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The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution. The density φ ( z ) {\displaystyle \varphi (z)} has its peak 1 / 2 π {\displaystyle 1/{\sqrt {2\pi }}} at z = 0 {\displaystyle z=0} and inflection points at z = + 1 {\displaystyle z=+1} and z = − 1 {\displaystyle z=-1} .

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However, many other distributions are bell-shaped (such as the Cauchy, Student's t, and logistic distributions). If Z {\displaystyle Z} is a standard normal deviate, then X = σ Z + μ {\displaystyle X=\sigma Z+\mu } will have a normal distribution with expected value μ {\displaystyle \mu } and standard deviation σ {\displaystyle \sigma } .The parameter μ {\displaystyle \mu } is the mean or expectation of the distribution (and also its median and mode), while the parameter σ {\displaystyle \sigma } is its standard deviation. The probability density must be scaled by 1 / σ {\displaystyle 1/\sigma } so that the integral is still1. Although the density above is most commonly known as the standard normal, a few authors have used that term to describe other versions of the normal distribution.

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Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases.The univariate probability distribution is generalized for vectors in the multivariate normal distribution and for matrices in the matrix normal distribution. Conversely, if X {\displaystyle X} is a normal deviate with parameters μ {\displaystyle \mu } and σ 2 {\displaystyle \sigma