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Standard Normal Distribution

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StandardNormalDistribution

A standard normal distribution is a normal distribution with zero mean (mu=0) and unit variance (sigma^2=1), given by the probability density function and distribution function

P(x)=1/(sqrt(2pi))e^(-x^2/2)
(1)
D(x)=1/2[erf(x/(sqrt(2)))+1]
(2)

over the domain x in (-infty,infty).

It has mean, variance, skewness, and kurtosis excess given by

mu=0
(3)
sigma^2=1
(4)
gamma_1=0
(5)
gamma_2=0.
(6)

The first quartile of the standard normal distribution occurs when D(x)=1/4, which is

x_(1/4)=-sqrt(2)erf^(-1)(1/2)
(7)
=-0.67448975019...
(8)

(OEIS A092678; Kenney and Keeping 1962, p. 134), where erf^(-1)(x) is the inverse erf function. The absolute value of this is known as the probable error.

See also

Erf, Normal Distribution, Normal Distribution Function, Probable Error, Probability Integral, Tetrachoric Function

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References

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 129 and 134, 1962.Sloane, N. J. A. Sequence A092678 in "The On-Line Encyclopedia of Integer Sequences."

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Standard Normal Distribution

Cite this as:

Weisstein, Eric W. "Standard Normal Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StandardNormalDistribution.html

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