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In
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 ...
, a continuity correction is an adjustment that is made when a discrete distribution is approximated by a continuous distribution.


Examples


Binomial

If a random variable ''X'' has a binomial distribution 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 random variable with the same expected value 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 numbe ...
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 co ...
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 having the ability to evaluate probability distribution functions accurately, continuity corrections played an important role in the practical application of statistical tests 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 hypoth ...
, involving the binomial distribution, 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 be ...
. 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.


See also

* Yates's correction for continuity * Wilson score interval with continuity correction


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