Normal Distribution and z-Scores: How to Read a Percentile

Many natural and measured quantities form a bell: common in the middle, rare at the tails, roughly symmetric, fully determined by the mean μ (centre) and standard deviation σ (spread).

A larger σ flattens the curve. Same mean and same σ means the same whole curve — which is why a mean alone is never enough.

• About 68.3% of values fall within μ ± 1σ
• About 95.4% within μ ± 2σ
• About 99.7% within μ ± 3σ

So "two standard deviations above the mean" leaves roughly 2.3% above you ((100 − 95.4) ÷ 2 ≈ 2.3%). Remember to halve it — you want one tail, not both.

z = (x − μ) ÷ σ counts standard deviations from the mean. Common conversions:

• z = 0 → 50th percentile
• z = 1 → 84.1st
• z = 1.5 → 93.3rd
• z = 1.96 → 97.5th (the two-sided 95% cutoff)
• z = −1 → 15.9th

Class mean 70, SD 10, you scored 85:

z = (85 − 70) ÷ 10 = 1.5 → about the 93.3rd percentile, above roughly 93% of the class.

In another class with mean 75 and SD 5, the same 85 gives z = 2 → 97.7th percentile. The same raw score means something different in each class — which is why raw marks alone cannot rank people.

• Strong skew: income, house prices and dwell time are right-skewed; normal-based percentiles mislead badly.
• Small samples: an SD estimated from a dozen points is unstable, so z-scores are too.
• Outliers: one extreme value inflates both mean and SD and flattens everyone else’s z.
• Non-independent data: repeated measures on the same people change the standard error and assumptions.

Safer with enough data: use the empirical percentile (rank after sorting) — it assumes no distribution at all.