*33*

In statistics, a **z-score** tells us how many standard deviations away a given value lies from the mean. We use the following formula to calculate a z-score:

**z**Â = (X â€“Â Î¼) / Ïƒ

where:

- X is a single raw data value
- Î¼ is the mean
- Ïƒ is the standard deviation

A z-score for an individual value can be interpreted as follows:

**Positive z-score:Â**The individual value is greater than the mean.**Negative z-score:Â**The individual value is less than the mean.**A z-score of 0:**Â The individual value is equal to the mean.

The larger the absolute value of the z-score, the further away an individual value lies from the mean.

The following example shows how to calculate and interpret z-scores.

**Example: Calculate and Interpret Z-Scores**

Suppose the scores for a certain exam are normally distributed with a mean of 80 and a standard deviation of 4.

**Question 1: **Find the z-score for an exam score of 87.

We can use the following steps to calculate the z-score:

- The mean is Î¼ = 80
- The standard deviation is Ïƒ = 4
- The individual value weâ€™re interested in is X = 87
- Thus, z = (X â€“ Î¼) / ÏƒÂ =Â (87 â€“ 80) /4 =
**1.75**.

This tells us that an exam score of 87 liesÂ **1.75 standard deviations above the mean**.

**Question 2: **Find the z-score for an exam score of 75.

We can use the following steps to calculate the z-score:

- The mean is Î¼ = 80
- The standard deviation is Ïƒ = 4
- The individual value weâ€™re interested in is X = 75
- Thus, z = (X â€“ Î¼) / ÏƒÂ =Â (75 â€“ 80) /4 = â€“
**1.25**.

This tells us that an exam score of 75 lies **1.25 standard deviations below the mean**.

**Question 3: **Find the z-score for an exam score of 80.

We can use the following steps to calculate the z-score:

- The mean is Î¼ = 80
- The standard deviation is Ïƒ = 4
- The individual value weâ€™re interested in is X = 80
- Thus, z = (X â€“ Î¼) / ÏƒÂ =Â (80 â€“ 80) /4 =
**0**.

This tells us that an exam score of 80 isÂ **exactly equal to the mean**.

**Why Are Z-Scores Useful?**

Z-scores are useful because they give us an idea of how an individual value compares to the rest of a distribution.

For example, is an exam score of 87 good? Well, that depends on the mean and standard deviation of all exam scores.

If the exam scores for the whole population are normally distributed with a mean of 90 and a standard deviation of 4, we would calculate the z-score for 87 to be:

z = (X â€“ Î¼) / ÏƒÂ =Â (87 â€“ 90) /4 = **-0.75**.

Since this value is negative, it tells us that an exam score of 87 is actuallyÂ *below* the average exam score for the population. Specifically, an exam score of 87 isÂ **0.75 standard deviations below the mean**.

In a nutshell, z-scores give us an idea of how individual values compare to the mean.

**How to Calculate Z-Scores in Practice**

The following tutorials show step-by-step examples of how to calculate z-scores in different statistical software:

How to Calculate Z-Scores in Excel

How to Calculate Z-Scores in R

How to Calculate Z-Scores in Python

How to Calculate Z-Scores in SPSS