What Is Z-Score?
A z-score standardizes a value by describing how many standard deviations it lies above or below the mean of its distribution. It converts measurements from different scales into comparable units, which is why it is also called a standard score.
A positive z-score means the value is above the mean, and a negative z-score means it is below. Under the normal distribution, z-scores link directly to percentiles, making them useful for comparing an individual result with a reference group and for locating values on the standard normal curve.
Formula
z = (x − μ) / σ
where:
x = the value being standardized
μ = population mean
σ = population standard deviation
Applications
- Comparing exam scores that come from different test versions
- Locating a measurement on the normal distribution to find its percentile
- Identifying outliers as values with unusually large |z|
- Standardizing variables before combining them in an analysis
Sources
- Moore, D. S., McCabe, G. P., & Craig, B. A. (2021). Introduction to the Practice of Statistics (10th ed.). Macmillan.
- Montgomery, D. C., & Runger, G. C. (2018). Applied Statistics and Probability for Engineers (7th ed.). Wiley.