Z-Score Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Z-Score.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A z-score measures how many standard deviations a data value is above or below the mean: z = (x - \mu)/\sigma.

A universal measuring stick—z = 2 means '2 SDs above average.'

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Z-scores let you compare values from different distributions.

Common stuck point: A z-score of +2 means the value is 2 standard deviations above the mean — it does not mean 2% probability or 2 units away on the original scale.

Sense of Study hint: Try saying it aloud: 'My value is ___ away from the mean, and one SD is ___.' Divide the first blank by the second.

Worked Examples

Example 1

easy
A student scored 82 on an exam where the mean was 74 and the standard deviation was 8. What is the student's z-score?

Solution

  1. 1
    Recall the z-score formula: z = \frac{x - \mu}{\sigma}, which measures how many standard deviations x is from the mean.
  2. 2
    Identify given values: x = 82, \mu = 74, \sigma = 8.
  3. 3
    Substitute and calculate: z = \frac{82 - 74}{8} = \frac{8}{8} = 1.0

Answer

z = 1.0
A z-score of 1.0 means the student scored exactly one standard deviation above the mean. Z-scores allow comparison across different scales.

Example 2

medium
On Test A, Maria scored 78 (\mu = 70, \sigma = 5). On Test B, she scored 85 (\mu = 80, \sigma = 10). On which test did she perform relatively better?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A data point has value 45 in a distribution with \mu = 50 and \sigma = 4. Find its z-score.

Example 2

medium
In a class, test scores have mean 70 and standard deviation 8. What raw score corresponds to a z-score of 1.25?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

meanstandard deviation