Z-Score Formula

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

The Formula

z=x−μσz = \frac{x - \mu}{\sigma}

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

Quick Example

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

Notation

zz is the standard score; Z∼N(0,1)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−μ)/σz = (x - \mu)/\sigma.

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

Formal View

z=x−μσz = \frac{x - \mu}{\sigma}; equivalently, if X∼N(μ,σ2)X \sim N(\mu, \sigma^2) then Z=X−μσ∼N(0,1)Z = \frac{X - \mu}{\sigma} \sim N(0, 1)

Worked Examples

Example 1

easy
A student scored 8282 on an exam where the mean was 7474 and the standard deviation was 88. What is the student's z-score?

Answer

z=1.0z = 1.0

First step

1
Recall the z-score formula: z=x−μσz = \frac{x - \mu}{\sigma}, which measures how many standard deviations xx is from the mean.

Full solution

  1. 2
    Identify given values: x=82x = 82, μ=74\mu = 74, σ=8\sigma = 8.
  2. 3
    Substitute and calculate: z=82−748=88=1.0z = \frac{82 - 74}{8} = \frac{8}{8} = 1.0
A z-score of 1.01.0 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 7878 (μ=70\mu = 70, σ=5\sigma = 5). On Test B, she scored 8585 (μ=80\mu = 80, σ=10\sigma = 10). On which test did she perform relatively better?

Example 3

easy
A car gets 3232 mpg. The fleet has μ=28\mu = 28 mpg and σ=4\sigma = 4 mpg. Compute the z-score and describe what it means.

Common Mistakes

  • Forgetting to divide by σ\sigma — x−μx-\mu alone is a raw deviation, not a z-score.
  • Dropping the sign — a negative z means below the mean; the sign carries direction.
  • Using the wrong distribution's mean and SD — standardize each value with its own data set's μ\mu and σ\sigma.

Why This Formula Matters

The z-score is the universal ruler of statistics: it strips away units so a 700 SAT and a 30 ACT can be compared fairly, and it is the key into the standard normal table for probabilities. Forgetting to divide by the SD leaves you with a raw distance that means nothing across data sets. Recognizing it by "Am I expressing this value as a number of standard deviations from its mean?" — rather than by familiar numbers — is what lets a student tell it apart from percentile and standard deviation and raw deviation in a mixed problem set.

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−μ)/σz = (x - \mu)/\sigma.

How do you use the Z-Score formula?

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

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

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

Why is the Z-Score formula important in Math?

The z-score is the universal ruler of statistics: it strips away units so a 700 SAT and a 30 ACT can be compared fairly, and it is the key into the standard normal table for probabilities. Forgetting to divide by the SD leaves you with a raw distance that means nothing across data sets. Recognizing it by "Am I expressing this value as a number of standard deviations from its mean?" — rather than by familiar numbers — is what lets a student tell it apart from percentile and standard deviation and raw deviation in a mixed problem set.

What do students get wrong about Z-Score?

The procedure for z-score is the easy part; the trap is forgetting to divide by σ\sigma. Asking "Am I expressing this value as a number of standard deviations from its mean?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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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