Z-Score Calculator
Calculate a z-score from a raw score, mean, and standard deviation. Convert z-scores to percentiles, tail probabilities, between-z area, or reverse solve from a percentile. This Z-Score Calculator uses the standard normal distribution to convert raw values into standardized scores.
How to Use the Z-Score Calculator
Choose a calculation mode, enter the known values, then review the standardized score, percentile, and probability context.
Choose Z from score, Probability lookup, Percentile to z, or Dataset z-scores.
Enter the raw score, z-score, percentile, or dataset values requested by the selected mode.
For probability lookup, choose left-tail, right-tail, two-tailed, or between-z area.
Use the result summary and curve shading to interpret the value on a standard normal scale.
Why Convert to a Z-Score?
A z-score puts values from different scales onto one comparable standard normal scale.
- Compare scores: Standardize values measured in different units or distributions.
- Find percentiles: Use the standard normal CDF to estimate left-tail area.
- Evaluate unusual values: Values near +/-2 or +/-3 are farther from the mean in a normal distribution.
Z-Score Formula
A z-score converts a raw value to a standard normal scale where the mean is 0 and the standard deviation is 1. The z-score formula shows how far a raw value is from the mean in standard deviation units.
Example: Calculate a Z-Score from a Test Score
If a test score is 85, the class mean is 75, and the standard deviation is 8, the calculator standardizes the raw score and then maps it to a normal percentile.
Input values
x = 85, mean = 75, SD = 8
| Metric | Value |
|---|---|
| Formula | z = (85 - 75) / 8 |
| Z-score | 1.25 |
| Percentile | 89.44% |
| Right-tail area | 10.56% |
| Interpretation | 1.25 SD above mean |
Common Z-Score Values
Use these standard normal values as quick checks for percentile, right-tail probability, and common critical values. Percentile and tail-area results can be checked against a standard normal cumulative probability table.
| Z-score | Percentile | Right-tail area | Common use |
|---|---|---|---|
| -1 | 15.87% | 84.13% | 1 SD below mean |
| 0 | 50% | 50% | Mean |
| 1 | 84.13% | 15.87% | 1 SD above mean |
| 1.645 | 95% | 5% | One-tailed 95% |
| 1.96 | 97.5% | 2.5% | Two-tailed 95% critical value |
| 2.576 | 99.5% | 0.5% | Two-tailed 99% critical value |
Frequently Asked Questions
Common questions about z-scores, percentiles, and standardization.
What is a z-score?
A z-score tells how many standard deviations a value is above or below the mean. A z-score of 2 means the value is two standard deviations above the mean.
How do I calculate a z-score?
Use z = (x - mean) / standard deviation. The standard deviation must be greater than zero.
What does a negative z-score mean?
A negative z-score means the value is below the mean. For example, z = -1.5 means the value is 1.5 standard deviations below the mean.
Is z-score the same as percentile?
No. The z-score is a standardized distance from the mean. The percentile is the cumulative area to the left of that z-score in a normal distribution.
When should I use sample standard deviation?
Use sample standard deviation when your data are a sample from a larger population. Use population standard deviation only when your list contains the whole population. When calculating z-scores from a sample dataset, this calculator uses sample standard deviation so the dataset is standardized correctly.
Can I use this for non-normal data?
You can standardize any numeric value with a z-score, but percentile interpretations assume an approximately normal distribution.
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