Z-score Calculator
Calculate z-scores, raw scores, and probabilities from the standard normal distribution. Used in statistics, testing and quality control.
How to use the Z-score Calculator
- Enter your inputs into the Z-score Calculator above.
- Results update instantly as you type — no submit button needed.
- Adjust any value to see how the result changes in real time.
The z-score formulas
z = (x − μ) / σ · · · x = μ + zσ · · · P(Z < z) = Φ(z) (standard normal CDF)
A z-score tells you how many standard deviations a value is from the mean. Z=0 is the mean; Z=1.96 is roughly the 97.5th percentile (95% confidence threshold).
Worked example
SAT scores have mean 1050, SD 200. A score of 1300 has z = (1300−1050)/200 = 1.25. This puts the score at the 89.4th percentile (P(Z<1.25) ≈ 0.8944).
Frequently asked questions
What is the 68-95-99.7 rule?
For normal distributions: 68% of values fall within ±1 SD (z between −1 and 1); 95% within ±2 SD; 99.7% within ±3 SD. Useful for quick mental checks.
What z-score is "extreme"?
Conventional thresholds: |z| > 2 is unusual (5% of values); |z| > 3 is rare (0.3%); |z| > 4 is essentially never under normal assumption.
Does z-score require normal distribution?
The z formula works for any distribution. But interpreting it as a percentile from the standard normal table only works if the data is actually normal.