Z-score Calculator

Z-scores and the normal distribution: A complete guide

Z-scores standardize values so you can compare across datasets and scales. A Z-score tells you how many standard deviations a value lies from the mean. Positive Z means above the mean, negative Z means below, and a Z near 0 is close to average.

What is a Z-score?

The Z-score is computed as \(Z=\frac{X-\mu}{\sigma}\), where \(X\) is your value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. Because the formula rescales by \(\sigma\), Z reveals relative standing independent of units, enabling comparisons across different distributions.

Percentiles and probabilities

Using the standard normal distribution, the cumulative distribution function (CDF) converts Z to a percentile (the proportion of values at or below \(X\)). Tail probabilities show how extreme a value is: left tail \(P(X\le x)\), right tail \(P(X\ge x)\), and two-tailed \(2\cdot\min(\text{left},\text{right})\).

When to use Z-scores

• Comparisons across scales: Standardize exam scores, quality measures, or financial metrics.
• Outlier detection: Large \(|Z|\) indicates values far from the mean.
• Probability and hypothesis testing: Map Z to p-values under normal assumptions.
• Feature scaling: In data science, Z-scores (standardization) aid model training and interpretation.

Assumptions and cautions

• Normality: Z-to-probability mapping assumes normality; for means, the central limit theorem can help.
• Known σ vs. t-scores: If \(\sigma\) is unknown and sample is small, t-scores are more appropriate.
• Context matters: Domains like finance or biology may have skewed distributions; check robustness before interpreting tails.

Worked example