1. What is the Inverse Normal Distribution?

The inverse normal distribution (also called the probit function) calculates the z-score corresponding to a given cumulative probability in a standard normal distribution. It's the reverse operation of the normal cumulative distribution function.

2. How Does the Calculator Work?

The calculator uses the inverse normal function:

Where:

• \( \Phi^{-1} \) — Inverse of the standard normal cumulative distribution function
• \( p \) — Cumulative probability (0 to 1)
• \( z \) — Standard score (z-score)

Explanation: The calculation uses rational approximations with different formulas for different regions of the probability range for optimal accuracy.

3. Importance of Z-Score Calculation

Details: Inverse normal calculations are essential in statistics for determining critical values, constructing confidence intervals, and performing hypothesis testing when working with normally distributed data.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is the z-score for p=0.5?
A: The z-score for p=0.5 (median) is always 0 in a standard normal distribution.

Q2: What are common z-score values?
A: Common values include ±1.96 (for 95% CI), ±2.58 (for 99% CI), and ±1.645 (for 90% CI).

Q3: Can I use this for non-standard normal distributions?
A: Yes, after obtaining the z-score, you can convert it to any normal distribution using: X = μ + zσ.

Q4: What's the difference between inverse normal and normal CDF?
A: Normal CDF gives probability from z-score, while inverse normal gives z-score from probability.

Q5: How accurate is this calculation?
A: The algorithm provides approximately 7 decimal digits of accuracy.