y=Φ(x)=12(1+erf⁡x2)y = \Phi(x) = \dfrac{1}{2}\left(1 + \operatorname{erf}\dfrac{x}{\sqrt{2}}\right)

Graph of the Standard Normal Distribution Function y=Φ(x)y = \Phi(x)

y=Φ(x)y = \Phi(x) is the distribution function of the standard normal1. It gives the probability that a standard normal random variable is at most xx.

Φ(x)=∫−∞x12πe−t2/2 dt\Phi(x) = \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}}e^{-t^2/2}\,dt

Written with the error function erf⁡\operatorname{erf} it also takes the following form.

Φ(x)=12(1+erf⁡x2)\Phi(x) = \frac{1}{2}\left(1 + \operatorname{erf}\frac{x}{\sqrt{2}}\right)

The 2\sqrt{2} appears because of a difference in the definitions: erf⁡\operatorname{erf} uses e−t2e^{-t^2} while the normal distribution uses e−t2/2e^{-t^2/2}.

Domain and range

  • The domain is all real numbers
  • The range is the open interval (0,1)(0, 1)
  • It increases monotonically
  • The lines y=0y = 0 and y=1y = 1 are horizontal asymptotes

Being a probability it stays between 00 and 11 and reaches neither.

Symmetry

Since the density of the standard normal is even, Φ(−x)=1−Φ(x)\Phi(-x) = 1 - \Phi(x). The graph therefore has point symmetry about (0,12)\left( 0, \dfrac{1}{2} \right).

Monotonicity and concavity

The derivative is the density itself.

φ(x)=12πe−x2/2\varphi(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}

It is always positive, so the function increases monotonically with no extrema. The slope is greatest at the origin, where it is 12π≈0.399\dfrac{1}{\sqrt{2\pi}} \approx 0.399. The second derivative is −xφ(x)-x\varphi(x), whose sign changes at x=0x = 0, so (0,12)\left( 0, \dfrac{1}{2} \right) is the only inflection point.

Not expressible in elementary functions

No antiderivative of e−t2/2e^{-t^2/2} exists among the elementary functions. That is why Φ\Phi is given a symbol of its own, with printed tables once and numerical routines now. Writing it with erf⁡\operatorname{erf} does not reduce it to elementary functions either; it merely restates it through another special function.

Values worth remembering

xxΦ(x)\Phi(x)Probability of the matching range
11≈0.8413\approx 0.841368.27%68.27\%
1.6451.645≈0.9500\approx 0.9500the one-sided 5%5\% critical value
1.9601.960≈0.9750\approx 0.9750the two-sided 95%95\% confidence interval
22≈0.9772\approx 0.977295.45%95.45\%
33≈0.99865\approx 0.9986599.73%99.73\%

The last column, read together with the symmetry, gives the probability of falling within so many standard deviations of the mean: the familiar 6868–9595–99.799.7 rule2.

Comparison with the logistic function

The logistic function 11+e−x\dfrac{1}{1 + e^{-x}} draws a similar S. Matching the slopes by writing 11+e−1.702x\dfrac{1}{1 + e^{-1.702x}}, its greatest difference from Φ(x)\Phi(x) is only about 0.00950.0095, and the graphs all but coincide. That agreement is why probit regression and logistic regression give such similar results in statistics.

Applications

By the central limit theorem the distribution of a sum of independent random variables approaches the normal under suitable conditions, so Φ\Phi turns up everywhere in statistics.

  • The computation of a pp value
  • The construction of a confidence interval
  • The control limits of quality control

Its inverse Φ−1\Phi^{-1} is called the probit and is used as a link function that stretches a probability out onto the whole real line.

  1. Normal distribution, Wikipedia
  2. 68-95-99.7 rule, Wikipedia