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The approximation to the standard normal distribution can be improved by the use of a continuity correction: S c = |S| – 1. Thus 1 is subtracted from a positive S value and 1 is added to a negative S value. The z-score equivalent is then given by = ()
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 = (< +) (+ /)
Yates's correction should always be applied, as it will tend to improve the accuracy of the p-value obtained. [citation needed] However, in situations with large sample sizes, using the correction will have little effect on the value of the test statistic, and hence the p-value.
The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution. This is a special case when μ = 0 {\textstyle \mu =0} and σ 2 = 1 {\textstyle \sigma ^{2}=1} , and it is described by this probability density function (or density): φ ( z ) = e − z 2 2 2 π . {\displaystyle \varphi (z ...
Diagram showing the cumulative distribution function for the normal distribution with mean (μ) 0 and variance (σ 2) 1. These numerical values "68%, 95%, 99.7%" come from the cumulative distribution function of the normal distribution. The prediction interval for any standard score z corresponds numerically to (1 − (1 − Φ μ,σ 2 (z)) · 2).
By the Central Limit Theorem, as n increases, the Irwin–Hall distribution more and more strongly approximates a Normal distribution with mean = / and variance = /.To approximate the standard Normal distribution () = (=, =), the Irwin–Hall distribution can be centered by shifting it by its mean of n/2, and scaling the result by the square root of its variance:
The exact binomial test gives p = 0.053 and McNemar's test with continuity correction gives = 3.68 and p = 0.055. The asymptotic McNemar's test gives = 4.55 and p = 0.033 and the mid-P McNemar's test gives p = 0.035. Both the McNemar's test and mid-P version provide stronger evidence for a statistically significant treatment effect in this ...
Illustration of the Kolmogorov–Smirnov statistic. The red line is a model CDF, the blue line is an empirical CDF, and the black arrow is the KS statistic.. In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric test of the equality of continuous (or discontinuous, see Section 2.2), one-dimensional probability distributions.