Z-Score Calculator

About This Calculator

A z-score (also called a standard score) tells you how many standard deviations an observation lies above or below the mean of its distribution. Use the forward mode to convert any value (x) into a z-score and read off the corresponding p-value and percentile. Use the inverse mode to answer the reverse question: "what z-score corresponds to the 97.5th percentile?"

How It Works

Enter your observation (x), the population mean (μ), and the standard deviation (σ). The calculator applies the formula z = (x − μ) / σ and then passes z through the standard-normal CDF to compute the left-tail probability (p-value) and percentile. In inverse mode, enter a percentile to find the z-score using the Beasley–Springer–Moro rational approximation, which achieves less than 1.2 × 10⁻⁷ absolute error.

The Formula

Frequently Asked Questions

What is a z-score?

A z-score is the number of standard deviations an observation is from the mean. A z-score of 0 means the value equals the mean. A z-score of +1 means one standard deviation above the mean; −2 means two standard deviations below it.

What does the p-value from a z-score mean?

The p-value shown here is the left-tail probability — the fraction of a standard normal distribution that falls below z. For example, a z-score of 1.96 gives a p-value of 0.975, meaning 97.5% of values fall below it (in a standard normal distribution).

How do I use the inverse z-score?

Switch to Inverse mode and enter a percentile (0–100). The calculator returns the z-score that cuts off that percentile in the standard normal distribution. For example, the 97.5th percentile corresponds to z ≈ 1.96, which is the critical value for a 95% two-tailed confidence interval.

What z-scores correspond to common confidence intervals?

90% CI: z* = 1.645 (each tail = 5%). 95% CI: z* = 1.96 (each tail = 2.5%). 99% CI: z* = 2.576 (each tail = 0.5%).