Can a Z-Score Be Negative?


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 of a negative z-score

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.

Z-score equal to zero

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.

Positive z-score example

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

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