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The standard normal distribution is a normal distribution with mean 0 and variance 1. Normally distributed random variables are frequently converted to standard normal random variables by subtracting the mean μ, then dividing by the standard deviation σ. This allows random variables that are normally distributed but with different mean and variance to be compared to each other.

Probability Density Function

f ( x ) = 1 2 π e − x 2 / 2

Support

− ∞ < x < ∞

Mean

Variance

Example
The height of adult men in the US is normally distributed with mean 69.1 inches and standard deviation 2.8 inches. Let Y be the height of a randomly chosen man, and let X = (Y - 69.1)/2.8. Then X is a standard normal random variable.
The GPA of college students has a normal distribution with mean 3.15 and standard deviation 0.2. Let Y be the GPA of a randomly chosen student, and let X = (Y - 3.15)/0.2. Then X is a standard normal random variable.
The length of adult women's feet in the US is normally distributed with mean 9.58 inches and standard deviation 0.51 inches. Let Y be the foot length of a randomly chosen woman, and let X = (Y - 9.58)/0.51. Then X is a standard normal random variable.

X ~ Standard Normal

Chart of the standard normal distribution Chart area for displaying the standard normal pdf, cdf, visualization, and simulation

E(X) = , Var(X) =

An approximation often used with the standard normal distribution is that the probability of being within one, two, or three standard deviations of the mean are 68%, 95%, and 99.7% respectively.

The illustration above shows a sample of n points chosen independently and at random from a uniform(-√(3n), √(3n)) distribution. The population has mean 0 and variance n while the sample mean X̅ (shown as a light blue line) has mean 0 and variance 1. By the Central Limit Theorem, X̅ converges to a standard normal distribution as n approaches infinity.

Note that the Central Limit Theorem holds for the sample mean of any distribution, as long as the variance is finite.

The simulation above shows a sample of n points (marked red) chosen independently and at random from a uniform(-√(3n), √(3n)) distribution with mean 0 and variance n. The light blue line shows the sample mean X̅, which itself has mean 0 and variance 1. By the Central Limit Theorem, X̅ converges to a standard normal distribution as n approaches infinity. The histogram accumulates the results of each simulation.

Note that the Central Limit Theorem holds for the sample mean of any distribution, as long as the variance is finite. A uniform distribution has been used here for simplicity.

Y = X² ~ Chi-Squared(1) Σ Xᵢ ~ Normal(0, n) μ + σX ~ Normal(μ, σ²) X₁/X₂ ~ Standard Cauchy

Chart of the related distribution Chart area for displaying the related pdf, cdf, visualization, and simulation

E(Y) = , Var(Y) =

Note that the support of X is (-∞, ∞), so the support of Y is [0, ∞).

This follows since the normal distribution is a location-scale family of distributions.

This follows since the normal distribution is a location-scale family of distributions.

The standard Cauchy distribution is therefore a ratio distribution. Such distributions are often heavy-tailed, and in this case the expected value and variance are undefined.