1. What is Inverse Normal Score?

The inverse normal score, also known as the quantile function or probit function, calculates the z-score corresponding to a given cumulative probability in a standard normal distribution. It is the inverse operation of the cumulative distribution function.

2. How Does the Calculator Work?

The calculator uses the inverse normal function:

Where:

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

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

3. Importance of Inverse Normal Score

Details: The inverse normal score is crucial in statistics for hypothesis testing, confidence interval calculation, and transforming data to normality. It's widely used in quality control, finance, and scientific research.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is the range of possible z-scores?
A: z-scores can range from approximately -3.9 to 3.9 for probabilities between 0.00005 and 0.99995 in a standard normal distribution.

Q2: What does a z-score of 0 represent?
A: A z-score of 0 corresponds to a probability of 0.5, which is the mean of the standard normal distribution.

Q3: When is the inverse normal score used?
A: It's used in statistical testing, quality control processes, risk management, and any application requiring transformation between probabilities and standard normal deviates.

Q4: Are there limitations to this calculation?
A: The calculation assumes a perfect standard normal distribution. Extreme probabilities (very close to 0 or 1) may have reduced precision due to computational limitations.

Q5: How is this different from regular z-score calculation?
A: Regular z-score calculation transforms data to standard units, while inverse normal score calculates the z-score corresponding to a specific cumulative probability.