# Inverse Normal Distribution Calculator

Free inverse normal distribution calculator. Find the z-score or x-value for any cumulative probability using the inverse CDF (quantile function) of the normal distribution.

## What this calculates

Find the z-score or x-value that corresponds to a given cumulative probability. This is the inverse of the normal CDF, also called the quantile function or probit function.

## Inputs

- **Cumulative Probability (p)** — min 0.0001, max 0.9999 — The cumulative probability P(X < x). Must be between 0 and 1.
- **Mean (μ)** — The mean of the normal distribution. Use 0 for standard normal.
- **Standard Deviation (σ)** — min 0.0001 — The standard deviation. Use 1 for standard normal.

## Outputs

- **Z-Score** — The z-score corresponding to the given probability.
- **X Value** — The x-value in the specified distribution (x = μ + zσ).
- **Right-Tail Probability** — P(X > x) = 1 - p.
- **Interpretation** — formatted as text — What the result means in context.

## Details

The inverse normal distribution answers the question: "What value has a given percentage of the distribution below it?"

**What it does:**

Given a probability p, it finds the value x such that P(X < x) = p. For the standard normal distribution (mean = 0, SD = 1), this gives you the z-score directly.

**Common Inverse Normal Values:**

| Probability | Z-Score | Use Case |
|-------------|---------|----------|
| 0.90        | 1.2816  | 90% CI lower z |
| 0.95        | 1.6449  | One-tailed 5% significance |
| 0.975       | 1.9600  | 95% CI / two-tailed 5% |
| 0.99        | 2.3263  | One-tailed 1% significance |
| 0.995       | 2.5758  | 99% CI / two-tailed 1% |

**Converting to Any Normal Distribution:**

x = mean + z x SD

For example, if IQ scores have mean 100 and SD 15, the 95th percentile is: x = 100 + 1.6449 x 15 = 124.67.

**Where it is used:** Setting confidence intervals, finding critical values for hypothesis tests, determining percentile cutoffs, and quality control limits.

## Frequently Asked Questions

**Q: What is the inverse normal distribution?**

A: The inverse normal distribution (also called the quantile function or probit function) reverses the normal CDF. Instead of asking 'what is the probability below this value?', it asks 'what value has this probability below it?' For example, the inverse of 0.975 in the standard normal is 1.96, the familiar z-score used in 95% confidence intervals.

**Q: Why can I only enter probabilities between 0 and 1?**

A: The normal distribution extends to negative and positive infinity. A probability of 0 would correspond to negative infinity, and 1 would correspond to positive infinity. In practice, values very close to 0 or 1 (like 0.0001 or 0.9999) still produce finite z-scores, but exactly 0 and 1 are undefined.

**Q: How is this related to confidence intervals?**

A: Confidence intervals use the inverse normal to find critical values. For a 95% confidence interval, you need the z-score where 97.5% of the distribution falls below (because 2.5% is in each tail). That gives z = 1.96. For 99% confidence, you need the 99.5th percentile, giving z = 2.576.

**Q: How accurate is this calculator?**

A: This calculator uses the Beasley-Springer-Moro rational approximation algorithm, which provides accuracy to at least 6 significant digits across the full range of probabilities. For most statistical applications, this exceeds the precision needed.

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Source: https://vastcalc.com/calculators/statistics/inverse-normal-distribution
Category: Statistics
Last updated: 2026-04-08