When to use it
Use the normal distribution to compute probabilities and percentiles for data that is approximately bell-shaped — heights, measurement errors, standardized test scores — or for sample means, which are approximately normal by the Central Limit Theorem even when the raw data is not.
z-scores and the empirical rule
A z-score rescales any value to “number of standard deviations from the mean”: z = (x − μ) / σ. This lets you compare values from different normal distributions and read probabilities from a single standard table. To standardize a specific value, use the z-score calculator, or look up areas in the z-table.
Worked example
IQ scores are modelled as normal with μ = 100, σ = 15. The probability of scoring below 130 is the area to the left of z = 2, about 0.977 — so roughly 2.3% score above 130. Enter μ, σ, and a cutoff above to compute any such area.
Common mistakes
- Assuming normality — skewed or bounded data (counts, proportions, waiting times) is often not normal; check a histogram first.
- Confusing the density (PDF) with probability (CDF) — a single point has zero probability for a continuous variable; probabilities are areas under the curve.
- One-tail vs two-tail errors when reading “extreme” probabilities.
If your data is not normal
Consider a transformation (log, square-root), or use distribution-free methods. You can explore how other distributions behave in the interactive distribution explorer. Details are on the methodology page.