How many standard deviations from the mean: The z-value (or z-score) of x is z = (x − μ)/σ — how many standard deviations x lies above (z > 0) or below (z < 0) the mean.
A z-value counts how many standard deviations a value is from the mean; the shaded tail is its probability.
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IB-style question — standardise
X ~ N(60, 8²).
Find the z-value of x = 72 and of x = 50.
Step by step
- z = (x − μ)/σ for 72.
- And for 50.
Final answer
z = 1.5 (72 is 1.5σ above the mean); z = −1.25 (50 is 1.25σ below).
Sign tells you the side: Positive z is above the mean, negative z is below — the sign matters.
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Bigger z = better relative position: z-values let you compare results from different normal distributions on a common scale.
The value with the larger z is further above its own mean — a better relative result.
IB-style question — which is better?
On Test A (mean 70, sd 5) a student scores 80.
On Test B (mean 60, sd 10) they score 75.
Relative to each class, which result is better?
Step by step
- z for each.
- Compare.
Final answer
Test A is the better result — its z-value (2) is higher than Test B's (1.5).
Raw scores can mislead: 75 looks lower than 80 but you must compare z-values, not the raw marks, across different tests.
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Standardising gives N(0, 1): Standardising every value turns X ~ N(μ, σ²) into the standard normal Z ~ N(0, 1) (mean 0, sd 1). z = 0 sits at the mean; the same z gives the same probability position in any normal distribution — the key to the inverse normal (next).
IB-style question — interpret a z-value
A value has z = 0.
What can you say about it?
And if z = −2?
Step by step
- z = 0 means x equals the mean.
- z = −2 means 2σ below the mean.
Final answer
z = 0 is exactly the mean; z = −2 is two standard deviations below the mean.
z is unitless: A z-value has no units — it's a count of standard deviations, so it compares across any quantities.