Normal Distribution Function

A normalized form of the cumulative Gaussian Distribution function giving the probability that a variate assumes a value in the range $[0, x]$,

$$\Phi(x) \equiv Q(x) \equiv {1\over\sqrt{2\pi}} \int^x_0 e^{-t^2/2}\, dt.$$ (1)

It is related to the Probability Integral

$$\alpha(x) \equiv {1\over\sqrt{2\pi}} \int^x_{-x} e^{-t^2/2}\, dt$$ (2)

by

$$\Phi(x)={\textstyle{1\over 2}}\alpha(x).$$ (3)

Let $u \equiv t/\sqrt{2}$ so $du = dt/\sqrt{2}$. Then

$$\Phi(x) = {1\over \sqrt{\pi}}\int_0^{x/\sqrt{2}} e^{-u^2}\,d... ...r 2}}\mathop{\rm erf}\nolimits \left({x\over \sqrt{2}}\right).$$ (4)

Here, Erf is a function sometimes called the error function. The probability that a normal variate assumes a value in the range $[x_1, x_2]$ is therefore given by

$$\Phi(x_1,x_2)={1\over 2}\left[{\mathop{\rm erf}\nolimits \le... ...op{\rm erf}\nolimits \left({x_1\over \sqrt{2}}\right)}\right].$$ (5)

Neither $\Phi(z)$ nor Erf can be expressed in terms of finite additions, subtractions, multiplications, and root extractions, and so must be either computed numerically or otherwise approximated.

Note that a function different from $\Phi(x)$ is sometimes defined as ``the'' normal distribution function

$$\Phi'(x)\equiv {1\over 2}\left[{1+\mathop{\rm erf}\nolimits ... ...{x\over\sqrt{2}}\right)}\right]={\textstyle{1\over 2}}+\Phi(x)$$ (6)

(Beyer 1987, p. 551), although this function is less widely encountered than the usual $\Phi(x)$.

The value of $a$ for which $P(x)$ falls within the interval $[-a, a]$ with a given probability $P$ is a related quantity called the Confidence Interval.

For small values $x\ll 1$, a good approximation to $\Phi(x)$ is obtained from the Maclaurin Series for Erf,

$$\Phi(x)={1\over\sqrt{2\pi}}(2x-{\textstyle{1\over 3}}x^3+{\textstyle{1\over 20}}x^5-{\textstyle{1\over 168}}x^7+\ldots).$$ (7)

For large values $x\gg 1$, a good approximation is obtained from the asymptotic series for Erf,

$$\Phi(x)={1\over 2}+{e^{-x^2/2}\over\sqrt{2\pi}} (x^{-1}-x^{-3}+3x^{-5}-15x^{-7}+105x^{-9}+\ldots).$$ (8)

The value of $\Phi(x)$ for intermediate $x$ can be computed using the Continued Fraction identity

$$\int_0^x e^{-u^2}\,du = {\sqrt{\pi}\over 2}-{{\textstyle{1\o... ...+{\strut\displaystyle 4\over\strut\displaystyle x+\ldots}}}}}.$$ (9)

A simple approximation of $\Phi(x)$ which is good to two decimal places is given by

$$\Phi_1(x) \approx \cases{ 0.1x(4.4-x) & for $0 \leq x \leq ... ...cr 0.49 & for $2.2 < x < 2.6$\cr 0.50 & for $x\geq 2.6$.\cr}$$ (10)

Abramowitz and Stegun (1972) and Johnson and Kotz (1970) give other functional approximations. An approximation due to Bagby (1995) is

$$\Phi_2(x)={\textstyle{1\over 2}}\{1-{\textstyle{1\over 30}}[... ...rt{2}\,)}+(7+{\textstyle{1\over 4}}\pi x^2) e^{-x^2}]\}^{1/2}.$$ (11)

The plots below show the differences between $\Phi$ and the two approximations.

The first Quartile of a standard Normal Distribution occurs when

$$\int_0^t \Phi(z)\,dz={\textstyle{1\over 4}}.$$ (12)

The solution is $t=0.6745\dots$. The value of $t$ giving ${\textstyle{1\over 4}}$ is known as the Probable Error of a normally distributed variate.

See also Confidence Interval, Erf, Erfc, Fisher-Behrens Problem, Gaussian Distribution, Gaussian Integral, Hh Function, Normal Distribution, Probability Integral, Tetrachoric Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 931-933, 1972.

Bagby, R. J. ``Calculating Normal Probabilities.'' Amer. Math. Monthly 102, 46-49, 1995.

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987.

Johnson, N.; Kotz, S.; and Balakrishnan, N. Continuous Univariate Distributions, Vol. 1, 2nd ed. Boston, MA: Houghton Mifflin, 1994.