Inverse Normal Distribution Calculator

Probability (p):

Tail Area:

Mean (μ):

Standard Deviation (σ):

1. What is the Inverse Normal Distribution Calculator?

Definition: The Inverse Normal Distribution Calculator computes the x-value and Z-score for a given probability in a normal distribution, based on the specified tail area, mean, and standard deviation.

Purpose: This tool helps users determine the value of a random variable corresponding to a specific probability, useful in statistical analysis, quality control, and risk assessment.

2. How Does the Calculator Work?

The calculator uses the following formulas:

\( Z = \text{invNorm}(p) \text{ (for } P(X < x) = p\text{)} \)

\( Z = \text{invNorm}(1 - p) \text{ (for } P(X > x) = p\text{)} \)

\( Z = \text{invNorm}(1 - p/2) \text{ (for } P(|X - \mu| > x) = p\text{)} \)

\( Z = \text{invNorm}((1 + p)/2) \text{ (for } P(|X - \mu| < x) = p\text{)} \)

\( x = \mu + Z \cdot \sigma \)

Where:

\( p \): Probability (0 to 1);
\( \mu \): Mean;
\( \sigma \): Standard deviation;
\( Z \): Z-score;
\( x \): x-value.

Steps:

Enter the probability (p), tail area, mean (μ), and standard deviation (σ).
Calculate the Z-score using the inverse CDF for the selected tail area.
Calculate the x-value: mean plus Z-score times standard deviation (or absolute distance for two-tailed cases).
Display results, using scientific notation for values with absolute value less than 0.0001, otherwise to four decimal places.

3. Importance of the Inverse Normal Distribution Calculation

Calculating the inverse normal distribution is essential for:

Statistical Analysis: Determines thresholds for probabilities in hypothesis testing and confidence intervals.
Risk Assessment: Identifies critical values for risk management in finance and engineering.
Quality Control: Sets boundaries for acceptable product measurements based on specified probabilities.

4. Using the Calculator

Example: Calculate the x-value for a normal distribution with mean 100, standard deviation 0.00001, where \( P(X < x) = 0.00001 \):

Input: Probability: 0.00001; Tail Area: P(X < x) = p; Mean: 100; Standard Deviation: 0.00001.
Z-score: \( \text{invNorm}(0.00001) \approx -4.2650 \).
x-value: \( 100 + (-4.2650) \times 0.00001 \approx 99.99995735 \).
Result: x-value: 9.999957e-2; Z-score: -4.2650.

5. Frequently Asked Questions (FAQ)

Q: What is the inverse normal distribution?
A: The inverse normal distribution finds the x-value for a given probability in a normal distribution, useful for determining thresholds or critical values.

Q: What is a Z-score?
A: A Z-score measures how many standard deviations a value is from the mean in a normal distribution.

Q: When should I use this calculator?
A: Use it when you know a probability and need the corresponding x-value, such as in statistical testing, quality control, or risk analysis.