Every bell curve in the universe — heights, IQ, errors, exam scores — is the same shape, just shifted and stretched. Squeeze any of them to mean 0 and SD 1 and they all become *one* curve: the standard normal.
The standard normal distribution is the normal distribution with mean 0 and standard deviation 1. Any normal variable can be converted to it by computing a z-score.
Z-tables, hypothesis testing, confidence intervals, comparing scores from different scales, the central limit theorem — the standard normal is the reference ruler of statistics.
z = how many standard deviations x is from the mean. Subtract the mean (shift to 0), divide by σ (stretch to 1).
Exam scores: mean 70, SD 10. You scored 85. What's your z-score?
You scored z = 1.2 on a maths test and z = 1.5 on an English test (different scales). Which was your stronger performance?
The English result — a higher z-score means you stood further above that subject's average, regardless of the raw point scales. Z-scores make incomparable scales comparable.
Z-scores only make sense for roughly normal data. Computing a z-score on a wildly skewed distribution gives a number, but '2 SDs above the mean' won't correspond to the usual ~2.5% tail probability.
The 68-95-99.7 rule lives here: ~68% of the standard normal lies in [−1, 1], ~95% in [−2, 2], ~99.7% in [−3, 3]. Memorise those three and you can eyeball most z-score questions.
- Standard normal = bell curve with mean 0, SD 1.
- z = (x − μ)/σ converts any normal value to it.
- Z-scores let you compare values from different scales — and read tail probabilities from one table.