*12*

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

**z** = (X – μ) / σ

where X is the value we are analyzing, μ is the mean, and σ is the standard deviation.

**A z-score can be positive, negative, or equal to zero.**

A positive z-score indicates that a particular value is greater than the mean, a negative z-score indicates that a particular value is less than the mean, and a z-score of zero indicates that a particular value is equal to the mean.

A few examples should make this clear.

**Examples: Calculating a Z-Score**

Suppose we have the following dataset that shows the height (in inches) of a certain group of plants:

5, 7, 7, 8, 9, 10, 13, 17, 17, 18, 19, 19, 20

The sample mean of this dataset is **13** and the sample standard deviation is **5.51**.

**1. Find the z-score for the value “8” in this dataset.**

Here is how to calculate the z-score:

**z** = (X – μ) / σ = (8 – 13) / 5.51 = **-0.91**

This means that the value “8” is 0.91 standard deviations *below *the mean.

**2. Find the z-score for the value “13” in this dataset.**

Here is how to calculate the z-score:

**z** = (X – μ) / σ = (13 – 13) / 5.46 = **0**

This means that the value “13” is exactly equal to the mean.

**3. Find the z-score for the value “20” in this dataset.**

Here is how to calculate the z-score:

**z** = (X – μ) / σ = (20 – 13) / 5.46 = **1.28**

This means that the value “20” is 1.28 standard deviations *above* the mean.

**How to Interpret Z-Scores**

A Z Table tells us what percentage of values fall below certain Z-scores. A few examples should make this clear.

**Example 1: Negative Z-Scores**

Earlier, we found that the raw value “8” in our dataset had a z-score of **-0.91**. According to the Z Table, 18.14% of values fall below this value.

**Example 2: Z-Scores equal to zero**

Earlier, we found that the raw value “13” in our dataset had a z-score of **0**. According to the Z Table, 50.00% of values fall below this value.

**Example 3: Positive Z-Scores**

Earlier, we found that the raw value “20” in our dataset had a z-score of **1.28**. According to the Z Table, 89.97% of values fall below this value.

**Conclusion**

Z-scores can take on any value between negative infinity and positive infinity, but most z-scores fall within 2 standard deviations of the mean. There’s actually a rule in statistics known as the Empirical Rule, which states that for a given dataset with a normal distribution:

**68%**of data values fall within one standard deviation of the mean.**95%**of data values fall within two standard deviations of the mean.**99.7%**of data values fall within three standard deviations of the mean.

The higher the absolute value of a z-score, the further away a raw value is from the mean of the dataset. The lower the absolute value of a z-score, the closer a raw value is to the mean of the dataset.

**Related Topics:**

Empirical Rule Calculator

How to Apply the Empirical Rule in Excel