Table of Contents
What Is the Inverse Normal Distribution?
The inverse normal distribution function (also called the quantile function or probit function) returns the value x such that the cumulative probability P(X ≤ x) equals a given probability p. It is the reverse of the normal CDF.
This is commonly used to find critical values for hypothesis tests, confidence intervals, and to convert percentiles to raw scores. For example, if you need the z-score that leaves 2.5% in the upper tail, you find the inverse normal at p = 0.975.
Formula
Where Φ⊃-¹ is the inverse standard normal CDF, μ is the mean, and σ is the standard deviation.
Common Values
| Probability | Z-Score | Use |
|---|---|---|
| 0.90 | 1.2816 | 90th percentile |
| 0.95 | 1.6449 | 95th percentile / one-tail 5% |
| 0.975 | 1.9600 | Two-tail 5% critical value |
| 0.99 | 2.3263 | 99th percentile |
| 0.995 | 2.5758 | Two-tail 1% critical value |
Frequently Asked Questions
What is the relationship between z-score and percentile?
A z-score of 0 corresponds to the 50th percentile (the mean). Positive z-scores correspond to percentiles above 50%, and negative z-scores to percentiles below 50%. Each z-score maps to a unique percentile via the normal CDF.
How is this used in confidence intervals?
For a 95% confidence interval, you need the z-score at p = 0.975 (leaving 2.5% in each tail). This gives z = 1.96, so the interval is x̄ ± 1.96 × SE.
Can I use this for non-normal distributions?
No, this calculator assumes a normal distribution. For other distributions, you would need the specific inverse CDF for that distribution (e.g., inverse t, inverse chi-square).