SL 4.12—Z values, inverse normal to find mean and standard deviation: IB DP Math AA Notes

What is a Z-value (Standard Score)?

Z-value (Z-score)

A z-value represents the number of standard deviations a particular data point lies away from the mean of its distribution. It places any value from any normal distribution onto a single, universal scale.

The formula for calculating a z-value is:

$z = \frac{x - μ}{σ}$

Where:

$x$ is the raw data value
$μ$ is the mean of the distribution
$σ$ is the standard deviation of the distribution

Standardization is powerful because it lets us compare values from completely different normal distributions on the same scale — the standard normal distribution $Z \sim N (0, 1)$ , which has mean 0 and standard deviation 1.

Analogy

Think of z-scores like converting currencies. Whether you start with euros, dollars, or yen, converting everything to a single currency lets you compare amounts directly. Z-scores are the "universal currency" of normal distributions.

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What is a Z-value (Standard Score)?

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