The **standard normal distribution** is a
special case of the normal distribution. For the standard normal
distribution the value of the mean is equal to zero (\(\mu = 0\)) and the value of the standard
deviation is equal to 1 (\(\sigma =
1\)).

Thus, by plugging in \(\mu = 0\) and
\(\sigma = 1\) in the PDF of the normal
distribution, the equation simplifies to:

\[\begin{align}
f(x)& = \frac{1}{\sigma \sqrt{2
\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} \\
& =\frac{1}{1 \times \sqrt{2
\pi}}e^{-\frac{1}{2}\left(\frac{x-0}{1}\right)^2} \\
& = \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}
\end{align}\]

A random variable that follows the standard normal distribution is
denoted by \(z\). Consequently, the
units for the standard normal distribution curve are denoted by \(z\) and are called the **\(z\)-values** or **\(z\)-scores**. They are also called
**standard units** or **standard scores**.

The **cumulative distribution function (CDF)** of the
standard normal distribution, corresponding to the area under the curve
for the interval \((-\infty, z]\), is
usually denoted with the capital Greek letter \(\phi\) and given by:

\[F(x<z) = \phi (z) =
\frac{1}{\sqrt{2\pi}}
\int_{-\infty}^{z}e^{-\frac{1}{2}x^2}dx\]

where \(e \approx 2.71828\) and
\(\pi \approx 3.14159\).

#### Basic Properties of the Standard Normal Curve

The standard normal curve is a special case of the normal
distribution and thus also a probability distribution curve. Therefore,
basic properties of the normal distribution hold true for the standard
normal curve as well (Weiss 2010).

- The total area under the standard normal curve is 1 (this property
is shared by all density curves).
- The standard normal curve extends indefinitely in both directions,
approaching, but never touching, the horizontal axis as it does so.
- The standard normal curve is bell shaped and centered at \(z=0\). Almost the whole area under the
standard normal curve lies between \(z=-3\) and \(z=3\).

The \(z\)-values on the right side
of the mean are positive and those on the left side are negative. The
\(z\)-value for a point on the
horizontal axis gives the distance between the mean (\(z=0\)) and that point in terms of the
standard deviation. For example, a point with a value of \(z=2\) is two standard deviations to the
right of the mean. Similarly, a point with a value of \(z=-2\) is two standard deviations to the
left of the mean.