# Normal distribution

By Dan Kernler [CC BY-SA 4.0 ], from Wikimedia Commons ## Test comparison

Consider the following test data. Each form was normally distributed.

Student Raw score Form Mean Standard deviation
Jonce 90 A 80 15
Tashika 85 B 75 5

1. For each test form, what are the test scores that contain the middle 68% of the data?

2. For each test form, what score would a student need to be at the 84th percentile?

3. What are the standard scores for each student? (see next section)

4. Which student scored better than other test takers? Why?

## Standard scores (aka. z-scores)

A standard score (or z-score) measures how many standard deviations a score is from the mean. It is computed by…

$$z = \dfrac{x - \mu}{\sigma}$$

… where $$x$$ is a data point, $$\mu$$ is the mean, $$\sigma$$ is the standard deviation. In the normal distribution above, $$\mu$$ has $$z=0$$, $$\mu + \sigma$$ has $$z = 1$$, and so on.

### Curving a test

An exam was normally distributed with μ = 75 and σ = 17.6. The professor wants to base grades on standard deviations. Please complete the following:

Grade Z-score Exam score needed Percentile
A 2
B 1
C 0
D -1

Note: We use the 68-95-99.7 rule because more accurate results require conversion tables or calculus. Also see Percentile calculator.