Normal distribution

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

Empirical Rule

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.