Inverse Normal Distribution Calculator
Find the x-value for a given probability in a normal distribution
How to Use
- Enter the cumulative probability (between 0 and 1)
- Enter the mean of the distribution
- Enter the standard deviation of the distribution
- Click calculate to find the corresponding x-value
What is Inverse Normal Distribution?
The inverse normal distribution, also known as the quantile function or probit function, finds the x-value that corresponds to a given cumulative probability in a normal distribution. It's the inverse of the cumulative distribution function (CDF).
While the normal distribution CDF tells you the probability of a value being less than or equal to x, the inverse normal distribution tells you the value x for which a given probability is achieved.
Mathematical Formula
The inverse normal distribution uses the formula:
x = μ + σ × Φ⁻¹(p)
Where:
- x is the value we want to find
- μ is the mean of the distribution
- σ is the standard deviation
- Φ⁻¹ is the inverse standard normal CDF
- p is the cumulative probability
Common Applications
- Hypothesis testing - finding critical values
- Confidence intervals - determining bounds
- Quality control - setting specification limits
- Risk assessment - finding value-at-risk thresholds
- Statistical process control - establishing control limits
Example Calculations
Example 1: Find the 95th percentile of a normal distribution with μ=100 and σ=15:
Probability = 0.95, Mean = 100, Standard Deviation = 15
Result: x ≈ 124.67 (95% of values fall below this point)
Example 2: Find the value at the 25th percentile:
Probability = 0.25, Mean = 50, Standard Deviation = 10
Result: x ≈ 43.26 (25% of values fall below this point)
Frequently Asked Questions
- What's the difference between normal distribution and inverse normal distribution?
- Normal distribution (CDF) gives you the probability for a given x-value, while inverse normal distribution gives you the x-value for a given probability. They are mathematical inverses of each other.
- Why can't I use probability values of 0 or 1?
- The normal distribution extends to infinity in both directions, so probabilities of exactly 0 or 1 would correspond to x-values of negative or positive infinity, which are not meaningful in practical applications.
- How accurate is this calculator?
- This calculator uses the Beasley-Springer-Moro algorithm, which provides high accuracy (typically within 1.15 × 10^-9) for all probability values. This is more than sufficient for most statistical applications.
- What is a z-score and how is it related?
- A z-score is the number of standard deviations a value is from the mean. The inverse normal distribution first calculates the z-score for your probability, then converts it to the actual x-value using x = μ + σ × z.