Standardization¶

Definition¶

Let $X$ be a random variable with mean $\mu$ and standard deviation $\sigma>0$ $0$ . The standardization of $X$ is

Z=\frac{X-\mu}{\sigma}.

Interpretation¶

Standardization expresses a value relative to its mean and spread. It makes variables with different units or scales easier to compare.

A normal density with mean 5 and standard deviation 2, shifted and squeezed into the standard normal density centred at 0 with standard deviation 1.

The illustrated calculation gives $Z=\dfrac{X-\mu}{\sigma}$ $\frac{X - μ}{σ}$ . Subtracting $\mu$ centres the distribution at $0$ and dividing by $\sigma$ rescales the spread to $1$ , so $\mathbb{E}[Z]=0$ and $\operatorname{Var}(Z)=1$ $1$ . The $z$ -score measures the distance from the mean in units of $\sigma$ .

Change $\mu$ , $\sigma$ and $x$ : the standardized value $z=(x-\mu)/\sigma$ cuts off exactly the same probability under the standard normal.

Expectation supplies the centring value $\mu$ . Subtracting that value gives mean zero; dividing by the positive standard deviation then gives variance one without asserting that the transformed law is normal.

Key Results¶

Standardization subtracts the mean and divides by the standard deviation.
The standardized variable has mean zero and variance one when the original mean and variance exist.
For normal variables, standardization produces a standard normal variable.

Examples¶

A student's score can be standardized relative to the class mean and standard deviation.
A product weight can be standardized to see how many standard deviations it lies from the target.
Standardizing normal measurement errors allows probabilities to be computed from a standard normal table.

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 5

Aliases: z-transformation, z transformation, standardisation, standardized random variable Keywords: standard score, z-score Symbols: mu, sigma

Last updated: 17 October 2026 21:53