# Z-Score & Normal Distribution Calculator — Probability & Percentile

> Convert a raw score to a z-score and find left, right, between and two-tailed probabilities, or reverse-solve a value from a percentile, with a shaded normal curve.

- **Category:** Math
- **Interactive calculator:** https://youcalc.com/en/math/z-score-normal-distribution/
- **Price:** Free, no sign-up required

## Overview

This calculator converts a raw value into a z-score and shows how that value compares to the rest of a normally distributed population. Use it to find the probability of a result falling below, above, or between two values, to read off a percentile rank, or to reverse-solve a score from a target percentile.

## How to read your result

The headline figure is the z-score, which tells you how many standard deviations the value sits above or below the mean. The stat strip also shows the corresponding percentile. Below the inputs a shaded normal curve highlights the area of interest — left-shaded for a single value, between-shaded when you enter two values. The result card lists the left-tail, right-tail, and two-tailed probabilities so you can pick the one that matches your question.

## Method

The z-score formula is z = (x − μ) / σ, where x is the observed value, μ the population mean, and σ the standard deviation. Cumulative probabilities come from the standard normal CDF Φ(z), computed with the Abramowitz & Stegun erf approximation (7.1.26, maximum error ~1.5 × 10⁻⁷). The inverse CDF Φ⁻¹(p) uses Acklam's rational approximation to map a percentile back to a z-score. The raw value is then recovered as x = μ + z·σ.

## Example

- **Setup:** An IQ score of 130 on a test with mean 100 and standard deviation 15.
- **Result:** The z-score is 2.00. The left-tail probability is 0.9772, meaning 97.72 % of the population scores below 130. The right-tail probability is 0.0228 and the two-tailed probability is 0.0455.

## Frequently asked questions

### What does a z-score tell me?

A z-score measures distance from the mean in units of standard deviation. A z of 0 means the value equals the mean; a z of 1 means it is one standard deviation above; a z of −1 means one standard deviation below. This makes it possible to compare values from different distributions on the same scale.

### When should I use left-tail, right-tail, or two-tailed probability?

Use left-tail when asking "what fraction of the population scores less than X?" Use right-tail for "what fraction scores more than X?" Use two-tailed when testing whether a value is unusual in either direction — for example, in a hypothesis test with no predetermined direction.

### Does this assume a normal distribution?

Yes. All probabilities and percentiles here are calculated under the assumption of a perfect standard normal curve. For real data that is heavily skewed or has long tails, the results are approximate and a distribution-specific tool may be more appropriate.

## Related calculators

- [Standard Deviation Calculator](https://youcalc.com/en/math/statistics-standard-deviation/)
- [Permutations & Combinations Calculator](https://youcalc.com/en/math/permutations-combinations/)
- [Percentage Calculator](https://youcalc.com/en/math/percentage/)

## Sources

- https://mathworld.wolfram.com/z-Score.html
- https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/z-scores/a/z-scores-review

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Interactive version: https://youcalc.com/en/math/z-score-normal-distribution/ · From YouCalc — https://youcalc.com