User talk:RayLei

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Hello, RayLei, and Welcome to Wikipedia! Thank you for your contributions to this free encyclopedia. If you decide that you need help, check out Getting Help below, ask me on my talk page, or place {{Help me}} on your talk page and ask your question there. Please remember to sign your name on talk pages by using four tildes (~~~~) or by clicking if shown; this will automatically produce your username and the date. Also, please do your best to always fill in the edit summary field with your edits. Below are some useful links to facilitate your involvement. Happy editing! 70.24.251.71 (talk) 09:59, 5 March 2012 (UTC)
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Continuity Correction[edit]

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 trials 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.

Reason[edit]

Look at the Bin(10, 0.5) distribution, whose mean and variance are 5 and 2.5, respectively. How well does a N(5, 2.5) distribution approximate the Bin(10, 0.5)?


 File:Distribution of B(10,0.5).gif File:Distribution of N(5,2.5).gif

The probability measure of a continuous distribution, in particular normal, at one point is zero, so it is reasonable to approximate the lumps of probability at the integers by areas under the normal curve. Take P(X = 3), for example. The exact binomial probability is 0.1172.

If integrate the N(5, 2.5) density from 2 to 3 it is too low everywhere and the result is too small.

File:I-2 to 3.gif

Φ((3-5)/sqrt(2.5)) - Φ((2-5)/sqrt(2.5)) = Φ(-1.265) - Φ(-1.897) = 0.1030 - 0.0289 = 0.0741

On the other hand, if integrate the N(5, 2.5) density from 3 to 4 it is too high everywhere and the result is too large.

File:I-3 to 4.gif

Φ((4-5)/sqrt(2.5)) - Φ((3-5)/sqrt(2.5)) = Φ(-0.6325) - Φ(-1.265) = 0.2635 - 0.1030 = 0.1605

But if integrate from 2.5 to 3.5 the result is an much closer approximation.


File:I-2.5 to 3.5.gif

Φ((3.5-5)/sqrt(2.5)) - Φ((2.5-5)/sqrt(2.5)) = Φ(-0.9487) - Φ(-1.581) = 0.1714 - 0.0569 = 0.1145

More generally, one can use the integration under the normal from minus infinity to a+0.5 to approximate the binomial probability P(X <= a) and normal a-0.5 to infinity to approximate binomial P(x>=a), e.g. a = 3 and the exact binomial calculation gives P(x<=a)=0.1719 and P(x>=a)=0.9453 while the normal approximation (with continuity correction) is Φ((3.5-5)/sqrt(2.5)) = Φ(-0.9487) = 0.1714 and 1 - Φ((2.5-5)/sqrt(2.5)) = 1 - Φ(-1.581) = 1 - 0.0569 = 0.9431.

File:I-minus infinity to 3.5.gif File:I-2.5 to infinity.gif


Applications[edit]

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 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[edit]

References[edit]

  • 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.
  • Peter Macdonald. (1998-03-04). The Continuity Correction. In Statistics 2MA3Probability and Statistical Methods for Science. Retrieved 4 March 2012, from http://www.math.mcmaster.ca/peter/s2ma3/s2ma3_9798/cont_correct.html.

Category:Probability theory Category:Statistical tests Category:Computational statistics

nl:Continuïteitscorrectie

Requested move[edit]

User:RayLeiNewName – Place here your rationale for the proposed page name change, ideally referring to applicable naming convention policies and guidelines, and providing evidence in support where appropriate. RayLei (talk) 08:18, 4 March 2012 (UTC)

  • Commnet you need to read WP:SYMUD on how to create user drafts. It should not be built in your talk page, for instance. 70.24.251.71 (talk) 09:59, 5 March 2012 (UTC)