Z-Score Calculator
Compute Z-scores from raw scores, calculate raw score (x) from Z-score, or find normal distribution probabilities and tail areas for standard normal distributions.
Calculate Z-Score from Raw Score (x)
Enter the raw score (x), population mean (μ), and standard deviation (σ) to calculate the Z-score and normal tail probabilities.
Calculate Raw Score (x) from Z-Score
Enter the Z-score (z), population mean (μ), and standard deviation (σ) to find the corresponding raw score x.
Calculate Z-Score from Cumulative Probability
Enter a cumulative left-tail probability area P (between 0 and 1) to find the standard normal Z-score.
How To Use the Z-Score Calculator
A Z-score (or standard score) indicates how many standard deviations an observation or datum is above or below the mean value of a normal distribution.
- Z-Score Formula: $Z = \frac{x - \mu}{\sigma}$
- Raw Score Formula: $x = \mu + Z \times \sigma$
- Positive Z-score: The value lies above the distribution mean.
- Negative Z-score: The value lies below the distribution mean.
Frequently Asked Questions (FAQs)
A Z-score standardizes scores from different normal distributions so that they can be directly compared against one another, regardless of their original measurement scales.
Multiply the Z-score by the standard deviation (σ), then add the mean (μ) using the formula x = μ + (Z × σ).
Left-tail probability P(Z < z) represents the area under the normal curve to the left of z, whereas right-tail probability P(Z > z) represents the area to the right.
Yes, a negative Z-score indicates that the value is smaller than the population mean.
Z-scores are widely used in statistics, hypothesis testing, quality control, stock risk evaluation, standardized testing, and clinical diagnostics.