The normal distribution & z-scores
The normal distribution is symmetric and bell-shaped, fully determined by its mean μ and standard deviation σ. Standardising with z = (x − μ)/σ converts any normal variable to the standard normal, which is what makes a single table sufficient. The empirical rule — 68%, 95%, 99.7% within one, two and three SDs — is worth knowing as a sanity check.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Standardisez = (x − μ)/σ measures how many standard deviations x sits from the mean, and carries a sign.
- Look up the cumulative probabilityTables give Φ(z) = P(Z ≤ z). Sketch the region you want before reading anything off.
- Combine areas correctlyP(Z > z) = 1 − Φ(z); P(a < Z < b) = Φ(b) − Φ(a); and Φ(−z) = 1 − Φ(z) by symmetry.
- Work backwards for a valueGiven a probability, find z from the table and invert: x = μ + zσ.
Worked example
Heights are normal with μ = 170 cm and σ = 8 cm. Find P(height > 182).
- Standardise: z = (182 − 170)/8 = 1.5.
- We want P(Z > 1.5) = 1 − Φ(1.5).
- Φ(1.5) ≈ 0.9332.
- 1 − 0.9332 = 0.0668.
Answer. About 6.7% of people are taller than 182 cm — consistent with 182 being 1.5 SDs above the mean.
Where marks get dropped
These are the specific errors that cost credit on the normal distribution & z-scores questions — QED's rubric penalises each of them separately.
- Forgetting to divide by σ when standardising, which turns the z-score into a raw deviation.
- Reading the table as the upper tail. Standard tables give P(Z ≤ z); the upper tail needs 1 − Φ(z).
- Applying normal methods to obviously skewed data. Standardising does not make a distribution normal.
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The normal distribution & z-scores — frequently asked questions
What does a z-score of −2 mean?
The value sits two standard deviations BELOW the mean. Roughly 2.5% of a normal distribution lies below that point.
Why is the normal distribution so common?
Because of the central limit theorem: sums and averages of many independent contributions are approximately normal whatever the underlying distribution.
Can I use z-scores on non-normal data?
You can compute them as a standardised measure of position, but the probability interpretations require normality — or a large enough sample for the CLT.
The rest of Statistics
Describing data, distributions, estimation and hypothesis tests. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Mean, median & mode
- 2Variance, standard deviation & spread
- 3Shape, skew & outliers
- 4Boxplots, histograms & quartiles
- 5The normal distribution & z-scores
- 6Sampling, bias & the sampling distribution
- 7The central limit theorem
- 8Confidence intervals for a mean
- 9Hypothesis testing & p-values
- 10t-tests & comparing two means
- 11Chi-square tests for independence
- 12Correlation & least-squares regression
- 13Type I / Type II errors & power
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