Z-Score Calculator
Standardize a data point by subtracting the mean and scaling by the standard deviation. The calculator also estimates the percentile rank and probability above or below the value.
Understanding the z-score formula
A z-score measures how many standard deviations a value sits above or below the mean of a normally distributed variable. The transformation is , where is the raw value, is the mean, and is the standard deviation. Because the conversion is linear, positive z-scores lie above the mean and negative z-scores lie below the mean.
Once the z-score is known, the calculator estimates the percentile using the cumulative distribution function of the standard normal curve: . This integral returns the probability that a random observation falls below the target value. The upper-tail and two-tail probabilities are derived from the same CDF to support hypothesis tests and outlier checks.
Reference scenarios
| Context | Value | Mean | Std dev | Z-score | Percentile |
|---|---|---|---|---|---|
| Exam score above average | 87 | 75 | 8.5 | 1.41 | 92.0% |
| Manufacturing tolerance low reading | 19.82 mm | 20.00 mm | 0.10 mm | -1.80 | 3.6% |
| Hospital lab test slightly elevated | 142 mg/dL | 135 mg/dL | 6 mg/dL | 1.17 | 87.9% |
Next steps for data analysis
Pair this z-score calculator with the Normal Distribution Probability Calculator, Percentile to Z-Score Converter, and Standard Deviation Calculator to explore sampling variability, convert between percentiles and raw values, and double-check the spread of your data set.