Z-score Calculator
Input Score & Parameters
Results
| Z-score | — |
|---|---|
| Percentile / Left-tail Probability — P(Z < z) | — |
| Right-tail Probability — P(Z > z) | — |
Raw test scores live on different scales—a 720 on one exam and an 85 on another tell you little until you know each distribution. Z-scores translate a value into "how many standard deviations from the mean," which is why admissions offices and quality teams reach for them when comparing unlike measurements.
How to Use This Z-score Calculator
- Enter the raw score (x). This is the value you want to standardize—a test result, measurement, or KPI.
- Enter population mean (μ) and population standard deviation (σ). σ must be a positive number greater than zero.
- Press Calculate. The table shows the Z-score, left-tail probability P(Z < z) with percentile, and right-tail probability P(Z > z).
- Read the step-by-step panel. It walks through the subtraction, division, and normal CDF lookup. If you need σ from data first, run the Standard Deviation Calculator.
Z-score Formulas and Practical Applications
Picture a bell curve pinned at mean zero and spread one: every Z-score is a position on that universal ruler, no matter what units the original data used.
Standard score formula
z = (x − μ) / σ
Example: score x = 85, mean μ = 75, SD σ = 10 gives z = (85 − 75) / 10 = 1.0—one standard deviation above average.
Tail probabilities
Under a normal model, P(Z < z) is the cumulative area to the left—the percentile rank of your score. P(Z > z) is the complement: 1 − P(Z < z). A z of 1.0 leaves roughly 84% of the distribution below and 16% above.
Where Z-scores show up in practice
- Comparing a student's performance across subjects with different means and spreads.
- Flagging process measurements beyond two or three SD from target on a control chart.
- Linking raw scores to normal probabilities before building a confidence interval or reading a sample-size table.
Frequently Asked Questions
What is a Z-score?
It is the signed distance from the mean measured in standard-deviation units. Zero means exactly average; +2 means two SD above; −1.5 means 1.5 SD below.
What do the left-tail and right-tail probabilities mean?
Left-tail P(Z < z) doubles as a percentile when the normal model fits. Right-tail P(Z > z) answers "how rare is a score this high or higher?"
Should I use sample or population standard deviation?
This form labels the SD field as population σ. In coursework, instructors sometimes supply σ or treat a large-sample s as σ—follow your assignment rules. Compute s from data on the standard deviation page if needed.
Why must standard deviation be greater than zero?
Z divides by σ. Identical population values give σ = 0, and every score would sit on the mean—there is no spread to standardize against.
Can Z-scores compare scores from different tests?
Yes, when both distributions are roughly normal and you have trustworthy μ and σ for each. The Statistics Calculator helps derive descriptive stats from raw lists first.