# Continuity correction

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Approximation in mathematics

In [mathematics](/source/Mathematics), a **continuity correction** is an adjustment made when a [discrete object](/source/Discrete_mathematics) is approximated using a [continuous object](/source/Continuous_function).

## Examples

### Binomial

See also: [Binomial distribution § Normal approximation](/source/Binomial_distribution#Normal_approximation)

If a [random variable](/source/Random_variable) *X* has a [binomial distribution](/source/Binomial_distribution) with parameters *n* and *p*, i.e., *X* is distributed as the number of "successes" in *n* independent [Bernoulli trials](/source/Bernoulli_trial) with probability *p* of success on each trial, then

- P ( X ≤ x ) = P ( X < x + 1 ) {\displaystyle P(X\leq x)=P(X<x+1)}

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 ≤ x + 1 / 2 ) {\displaystyle P(Y\leq x+1/2)}

where *Y* is a [normally distributed](/source/Normal_distribution) random variable with the same [expected value](/source/Expected_value) and the same [variance](/source/Variance) 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](/source/Poisson_distribution) with expected value λ then the variance of *X* is also λ, and

- P ( X ≤ x ) = P ( X < x + 1 ) ≈ P ( Y ≤ x + 1 / 2 ) {\displaystyle P(X\leq x)=P(X<x+1)\approx P(Y\leq x+1/2)}

if *Y* is normally distributed with expectation and variance both λ.

## Applications

Before the ready availability of [statistical software](/source/Statistical_software) having the ability to evaluate [probability distribution](/source/Probability_distribution) functions accurately, continuity corrections played an important role in the practical application of [statistical tests](/source/Statistical_hypothesis_test) in which the [test statistic](/source/Test_statistic) has a discrete distribution: it had a special importance for manual calculations. A particular example of this is the [binomial test](/source/Binomial_test), involving the [binomial distribution](/source/Binomial_distribution), as in [checking whether a coin is fair](/source/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](/source/Yates's_correction_for_continuity)

- [Wilson score interval with continuity correction](/source/Binomial_proportion_confidence_interval#Wilson_score_interval_with_continuity_correction)

## References

- Devore, Jay L. (1995). *Probability and Statistics for Engineering and the Sciences* (Fourth ed.). Duxbury Press. [ISBN](/source/ISBN_(identifier)) [0-534-24264-2](https://en.wikipedia.org/wiki/Special:BookSources/0-534-24264-2).

- Feller, W. (1945). "On the normal approximation to the binomial distribution". *The Annals of Mathematical Statistics*. **16** (4): 319–329. [JSTOR](/source/JSTOR_(identifier)) [2236142](https://www.jstor.org/stable/2236142).

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Adapted from the Wikipedia article [Continuity correction](https://en.wikipedia.org/wiki/Continuity_correction) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Continuity_correction?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.