Continuity correction
Encyclopedia
In probability theory
, 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
for any x ∈ {0, 1, 2, ... n}. If np and n(1 − p) are large (sometimes taken to mean ≥ 5), then the probability above is fairly well approximated by
where Y is a 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 − p). This addition of 1/2 to x is a continuity correction.
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
if Y is normally distributed with expectation and variance both λ.
, 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.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, if a random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
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
Bernoulli trial
In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, "success" and "failure"....
s with probability p of success on each trial, then
for any x ∈ {0, 1, 2, ... n}. If np and n(1 − p) are large (sometimes taken to mean ≥ 5), then the probability above is fairly well approximated by
where Y is a normally distributed random variable with the same expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
and the same variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
as X, i.e., E(Y) = np and var(Y) = np(1 − p). This addition of 1/2 to x is a continuity correction.
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
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 and/or space if these events occur with a known average rate and independently of the time since...
with expected value λ then the variance of X is also λ, and
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 was a special importance for manual calculations. A particular example of this is the binomial testBinomial 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.-Common use:...
, 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.