Standardize values or compute z-scores for an entire dataset. Convert any value to standard deviations from the mean.
Last updated: March 2026
A z-score (also called a standard score) measures how many standard deviations a value is from the mean of a dataset. It's calculated as: z = (x − μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
Z-scores allow you to standardize values from different distributions, making them directly comparable. A z-score of 0 means the value equals the mean, positive scores are above the mean, and negative scores are below the mean. Z-scores are fundamental in statistics and are used in hypothesis testing, creating control charts, and identifying outliers.
In a normal distribution, approximately 68% of values fall within 1 standard deviation (z = ±1), 95% within 2 standard deviations (z = ±2), and 99.7% within 3 standard deviations (z = ±3). This is known as the empirical rule.
| z = 0 | Value equals the mean |
| z > 0 | Value is above the mean |
| z < 0 | Value is below the mean |
| |z| > 3 | Potential outlier (rare value) |
Scenario: Test scores have μ = 80 and σ = 10. What's the z-score for someone who scored 95?
Calculation
Interpretation: A score of 95 is 1.5 standard deviations above the mean, placing it in the upper tail of the distribution. This score is better than approximately 93% of scores.
Use z-scores when you want to compare values from different scales or distributions, identify outliers, or standardize data for statistical tests.
If you have data from an entire population, use the population standard deviation. If you have a sample, use the sample standard deviation (dividing by n-1). Most real-world scenarios use sample standard deviation.
Yes! A negative z-score simply means the value is below the mean. The sign indicates direction—negative means lower, positive means higher than average.
A z-score greater than 3 (or less than -3) is considered unusual. In a normal distribution, this occurs less than 0.3% of the time, so it often indicates an outlier.
A z-score measures standard deviations from the mean (standardized value), while a percentile shows the percentage of data below that value. A z-score of 0 corresponds to the 50th percentile.
Yes! Use the formula: x = μ + (z × σ). This is the reverse of the z-score calculation and restores the original scale.
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