# Normal probabilities and percentiles

To find a probability, convert to z and read the area. To find a percentile, work backwards: find the z for the area, then compute X = mu + z x sigma.

Statistics (university) · Part 04: Continuous distributions and the normal curve · Card 6 of 8 · Last updated 2026-10-19

## Analogy

Going from a score to a rank is a lookup. Going from a rank to a score is the same lookup run backwards.

## Example

**IQ scores**

Mean 100, SD 15
P(IQ > 130): z = 2, tail = 0.0228
90th percentile: z = 1.2816
X = 100 + 1.2816 x 15 = 119.2

## In practice

Used to set cut-off scores, such as the mark needed to be in the top 10 percent.

## Practice

### Flashcards

- **z value that cuts off 2.5% in the upper tail?** 1.96.

### 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

### What is a percentile?

The kth percentile is the value below which k percent of the data falls. The 90th percentile of IQ scores is about 119, meaning 90 percent of people score lower. The median is the 50th percentile. Percentiles are used in growth charts, exam rankings and income reports.

### How do I find the middle 95 percent of a normal distribution?

Use z values of -1.96 and +1.96, which leave 2.5 percent in each tail. Then convert back with X = mu + z x sigma. For mean 100 and SD 15, the interval is 100 plus or minus 29.4, which runs from 70.6 to 129.4.

## Related concepts

- [Standard normal and z-scores](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-4/statistics-university-part-4-concept-5)
- [Normal distribution](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-4/statistics-university-part-4-concept-4)
- [Normal approximation to the binomial](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-4/statistics-university-part-4-concept-7)
- [Sums of normal variables](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-4/statistics-university-part-4-concept-8)

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