Z-Score Calculator
Enter a data value, population mean, and standard deviation to calculate the z-score and p-value.
Z-Score — Formula & Guide
Formula
z = (x − μ) / σ
Where x = data point, μ = mean, σ = standard deviation.
Frequently Asked Questions
What does a z-score of 2 mean?
A z-score of 2 means the value is 2 standard deviations above the mean, placing it at approximately the 97.7th percentile — better than ~97.7% of all values in a normal distribution.
How do I standardize data using z-scores?
Apply z = (x − mean) / SD to every data point. This converts data with any scale to a standard scale (mean 0, SD 1), allowing you to compare values from different distributions.
Interpretation
z = 0: value equals the mean • z = +1: one SD above the mean • z = −1: one SD below the mean.
For a standard normal: 68% of data falls within z = ±1, 95% within z = ±1.96.
Related Calculators
How to Use Z-Scores
A z-score (standard score) measures how many standard deviations a value is from the mean: z = (x − μ) / σ. A z-score of 0 is exactly average. A z-score of +2 is two SDs above the mean; −1.5 is 1.5 SDs below. For a normal distribution, z-scores convert to percentiles: z = 1.645 corresponds to the 95th percentile. Use z-scores to compare values from different distributions — e.g., a student scoring 80 on a test with mean 70, SD 5 has z = 2.0; another scoring 650 on an SAT section with mean 500, SD 100 also has z = 1.5, so the first student performed relatively better.
Z-scores are the foundation of hypothesis testing, control charts in quality management, anomaly detection, and feature normalization in machine learning. In finance, z-scores power the Altman Z-Score bankruptcy prediction model. To find the probability that a normal random variable falls below a given z-score, consult the standard normal table or use a calculator — P(Z < 1.96) ≈ 0.975, which is why 1.96 appears in the 95% confidence interval formula.
Z-Score to Percentile Reference
| Z-Score | Percentile | Meaning |
|---|---|---|
| -2.576 | 0.5% | 99% CI lower tail |
| -1.960 | 2.5% | 95% CI lower tail |
| -1.000 | 15.9% | 1 SD below mean |
| 0.000 | 50.0% | Exactly average |
| +1.000 | 84.1% | 1 SD above mean |
| +1.960 | 97.5% | 95% CI upper tail |
| +2.576 | 99.5% | 99% CI upper tail |
How Z-Scores Are Applied in Statistics
A z-score (also called a standard score) measures how many standard deviations a data point lies above or below the mean of its distribution. A z-score of 0 means the value equals the mean; a score of positive 1 means it is one standard deviation above the mean; a score of negative 2 means it is two standard deviations below. Z-scores standardize values from different distributions so they can be compared on a common scale. This is why SAT and GRE scores are often converted to z-scores when comparing students tested in different years with slightly different raw score ranges. In quality control, manufacturers use z-scores to detect measurements outside acceptable control limits, typically set at plus or minus 3 standard deviations (a three-sigma boundary). In finance, z-scores quantify how unusual a daily return is relative to historical volatility. The Altman Z-score specifically predicts corporate bankruptcy probability. In medical statistics, z-scores for height and weight express where a child falls within growth charts relative to the reference population. To convert a z-score back to the original scale, multiply the z-score by the standard deviation and add the mean.
Z-Score and Percentile Reference Table (Standard Normal)
| Z-score | Percentile (approx.) | Interpretation |
|---|---|---|
| −3.0 | 0.13% | Extremely low |
| −2.0 | 2.3% | Very low |
| −1.0 | 15.9% | Below average |
| 0.0 | 50.0% | Average |
| +1.0 | 84.1% | Above average |
| +2.0 | 97.7% | Very high |
| +3.0 | 99.87% | Extremely high |
