Standard normal and z-scores
The standard normal has mean 0 and SD 1. Any normal value X is converted by Z = (X - mu) / sigma. Then a z table or calculator gives the area to the left, P(Z <= z).
Last updated
Analogy
Converting everything to the same ruler so a single table works for every bell curve.
Example
Heights mean 170, SD 8P(height < 182)z = (182 - 170) / 8 = 1.5Table: P(Z <= 1.5) = 0.9332About 93% are shorter
In practice
Lets one table answer questions about heights, test scores and machine weights alike.
Practice
Flashcards
- How do you standardise a normal value?Z = (X - mean) / SD.
Interview questions
- Scores are normal with mean 100 and SD 15. Find the probability of a score above 130 and the 90th percentile.
- z = 2 gives tail 0.0228
- 90th percentile has z about 1.28
- X = 100 + 1.28 x 15 = 119.2
- Standardise first
Questions people ask
How do I use a z table?
Convert your value to a z-score, then look up that z in the table to read the area to its left. For an area on the right, subtract the table value from 1. For a range, subtract the left area at the lower z from that at the upper z.
Why standardise before using normal tables?
There are infinitely many normal curves, one for each mean and SD, and no one can print them all. Standardising slides and rescales any normal curve into the single standard one, so one table covers all cases. Students then only need to learn one lookup method and one conversion formula.