Normal Distribution and Z-Scores

Manual: General · Subject: Statistics

Learn the shape of the normal distribution and how to standardize values with z-scores.

A common bell-shaped pattern

The normal distribution is a symmetric, bell-shaped distribution often used to model natural measurements such as heights and test scores.

A z-score tells how many standard deviations a value is from the mean: $z=\frac{x-\mu}{\sigma}$

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How to interpret $z$

A positive $z$ is above the mean, a negative $z$ is below the mean, and $z=0$ is exactly at the mean.

Example z-scores

Value	Interpretation
$z=1$	1 standard deviation above the mean
$z=-2$	2 standard deviations below the mean
$z=0$	At the mean

What does a z-score of $-1.5$ mean?

1.5 units above the mean 1.5 standard deviations below the mean The value is impossible The value is exactly the mean

What is the formula for a z-score?

Which feature is true of the normal distribution?

It is always skewed left It is symmetric It has no center It only applies to categories

Why are z-scores useful?