Inverse Normal Distribution Calculator
Find the Z-score or raw value (X) for any cumulative probability. Enter a probability along with your mean and standard deviation to instantly compute the inverse normal (quantile) value.
Distribution Inputs
Enter a cumulative probability, mean, and standard deviation
Enter a value strictly between 0 and 1 (e.g. 0.95 for the 95th percentile).
The average or expected value of the distribution. Use 0 for the standard normal distribution.
Must be greater than 0. Use 1 for the standard normal distribution.
Your Inverse Normal Result
Z-score, X-value, and distribution breakdown
Enter a probability, mean, and standard deviation above and click Calculate Inverse Normal to reveal your Z-score and X-value.
Common Z-Score Percentiles
These are some of the most frequently used cumulative probabilities and their corresponding Z-scores on the standard normal distribution, commonly used for confidence intervals and hypothesis testing.
| Cumulative Probability | Z-Score | Common Use | Confidence Level |
|---|---|---|---|
| 0.90 | 1.2816 | One-tailed 90% | 80% |
| 0.95 | 1.6449 | One-tailed 95% | 90% |
| 0.975 | 1.9600 | Two-tailed 95% | 95% |
| 0.99 | 2.3263 | One-tailed 99% | 98% |
| 0.995 | 2.5758 | Two-tailed 99% | 99% |
| 0.999 | 3.0902 | Quality control limit | 99.8% |
Inverse Normal Distribution FAQ
Everything you need to know about the inverse normal distribution, the quantile function, and how Z-scores relate to raw values.
The inverse normal distribution, also called the quantile function or inverse CDF, takes a cumulative probability as input and returns the corresponding Z-score or raw value (X) on a normal distribution. It is the opposite operation of the standard normal CDF, which converts a value into a probability.
To calculate the inverse normal distribution by hand, you typically look up the desired cumulative probability in a standard normal (Z) table to find the corresponding Z-score. Once you have the Z-score, you can convert it to a raw value using the formula X = mean + (Z multiplied by the standard deviation).
The normal CDF (cumulative distribution function) takes a value (X or Z) and returns a probability. The inverse normal does the opposite: it takes a probability and returns the corresponding value (X or Z). They are inverse functions of one another.
Yes. The inverse normal calculation first finds the Z-score for the standard normal distribution, which has a mean of 0 and a standard deviation of 1. This Z-score is then rescaled to any mean and standard deviation using the formula X = mean + (Z times standard deviation).
The cumulative probability entered must be greater than 0 and less than 1, since probabilities represent the area under the normal curve and cannot equal or exceed 100 percent or fall to 0 percent at a finite Z-score.
The inverse normal distribution is widely used in statistics, finance, and quality control. Common applications include calculating confidence interval bounds, determining percentile cutoffs for test scores, setting control limits in manufacturing, and computing Value at Risk in financial risk modelling.