1. What is the Inverse Normal Distribution?

The inverse normal distribution, also known as the quantile function of the normal distribution, calculates the value below which a given percentage of observations in a standard normal distribution fall.

2. How Does the Calculator Work?

The calculator uses the inverse normal formula:

Where:

• \( x \) — The calculated value
• \( \mu \) — Mean of the distribution
• \( \sigma \) — Standard deviation of the distribution
• \( p \) — Probability (between 0 and 1)
• \( \Phi^{-1} \) — Inverse of the standard normal CDF

Explanation: The function finds the z-score for a given probability and then scales it to the specified normal distribution.

3. Importance of Inverse Normal Calculation

Details: This calculation is essential for determining critical values, setting confidence intervals, and performing statistical hypothesis testing.

4. Using the Calculator

Tips: Enter the mean (μ), standard deviation (σ > 0), and probability (0 ≤ p ≤ 1). The calculator will return the value x such that P(X ≤ x) = p.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between normal and inverse normal?
A: Normal distribution gives probability for a value, while inverse normal gives the value for a probability.

Q2: What is Φ⁻¹(0.5)?
A: For p=0.5 (median), Φ⁻¹(0.5) = 0, so x = μ.

Q3: How is this related to z-scores?
A: When μ=0 and σ=1, the result is exactly the z-score for the given probability.

Q4: What if p=0 or p=1?
A: Theoretically, these would be ±∞, but in practice, very small/large numbers are used.

Q5: Can this be used for non-normal distributions?
A: No, this is specific to normal distributions. Other distributions have their own quantile functions.