1. What is the Inverse Normal Function?

The inverse normal function (also called the quantile function or probit function) converts a probability value to its corresponding z-score on the standard normal distribution. It's the inverse of the cumulative distribution function (CDF) of the normal distribution.

2. How Does the Calculator Work?

The calculator uses the inverse normal function:

Where:

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

Explanation: The function finds the z-score where the area under the standard normal curve to the left equals the input probability.

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.

4. Using the Calculator

Tips: Enter a probability value between 0 and 1, select the appropriate tail direction, and the calculator will return the corresponding z-score.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between left-tailed and right-tailed?
A: Left-tailed finds z where P(Z ≤ z) = p. Right-tailed finds z where P(Z ≥ z) = p (equivalent to P(Z ≤ z) = 1-p).

Q2: How is two-tailed different?
A: Two-tailed finds the symmetric z-scores where P(-z ≤ Z ≤ z) = p (central probability).

Q3: What are common z-score values?
A: Common critical values: ±1.96 (95% CI), ±2.58 (99% CI). For p=0.975, z≈1.96.

Q4: Can I use this for non-standard normal distributions?
A: For X~N(μ,σ²), first find z then transform: x = μ + zσ.

Q5: Why might my result differ from statistical tables?
A: This uses a numerical approximation. Tables may round to 2-3 decimal places or use different approximations.