What Is a Z-Score?
A z-score, also called a standard score, tells you how many standard deviations a particular value (\(x\)) is above or below the mean (\(\mu\)) of a distribution. A positive z-score means the value is above the mean; a negative z-score means it is below. A z-score of 0 means the value equals the mean. Because z-scores remove the original units, they let you compare scores from completely different scales — for example, comparing a test score to a height measurement.
How to Use This Calculator
Enter three numbers: the raw score (\(x\)) you want to standardize, the mean (\(\mu\)) of the dataset, and the standard deviation (\(\sigma\)). The calculator returns the z-score instantly. Make sure the standard deviation is greater than zero — division by zero is undefined.
The Formula Explained
The z-score is computed as $$z = \frac{\text{Raw Score }(x) - \text{Mean }(\mu)}{\text{Std Dev }(\sigma)}$$ First subtract the mean from your value to find the raw deviation, then divide by the standard deviation to express that deviation in standard-deviation units. A z of +1.5 means the value sits 1.5 standard deviations above the mean.
Worked Example
Suppose a student scores 85 on a test where the class mean is 70 and the standard deviation is 10. Then $$z = \frac{85 - 70}{10} = \frac{15}{10} = 1.5$$ The student scored 1.5 standard deviations above average — better than roughly 93% of the class under a normal distribution.
FAQ
What does a negative z-score mean? It means the value is below the mean. For example, \(z = -2\) is two standard deviations below average.
What is a "good" z-score? It depends on context, but in a normal distribution about 68% of values fall between \(z = -1\) and \(z = +1\), and about 95% between \(-2\) and \(+2\).
Can I convert a z-score back to a raw score? Yes. Rearranging the formula gives \(x = \mu + z\cdot\sigma\).
