How to Calculate Z-Score
Master the z-score formula with step-by-step examples. Learn how to find, interpret, and use z-scores in statistics.
✓ z = (x - μ) / σ explained
✓ Step-by-step examples
✓ Interpretation guide
Z-Score calculator
Compute the standard score and cumulative probability.
84.1% of the distribution falls below this value.
Z-Score: Step-by-Step Calculation
A z-score (or standard score) measures how far a data point is from the mean in units of standard deviation. It's one of the most fundamental concepts in statistics.
The Formula
z = (x - μ) / σ
x = individual data point
μ = population mean (average)
σ = population standard deviation
Example: Test Scores
Class average (μ): 75
Standard deviation (σ): 10
Your score (x): 92
z = (92 - 75) / 10 = 17 / 10 = 1.70
Your score is 1.70 standard deviations above the mean — better than ~95.5% of the class.
Interpreting Z-Scores
- z = 0: Exactly at the mean
- z = 1: 1 SD above mean (84.1th percentile)
- z = 2: 2 SD above (97.7th percentile)
- z = 3: 3 SD above (99.87th percentile)
- z = -1: 1 SD below (15.9th percentile)
- z = -2: 2 SD below (2.3rd percentile)
When to Use Z-Scores
- Comparing scores from different distributions (SAT vs ACT)
- Identifying outliers (|z| > 2 or 3)
- Quality control (Six Sigma uses ±6σ limits)
- Hypothesis testing and p-value calculation
⚠️ Assumption
Z-score percentiles assume a normal (bell curve) distribution. For heavily skewed data, z-scores still measure distance from the mean but the percentile interpretation may not be accurate.