# Binomial test

﻿
Binomial test

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 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.

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.

References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Binomial (disambiguation) — A binomial is a polynomial with two terms.Binomial may also refer to:In mathematics: *Binomial theorem, a theorem about powers of binomials *Binomial coefficient, numbers appearing in the expansions of powers of binomials *Binomial type, a… …   Wikipedia

• Binomial distribution — Probability distribution name =Binomial type =mass pdf cdf Colors match the image above parameters =n geq 0 number of trials (integer) 0leq p leq 1 success probability (real) support =k in {0,dots,n}! pdf ={nchoose k} p^k (1 p)^{n k} ! cdf =I {1… …   Wikipedia

• Binomial probability — typically deals with the probability of several successive decisions, each of which has two possible outcomes. Definition The probability of an event can be expressed as a binomial probability if its outcomes can be broken down into two… …   Wikipedia

• Binomial proportion confidence interval — In statistics, a binomial proportion confidence interval is a confidence interval for a proportion in a statistical population. It uses the proportion estimated in a statistical sample and allows for sampling error. There are several formulas for …   Wikipedia

• Binomial series — In mathematics, the binomial series is the Taylor series at x = 0 of the function f given by f(x) = (1 + x) α, where α ∈ C is an arbitrary complex number. Explicitly, and the binomial series is the power series… …   Wikipedia

• Pearson's chi-squared test — (χ2) is the best known of several chi squared tests – statistical procedures whose results are evaluated by reference to the chi squared distribution. Its properties were first investigated by Karl Pearson in 1900. In contexts where it is… …   Wikipedia

• McNemar's test — In statistics, McNemar s test is a non parametric method used on nominal data. It is applied to 2 × 2 contingency tables with a dichotomous trait, with matched pairs of subjects, to determine whether the row and column marginal… …   Wikipedia

• Pearson's chi-square test — Pearson s chi square ( chi;2) test is the best known of several chi square tests – statistical procedures whose results are evaluated by reference to the chi square distribution. Its properties were first investigated by Karl Pearson. In contexts …   Wikipedia

• Parametrischer Test — Die Artikel Statistischer Test und Signifikanztest überschneiden sich thematisch. Hilf mit, die Artikel besser voneinander abzugrenzen oder zu vereinigen. Beteilige dich dazu an der Diskussion über diese Überschneidungen. Bitte entferne diesen… …   Deutsch Wikipedia

• Nicht-parametrischer Test — Der Zweig der Statistik, der als parameterfreie Statistik bekannt ist, beschäftigt sich mit parameterfreien statistischen Modellen und parameterfreien statistischen Tests. Andere gebräuchliche Bezeichnungen sind nicht parametrische Statistik oder …   Deutsch Wikipedia