1. What is the Inverse Z-Score?

The inverse Z-score (also called the quantile function or probit function) returns the Z-value that corresponds to a given cumulative probability in a standard normal distribution. It's the reverse operation of finding the probability associated with a Z-score.

2. How Does the Calculator Work?

The calculator uses the inverse normal distribution function:

Where:

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

Explanation: For a given probability p, the function returns the Z-score where the area under the standard normal curve to the left of Z equals p.

3. Importance of Inverse Normal Calculation

Details: The inverse normal calculation is essential in statistics for determining critical values, setting 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 from the standard normal distribution.

5. Frequently Asked Questions (FAQ)

Q1: What does a probability of 0.975 return?
A: This returns approximately 1.96, which is the Z-score for the 97.5th percentile (commonly used for 95% confidence intervals).

Q2: Can I use this for non-standard normal distributions?
A: You can transform the result using X = μ + Zσ, where μ is the mean and σ is the standard deviation of your distribution.

Q3: What's the inverse of p=0.5?
A: The inverse of 0.5 is 0, as half of the standard normal distribution lies to the left of the mean.

Q4: How precise is this calculation?
A: The precision depends on the implementation. Professional statistical software uses sophisticated algorithms for high accuracy.

Q5: What's the relationship to p-values?
A: The inverse normal can convert p-values to Z-scores, which is useful in meta-analyses and other statistical methods.