Z-Score Calculator

A z-score tells you how many standard deviations a data point is from the mean — and from that, you can find what percentile it falls in. Essential for statistics, academic testing, quality control, and research.

How to Use This Calculator

1. Enter the raw score (x — the data point you're analyzing).
2. Enter the mean (μ — population or sample mean).
3. Enter the standard deviation (σ or s).
4. Click Calculate to see the z-score, percentile rank, and probability.

Z-Score Formula

z = (x − μ) / σ

• x = individual data point | μ = mean | σ = standard deviation
• Positive z: above average | Negative z: below average | z = 0: exactly average

Example Calculation

Exam score: 85 | Class mean: 72 | SD: 10

• z = (85 − 72) / 10 = 1.3
• Percentile: approximately 90.3% (scored better than 90% of students)
• Interpretation: Score is 1.3 standard deviations above the mean

Z-Score to Percentile Reference

• z = −2: ~2.3rd percentile
• z = −1: ~15.9th percentile
• z = 0: 50th percentile (exactly average)
• z = +1: ~84.1st percentile
• z = +2: ~97.7th percentile
• z = +3: ~99.9th percentile

Common Mistakes to Avoid

• Using sample SD when population SD is needed (or vice versa) — The z-score formula uses population SD for population data. For sample-based inference, use the t-distribution instead.
• Applying z-scores to non-normal distributions — Z-scores and the resulting percentiles assume a normal distribution. Highly skewed data may have very different percentile mappings.
• Confusing z-score with raw score — A z-score of 1.3 doesn't mean the score is 1.3. It means the score is 1.3 standard deviations above average.

Frequently Asked Questions

What is a good z-score?

It depends on context. In testing, z > 1 (top 84%) is good; z > 2 (top 2.3%) is excellent. In quality control, z-scores outside ±3 often trigger investigation (3σ rule).

What is the z-score used for in hypothesis testing?

In hypothesis testing, the z-score measures how far sample data deviates from the null hypothesis. If |z| > 1.96, the result is significant at the 95% confidence level (p < 0.05).

What is a t-score vs. z-score?

A t-score is used when the population SD is unknown and you're using the sample SD — particularly with small sample sizes (n < 30). For large samples (n > 30), the t-distribution approximates the z-distribution.

Conclusion

Z-scores standardize data to a common scale, making comparisons across different measurements possible. Whether you're placing yourself in a distribution, performing hypothesis testing, or analyzing quality data, the z-score is your foundation.

Related: Standard Deviation Calculator | Statistics Calculator | Confidence Interval Calculator