactly 51 6s in a sample of 235 if the true probability of a 6 on each trial is 1/6. We then find the probability of finding exactly 52, exactly 53, and so on up to 235, and add all these probabilities together. In this way, we obtain the probability of obtaining the observed result (51 6s) or a more extreme result (>51 6s) and in this example, the result is 0.02654425,which means we cannot reject the null hypothesis (two-tailed test) if we are working at the 5% significance level. The result calculated, 0.02654425 is significant for a one-tailed test of the observed number of 6s. In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ... The two-tailed test is the test of a given statistical hypothesis in which a value of the statistic that is either sufficiently small or sufficiently large will lead to rejection of the hypothesis tested. ...

For large samples such as this example, 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, Pearson's chi-square test and the G-test. However, for small samples these approximations break down, and there is no alternative to the binomial test. In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... Pearsons chi-square test (χ2) is one of a variety of chi-square tests – statistical procedures whose results are evaluated by reference to the chi-square distribution. ... In statistics, G-tests are likelihood-ratio or maximum likelihood statistical significance tests that are increasingly being used in situations where chi-square tests were previously recommended. ...

The most common use of the binomial test is in the case where the null hypothesis is that two categories are equally likely to occur. Tables are widely available to give the significance observed numbers of observations in the categories for this case. However, as the example above shows, the binomial test is not restricted to this case.

Where there are more than two categories, and an exact test is required, a test based on the multinomial distribution must be used instead of the binomial test. In probability theory, the multinomial distribution is a generalization of the binomial distribution. ...

References

• Abdi, H. (2007).[1] Binomial Distribution: Binomial and Sign Tests. In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.