probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

, a continuity correction is an adjustment that is made when a discrete

distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a varia ...

is approximated by a continuous distribution.

Examples

Binomial

If a

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...

''X'' has a

binomial distribution In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no qu ...

with parameters ''n'' and ''p'', i.e., ''X'' is distributed as the number of "successes" in ''n'' independent

Bernoulli trial In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is ...

s with probability ''p'' of success on each trial, then :

P(X\leq x) = P(X for any ''x'' ∈ {0, 1, 2, ... ''n''}.  If ''np'' and ''np''(1 − ''p'') are large (sometimes taken as both ≥ 5), then the probability above is fairly well approximated by

: P(Y\leq x+1/2) where ''Y'' is a

normally distributed In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is ...

random variable with the same

expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...

and the same

variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...

as ''X'', i.e., E(''Y'') = ''np'' and var(''Y'') = ''np''(1 − ''p''). This addition of 1/2 to ''x'' is a continuity correction.

Poisson

A continuity correction can also be applied when other discrete distributions supported on the integers are approximated by the normal distribution. For example, if ''X'' has a

Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known ...

with expected value λ then the variance of ''Y'' is also λ, and :

P(X\leq x)=P(X if ''Y'' is normally distributed with expectation and variance both λ.

Applications

Before the ready availability of

statistical software Statistical software are specialized computer programs for analysis in statistics and econometrics. Open-source * ADaMSoft – a generalized statistical software with data mining algorithms and methods for data management * ADMB – a softwa ...

having the ability to evaluate probability distribution functions accurately, continuity corrections played an important role in the practical application of

statistical tests A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis. Hypothesis testing allows us to make probabilistic statements about population parameters. ...

in which the test statistic has a discrete distribution: it had a special importance for manual calculations. A particular example of this is the

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 using sample data. Usage The binomial test is useful to test hypothe ...

, involving the

, as in

checking whether a coin is fair In statistics, the question of checking whether a coin is fair is one whose importance lies, firstly, in providing a simple problem on which to illustrate basic ideas of statistical inference and, secondly, in providing a simple problem that can ...

. Where extreme accuracy is not necessary, computer calculations for some ranges of parameters may still rely on using continuity corrections to improve accuracy while retaining simplicity.

References

* Devore, Jay L., ''Probability and Statistics for Engineering and the Sciences'', Fourth Edition, Duxbury Press, 1995. * Feller, W., ''On the normal approximation to the binomial distribution'', The Annals of Mathematical Statistics, Vol. 16 No. 4, Page 319–329, 1945. Theory of probability distributions Statistical tests Computational statistics

Examples

Binomial

Poisson

Applications

See also

References