Editing Continuity correction

{{Short description|Approximation in mathematics}}
In [[mathematics]], a '''continuity correction''' is an adjustment made when a [[Discrete mathematics|discrete object]] is approximated using a [[Continuous function|continuous object]].

==Examples==

===Binomial===
{{see also|Binomial distribution#Normal approximation}}

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

:<math>P(X\leq x) = P(X<x+1)</math>

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

:<math>P(Y\leq x+1/2)</math>

where ''Y'' is a [[normal distribution|normally distributed]] random variable with the same [[expected value]] and the same [[variance]] as ''X'', i.e., E(''Y'') = ''np'' and var(''Y'') = ''np''(1 &minus; ''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]] with expected value λ then the variance of ''X'' is also λ, and

:<math>P(X\leq x)=P(X<x+1)\approx P(Y\leq x+1/2)</math>

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 hypothesis test|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]], involving the [[binomial distribution]], as in [[checking whether a coin is fair]]. 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]]
*[[Binomial proportion confidence interval#Wilson score interval with continuity correction|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.

[[Category:Theory of probability distributions]]
[[Category:Computational statistics]]