Z-Score Calculator
This calculator computes the z-score (standard score) for a value in a normal distribution and converts it to an approximate percentile rank. A z-score tells you how many standard deviations a particular observation sits above or below the mean of its distribution. The formula is z = (X minus mu) divided by sigma, where X is your observed value, mu is the population mean, and sigma is the standard deviation. A positive z-score means the value is above average; a negative z-score means it is below average. Because z-scores are dimensionless, they let you compare values that come from distributions with different means and spreads, such as comparing a test score in mathematics with a score in reading. The percentile rank tells you what proportion of the distribution falls below your value. For a standard normal distribution, a z-score of 1.00 corresponds to approximately the 84th percentile, meaning 84% of observations fall below that point. To use the calculator, enter the observed value X, the mean mu, and the standard deviation sigma. You get back the z-score and the approximate percentile. The percentile is computed using a well-known polynomial approximation of the cumulative normal distribution function. This tool is useful for students, teachers, researchers, and anyone interpreting standardised test scores, medical reference ranges, or quality control data. Results assume a normal distribution and are estimates only.
How it works
The z-score formula is z = (X − μ) ÷ σ. The calculator subtracts the mean from your value and divides by the standard deviation. It then converts the z-score to a percentile using a rational approximation of the standard normal cumulative distribution function (CDF), which gives accurate results across the common range of z-scores. The deviation from the mean is simply X minus mu shown with its sign, and the direction tells you whether the value is above or below average.
Worked example
A student scores 75 on a test where the class mean is 70 and the standard deviation is 5. The z-score is (75 − 70) ÷ 5 = 1.00. Looking up z = 1.00 on the standard normal CDF gives a cumulative probability of approximately 0.8413, meaning the student is at the 84th percentile and scored better than roughly 84% of the class. The deviation from the mean is +5.00, confirming the score is above average.
Related calculators
- Normal Distribution Calculator: find P(X<x) and P(X>x) for any normal distribution directly.
- Variance Calculator: compute the standard deviation you need as an input to this calculator.
- Confidence Interval Calculator: use z-scores to build intervals around a sample mean.
- Percent Error Calculator: measure how far a result deviates from an accepted value.