Values on different scales can be compared by expressing distance from the mean in standard deviations.
A student scores 85 on Exam A and 92 on Exam B. The second score is numerically higher, but the exams may have different means and spreads.
A z-score states how many standard deviations an observation lies above or below the mean.
Key question
How can values from different scales be placed on one common scale and converted into normal probabilities?
What does a z-score mean?
Standard score
z = (x − μ) ÷ σ
| z-score | Meaning |
|---|---|
| 0 | equal to the mean |
| +1 | one standard deviation above the mean |
| −2 | two standard deviations below the mean |
Why standardize?
Raw scores are tied to their units and scales. z-scores place them on a common scale with mean 0 and standard deviation 1.
Exam A
score 85 · mean 70 · SD 10
z = 1.5
Exam B
score 92 · mean 85 · SD 5
z = 1.4
What standardization does not change
The ordering and shape of the distribution remain the same. Only location and scale are transformed.
How is probability obtained from a z-score?
Once a normal value is converted to z, the standard normal distribution can be used to calculate tail or interval probabilities.
| Question | Probability expression |
|---|---|
| Probability below x | P(Z ≤ z) |
| Probability above x | P(Z ≥ z) |
| Probability between two values | P(z₁ ≤ Z ≤ z₂) |
Symmetry
The standard normal distribution is symmetric around zero, so P(Z ≤ −z) = P(Z ≥ z).
Comparing scores from different exams
Exam A gives z = 1.5 and Exam B gives z = 1.4. The raw score is higher on B, but the relative performance is slightly stronger on A.
Probability interpretation
z = 1.5 corresponds to approximately the 93.3rd percentile of the standard normal distribution.
Change the mean, standard deviation, and observation
Adjust μ, σ, and x to see the z-score and tail probability update.
μ
70.0
σ
10.0
z
1.50
Probability
6.7%
Key lesson
A z-score turns a raw value into a relative position within its distribution.