Z-Score Formula

The Formula

z = \frac{x - \mu}{\sigma}

When to use: A universal measuring stick—z = 2 means '2 SDs above average.'

Quick Example

Mean = 100, SD = 15. Score of 130 has z = (130 - 100) / 15 = 2

Notation

z is the standard score; Z \sim N(0, 1) is the standard normal distribution

What This Formula Means

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.'

Formal View

z = \frac{x - \mu}{\sigma}; equivalently, if X \sim N(\mu, \sigma^2) then Z = \frac{X - \mu}{\sigma} \sim N(0, 1)

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?

Common Mistakes

  • Subtracting the mean from the standard deviation instead of from the raw score: computing \frac{\mu - x}{\sigma} instead of \frac{x - \mu}{\sigma}
  • Interpreting a negative z-score as an error — it simply means the value is below the mean
  • Forgetting to divide by the standard deviation — just computing x - \mu gives the deviation, not the z-score

Why This Formula Matters

Z-scores standardize measurements from different scales, enabling comparison of apples and oranges and looking up probabilities in standard normal tables.

Frequently Asked Questions

What is the Z-Score formula?

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

How do you use the Z-Score formula?

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

What do the symbols mean in the Z-Score formula?

z is the standard score; Z \sim N(0, 1) is the standard normal distribution

Why is the Z-Score formula important in Math?

Z-scores standardize measurements from different scales, enabling comparison of apples and oranges and looking up probabilities in standard normal tables.

What do students get wrong about Z-Score?

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.

What should I learn before the Z-Score formula?

Before studying the Z-Score formula, you should understand: mean, standard deviation.

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