Z-Score (Standard Score) Examples in Statistics

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

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

Concept Recap

The number of standard deviations a value is from the mean: z = \frac{x - \mu}{\sigma}.

Z-scores put everything on the same scale. A z-score of +2 means 'two standard deviations above average' - unusually high. A z-score of -1 means 'one SD below average' - somewhat low but normal.

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: A z-score converts any value to a standardized number of standard deviations from the mean, allowing fair comparison across different scales and units.

Common stuck point: Students interpret z-scores as percentages. A z-score of 2 does not mean 2% — you need a normal distribution table to convert it to a percentile.

Worked Examples

Example 1

medium
A student scores 78 on a test where \mu = 70 and \sigma = 4. Calculate her z-score and interpret it.

Solution

  1. 1
    Step 1: z = \frac{x - \mu}{\sigma} = \frac{78 - 70}{4} = 2.
  2. 2
    Step 2: A z-score of 2 means the student scored 2 standard deviations above the mean.
  3. 3
    Step 3: By the empirical rule, only about 2.5% of students scored higher.

Answer

z = 2. The student scored 2 standard deviations above the mean.
Z-scores standardise values to a common scale, telling us how many standard deviations a value is from the mean. This allows comparison across different distributions.

Example 2

hard
Alice scores 85 in Maths (\mu = 75, \sigma = 5) and 90 in English (\mu = 80, \sigma = 10). In which subject did she perform better relative to her class?

Practice Problems

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

Example 1

medium
A data value has a z-score of -1.5. If the distribution has \mu = 50 and \sigma = 8, find the original value.

Example 2

medium
In a class, test scores have mean \mu = 65 and standard deviation \sigma = 5. A student scores 55. Find the z-score and interpret it.
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Background Knowledge

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

standard deviation intromean