Mathematics.

descriptive statistics

Z-Scores

Statistics20 minDifficulty3 out of 10

You should know: variance and standard deviation, normal distribution

Overview

A z-score (or standard score) tells you how many standard deviations a data point lies above or below the mean of its distribution. It is computed by subtracting the mean from a value and dividing by the standard deviation, converting raw data into a common, unit-free scale. Z-scores let you compare values from different distributions (e.g., a test score and a height measurement) on equal footing, and they are the basis for looking up probabilities in the standard normal table. A z-score of 0 means the value equals the mean; positive z-scores lie above the mean, negative ones below.

Intuition

Standardizing via z-scores is like converting different currencies to a common one before comparing prices. A raw score of 85 means nothing without context, but a z-score of +2 immediately tells you the value is two standard deviations above average — unusually high regardless of what was originally measured. This is why z-scores let you compare a student's math test performance to their reading test performance even though the two tests have different scales.

Formal Definition

Definition

For a value x drawn from a population with mean μ and standard deviation σ (or a sample with mean x̄ and standard deviation s):

z=x−μσz = \frac{x - \mu}{\sigma}
Population z-score
z=x−xˉsz = \frac{x - \bar{x}}{s}
Sample z-score
x=μ+zσx = \mu + z\sigma
Recovering the raw value from a z-score

Worked Examples

  1. Apply the z-score formula.

    z=130−10015=3015=2z = \frac{130 - 100}{15} = \frac{30}{15} = 2

Answer: z = 2 (the value is 2 standard deviations above the mean).

Practice Problems

Difficulty 2/10

A population has μ = 50, σ = 5. Find the z-score for x = 60.

Difficulty 3/10

A population has μ = 20, σ = 4. What raw value corresponds to z = -0.75?

Difficulty 5/10

On Exam A (mean 70, sd 10), a student scores 85. On Exam B (mean 60, sd 5), the same student scores 68. On which exam did the student perform relatively better?

Quiz

A z-score of 0 means:
Why are z-scores useful for comparing values from two different distributions?
If μ = 40, σ = 5, and z = -2, what is the raw value x?

Summary

  • A z-score expresses how many standard deviations a value lies from the mean: z = (x − μ)/σ.
  • Positive z-scores are above the mean, negative below; z = 0 is exactly at the mean.
  • Z-scores standardize data from different distributions onto a common scale, enabling direct comparison and normal-table lookups.

References

  1. WebsiteWikipedia — Standard score