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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.

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?

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 sixes 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 0.0265443, which means we would have a 97+% chance of being right to reject the null hypothesis if we were only considering deviations above the expected (one-tailed test). However, clearly a die could roll too few sixes as easily as too many and we would be just as suspicious, so we should use the two-tailed test which considers the probability of having a particular effect size either above or below expectation. Here the effect size is 11.83, since that's how many more sixes there were than expected, with 51 found vs. 39.17 expected. So now we have to find the probability that that the die would roll a six 27 times or less (39.17 expected - 11.83 equal effect size). Summing over all the probabilities (< 28 6s) yields .0172037. When we add this to the first result, we get .0437480, which is significant at the 5% significance level. If the cost of a false accusation was too high, we might have a more stringent requirement, like 1% significance level, in which case we could not reject the null hypothesis of a fair die with sufficient certainty.

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.

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, the multinomial test, based on the multinomial distribution, must be used instead of the binomial test.

[edit] See also

Wikiversity has learning materials about Binomial test

• Binomial distribution
• P-value

[edit] In statistical software packages

Binomial tests are available in most software used for statistical purposes. E.g.

• In R the above example could be calculated with the following code:

(One-tail test) binom.test(51,235,(1/6),alternative="greater")

(Two-tail test) binom.test(51,235,(1/6),alternative="two.sided")

• In SPSS the test can be utilized through the menu Analyze > Nonparametric test > Binomial

• In Python use Scipy

(One-tail test) scipy.stats.binom.sf(51-1, 235, 1.0/6) # -1 is there to include 51 as well ;-)

(Two-tail test) scipy.stats.binom_test(51, 235, 1.0/6)

[edit] References

• Binomial significance testing Retrieved 03-07-2009