1. What is the Inverse Normal Distribution?

The inverse normal distribution (\(\Phi^{-1}\)), also called the quantile function or probit function, is the inverse of the standard normal cumulative distribution function (CDF). It maps probabilities to z-scores on the standard normal curve.

2. How Does the Calculator Work?

The calculator uses the following equation:

Where:

• \( p \) — Probability value (0 < p < 1)
• \( \text{erfinv} \) — Inverse error function
• \( \Phi^{-1} \) — Inverse standard normal CDF

Explanation: The calculation uses an approximation of the inverse error function to compute the z-score corresponding to a given cumulative probability.

3. Importance of Inverse Normal Calculation

Details: The inverse normal function is essential in statistics for determining critical values, constructing confidence intervals, and performing hypothesis testing.

4. Using the Calculator

Tips: Enter a probability value between 0 and 1 (exclusive). The calculator will return the corresponding z-score on the standard normal distribution.

5. Frequently Asked Questions (FAQ)

Q1: What is the range of valid inputs?
A: The probability p must be between 0 and 1 (exclusive). Values of 0 or 1 would return -∞ or +∞ respectively.

Q2: How accurate is this calculation?
A: The approximation used is accurate to about 6 decimal places for most values of p.

Q3: What is the relationship to the probit function?
A: The probit function is simply another name for the inverse standard normal CDF (\(\Phi^{-1}\)).

Q4: When would I need to use this function?
A: Common uses include finding critical values for hypothesis tests, converting percentiles to z-scores, and in probit regression models.

Q5: What's the inverse of p=0.5?
A: \(\Phi^{-1}(0.5) = 0\), since 0 is the median of the standard normal distribution.