What Is the Normal CDF Φ Called?

The function $\Phi$ is most commonly called the standard normal cumulative distribution function (standard normal CDF). It is also called the standard normal distribution function. The name tells you both which distribution is involved and what the function returns: cumulative probability up to a specified value.

The notation usually refers specifically to a normal random variable with mean $0$ and standard deviation $1$. For a normal distribution with a different mean or standard deviation, writers may still say “normal CDF,” but the symbol $\Phi$ conventionally denotes the standardized version.

Definition and interpretation

If $Z$ has the standard normal distribution, written $Z\sim N(0,1)$, then

$$\Phi(x)=P(Z\le x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^2/2}\,dt.$$

Thus, $\Phi(x)$ is the area under the standard normal density curve to the left of $x$. For instance, $\Phi(0)=0.5$ because the standard normal curve is symmetric about zero. For any real $x$, the result is a probability between $0$ and $1$.

The function is also related to the error function, a standard special function:

$$\Phi(x)=\frac{1}{2}\left(1+\operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)\right).$$

This identity is useful because many mathematical libraries provide erf, even if they do not provide a function explicitly named Phi. The standard normal density has no elementary antiderivative, which is why tables, numerical algorithms, or special functions are used to evaluate its CDF.

A worked example

Suppose $Z\sim N(0,1)$ and you want the probability that $Z$ is at most $1$. This is exactly the CDF evaluated at $1$:

$$P(Z\le 1)=\Phi(1)\approx 0.8413.$$

So about $84.13\%$ of the standard normal distribution lies at or below $1$. If instead you want the probability that $Z$ is greater than $1$, use the complementary probability:

$$P(Z>1)=1-\Phi(1)\approx 1-0.8413=0.1587.$$

For a nonstandard normal variable, convert the value to a standard score first. If $X\sim N(10,2^2)$, for example, then

$$P(X\le 12)=P\left(\frac{X-10}{2}\le\frac{12-10}{2}\right)=\Phi(1)\approx 0.8413.$$

Here $2$ is the standard deviation, not the variance. Standardizing is what lets you use the same function $\Phi$ for normal distributions with any mean and positive standard deviation.

Nearby names and notation

Notation or term	Meaning
$\Phi(x)$	Standard normal CDF: $P(Z\le x)$ for $Z\sim N(0,1)$
$\phi(x)$	Standard normal probability density function
$\Phi^{-1}(p)$	Inverse CDF, also called the standard normal quantile function or probit function
$Q(x)$	Often the upper-tail probability $1-\Phi(x)$
$\operatorname{erf}(x)$	Error function, related to $\Phi$ by a change of scale and offset

The exact label can depend on context. “Gaussian CDF” is another common phrase, since “Gaussian” means normal, but it may refer to a normal distribution with any mean and variance. Calling $\Phi$ the standard normal CDF removes that ambiguity.

The word probit needs particular care: it usually names the inverse function $\Phi^{-1}$, not $\Phi$ itself. In statistics, a probit model uses this inverse-normal link. Similarly, $\phi$ (lowercase) generally denotes the density, whereas $\Phi$ (uppercase) denotes its cumulative distribution function.

Common mistakes

• Calling $\Phi$ the density: The density is usually $\phi(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$. The CDF is its integral.
• Treating $\Phi(x)$ as an upper-tail probability: By convention, it is the probability to the left, $P(Z\le x)$. The upper tail is $1-\Phi(x)$, often denoted $Q(x)$.
• Using $\Phi$ without standardizing: For $X\sim N(\mu,\sigma^2)$, calculate $z=(x-\mu)/\sigma$ and then use $\Phi(z)$.
• Confusing the inverse with the CDF: $\Phi^{-1}(p)$ takes a probability and returns a standard-normal quantile; it does not return a probability from a value.

When you need a precise name in writing, use “the standard normal cumulative distribution function, denoted by $\Phi$.”