Z Score

TL;DR

• Expresses how far a data point lies from the dataset mean in units of the standard deviation.
• Enables comparison of values within a dataset and across different datasets by standardizing scores.
• Can be used to estimate how likely a value is to occur (examples in source: ~68% and ~95% bands).

Definition

A z-score, also known as a standard score, is a measure of how many standard deviations a particular data point is from the mean of a set of data. It is used to compare how different a particular data point is from the mean of the data set and to determine how likely it is to occur.

The z-score is computed by subtracting the mean from the observation and dividing by the standard deviation:

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

Explanation

A positive z-score indicates the data point lies above the mean; a negative z-score indicates it lies below the mean. Converting observations to z-scores standardizes values so that they can be compared directly, even when they come from different datasets with different means and standard deviations. Z-scores can also be interpreted in terms of how likely a value is to occur within the distribution (as illustrated in the examples).

Examples

Heights example

For a group whose mean height is 5 feet 6 inches and standard deviation is 2 inches:

• A person who is 6 feet tall:

$$z = \frac{72 - 66}{2} = 3$$

This means the person’s height is 3 standard deviations above the mean.

• A person who is 5 feet 2 inches tall:

$$z = \frac{62 - 66}{2} = -2$$

This means the person’s height is 2 standard deviations below the mean.

Test scores example

Comparing scores from two groups that took the same test:

• First group: mean 80, standard deviation 10. A student who scored 90:

$$z = \frac{90 - 80}{10} = 1$$

• Second group: mean 70, standard deviation 5. A student who scored 90:

$$z = \frac{90 - 70}{5} = 4$$

Although both students scored 90, the student in the second group has a higher z-score, indicating their score is more unusual relative to their group’s mean.

Likelihood example

For a dataset with mean 50 and standard deviation 10:

• A data point with z-score 1 is 1 standard deviation above the mean and “is likely to occur approximately 68% of the time.”
• A data point with z-score 2 is 2 standard deviations above the mean and “is likely to occur approximately 95% of the time.”
• A data point with z-score -1 is 1 standard deviation below the mean and “is likely to occur approximately 32% of the time.”
• A data point with z-score -2 is 2 standard deviations below the mean and “is likely to occur approximately 5% of the time.”

Use cases

• Comparing individual data points to the mean of a dataset.
• Standardizing and comparing data from different datasets or populations.
• Estimating how likely a particular data point is to occur within a distribution.

Related terms

• Standard score (synonym)
• Mean
• Standard deviation