The normal distribution and z-scores
Mark distances from the mean in standard deviations on a number line, use the 68–95–99.7 rule, work out a z-score, and find the probability in a tail.
- UK: Year 13
- Germany: grade 12
- Afghanistan: grade 12
- Palestine: grade 11
- Russia: grade 11
Guide
This studio follows Afghan grade 12 chapter 7 (the normal distribution, the area under its curve, and standardising) and the UK A-level normal distribution N(μ, σ²).
The number line is the horizontal axis of the bell curve. About 68% of values lie within one standard deviation of the mean, about 95% within two and about 99.7% within three. The z-score z = (x − μ) ÷ σ says how many standard deviations a value is from the mean.
Common mistakes worked on here: using σ for every distance, reading 68% as one side only, forgetting to divide by σ, and forgetting to halve the 5% outside two standard deviations.