Track 5 · Probability Distributions · Lesson 7 of 10
Standardising and normal calculations
Convert values to z-scores and use the standard normal distribution to solve forward (value → probability) and inverse (probability → value) problems.
A packing line fills 1 kg bags of flour with some variation, a university awards scholarships to the top 5% of an exam, and a race organiser wants to know what share of runners finish in under an hour. Every one of these questions is answered the same way: translate the value into standard deviations from the mean, then read one universal curve.
There are infinitely many normal distributions — one for every mean and standard deviation (see the normal distribution). Fortunately they all have exactly the same shape. Shift a normal curve so that its centre is at 0 and rescale the axis so that one standard deviation is one unit, and every one of them becomes the standard normal distribution, .
The tool for that shift-and-rescale is the z-score: how many standard deviations a value lies above (positive) or below (negative) the mean.
Included with a Formula Quest membership
It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:
- Standardise a value with z = (x − μ)/σ and interpret it in standard deviations
- Find lower-tail, upper-tail and interval probabilities for any normal distribution
- Solve inverse problems: find the value that cuts off a given percentage, using x = μ + zσ
- Compare values from different scales with z-scores
- Check a fitted normal model against the actual proportions in a dataset