Normal Scores

TL;DR

• Expresses a score’s distance from the dataset mean in standard-deviation units.
• Computed by subtracting the mean from the score and dividing by the standard deviation.
• Enables comparison of scores within a dataset or across different datasets.

Definition

Normal scores, also known as standard scores or z-scores, measure how far a particular score falls from the mean of a dataset. They are calculated by subtracting the mean from the score and dividing the result by the standard deviation.

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

Explanation

A normal score converts a raw value into a relative position within the distribution by:

• Taking the deviation of the score from the dataset mean (score − mean).
• Dividing that deviation by the dataset standard deviation.

This standardization places scores on a common scale in units of standard deviations, which makes it easier to interpret how unusually high or low a score is and to compare scores across different datasets or measures.

Examples

Math test example

A group of students took a math test with mean 80 out of 100 points.

• John’s score: 95
  John’s deviation: 95 - 80 = 15
  John’s normal score: 15 / 10 (standard deviation) = 1.5

• Jane’s score: 65
  Jane’s deviation: 65 - 80 = -15
  Jane’s normal score: -15 / 10 (standard deviation) = -1.5

These normal scores show that John is 1.5 standard deviations above the mean and Jane is 1.5 standard deviations below the mean.

Intelligence test example

A group of people took an intelligence test with mean 100 points.

• Bob’s score: 120
  Bob’s deviation: 120 - 100 = 20
  Bob’s normal score: 20 / 15 (standard deviation) = 1.33

• Sarah’s score: 80
  Sarah’s deviation: 80 - 100 = -20
  Sarah’s normal score: -20 / 15 (standard deviation) = -1.33

These normal scores show that Bob is 1.33 standard deviations above the mean and Sarah is 1.33 standard deviations below the mean.

Use cases

• Commonly used in psychology and education to assess individual performance relative to a group.
• Used in statistical analysis to standardize data and enable comparisons across groups or datasets.

Related terms

• Standard scores
• Z-scores