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Since probability tables cannot be printed for every normal distribution, as there are an infinite variety of normal distributions, it is common practice to convert a normal to a standard normal (known as a z-score) and then use the standard normal table to find probabilities.
is the standard normal probability density function, ... "A table of normal integrals". Communications in Statistics: Simulation and Computation. B9 (4): ...
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 ...
In statistics, the Q-function is the tail distribution function of the standard normal distribution. [ 1 ] [ 2 ] In other words, Q ( x ) {\displaystyle Q(x)} is the probability that a normal (Gaussian) random variable will obtain a value larger than x {\displaystyle x} standard deviations.
The Ewens's sampling formula is a probability distribution on the set of all partitions of an integer n, arising in population genetics. The Balding–Nichols model; The multinomial distribution, a generalization of the binomial distribution. The multivariate normal distribution, a generalization of the normal distribution.
The "68–95–99.7 rule" is often used to quickly get a rough probability estimate of something, given its standard deviation, if the population is assumed to be normal. It is also used as a simple test for outliers if the population is assumed normal, and as a normality test if the population is potentially not normal.
In order to create the studentized range distribution for normal data, we first switch from the generic f X and F X to the distribution functions φ and Φ for the standard normal distribution, and change the variable r to s·q, where q is a fixed factor that re-scales r by scaling factor s:
/ is the critical value of the standard normal distribution (e.g., 1.96 for a 95% confidence level). The MDE for when using the (two-sided) z-test formula for comparing two proportions, incorporating critical values for α {\displaystyle \alpha } and 1 − β {\displaystyle 1-\beta } , and the standard errors of the proportions: [ 1 ] [ 2 ]