Z-Normalize a List Calculator
Standardize a list of numbers to z-scores — subtract the mean and divide by the standard deviation so the data has mean 0 and SD 1.
How it works
z = (x − μ) ÷ σ · sample σ uses n − 1, population σ uses n
Z-normalization (standardization) rescales a list so it is centred on zero with a standard deviation of one, letting you compare values from different scales on a common footing. A z-score of +1 means the value sits one standard deviation above the mean; a negative z-score is below it. This is the standard preprocessing step before many statistical tests and machine-learning models, which assume features share a comparable scale.
Worked example
For 10, 12, 14, 16, 18 the mean is 14 and the sample SD is √10 = 3.162. The value 18 becomes (18 − 14) ÷ 3.162 = +1.265, and 10 becomes (10 − 14) ÷ 3.162 = −1.265. The middle value 14 maps to exactly 0.
Sources: Standard score — Wikipedia
Frequently asked questions
What is the difference between sample and population standard deviation here?
Sample SD divides by n − 1 (use it when your data is a sample of a larger group); population SD divides by n (use it when the list is the entire population). Sample SD is slightly larger.
Why does the calculator refuse a list of identical numbers?
Identical values have a standard deviation of zero, and dividing by zero makes every z-score undefined, so the transformation cannot be applied.
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This calculator is for educational purposes. Double-check important results before acting on them.