In statistics, the Q-function is the tail distribution function of the standard normal distribution. In other words, ${\displaystyle Q(x)}$ is the probability that a normal (Gaussian) random variable will obtain a value larger than ${\displaystyle x}$ standard deviations. Equivalently, ${\displaystyle Q(x)}$ is the probability that a standard normal random variable takes a value larger than ${\displaystyle x}$.

If ${\displaystyle Y}$ is a Gaussian random variable with mean ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2}}$, then ${\displaystyle X={\frac {Y-\mu }{\sigma }}}$ is standard normal and

where ${\displaystyle x={\frac {y-\mu }{\sigma }}}$.

Other definitions of the Q-function, all of which are simple transformations of the normal cumulative distribution function, are also used occasionally.

Because of its relation to the cumulative distribution function of the normal distribution, the Q-function can also be expressed in terms of the error function, which is an important function in applied mathematics and physics.

Formally, the Q-function is defined as

Thus,

where ${\displaystyle \Phi (x)}$ is the cumulative distribution function of the standard normal Gaussian distribution.