Inverse Normal Distribution Calculator
Find x-values and z-scores from probabilities using the inverse normal function
Area Under the Curve
Must be between 0 and 1 (exclusive)
The probability p is the area to the left of x (left-tailed p-value).
Distribution Parameters
Average value of the distribution
Must be positive (σ > 0)
x-value & Z-score
Distribution: X ~ N(0, 1²)
Condition: P(X < x) = 0.5
Formula: x = μ + Z × σ = 0 + Z × 1
IQ Score Examples
Left-tailed Example
Question: What IQ score puts someone in the bottom 20% of the population?
Parameters: μ = 100, σ = 15, p = 0.20
Condition: P(X < x) = 0.20
Answer: x = 87.38 (Z-score = -0.8416)
Interpretation: 20% of people have IQ ≤ 87.38
Right-tailed Example
Question: What IQ score puts someone in the top 2.5%?
Parameters: μ = 100, σ = 15, p = 0.025
Condition: P(X > x) = 0.025
Answer: x = 129.4 (Z-score = 1.96)
Interpretation: 2.5% of people have IQ ≥ 129.4
Two-tailed Example
Question: What range contains the middle 80% of IQ scores?
Parameters: μ = 100, σ = 15, p = 0.80
Condition: P(|X - μ| < x) = 0.80
Answer: x₁ = 80.78, x₂ = 119.22
Interpretation: 80% of people have IQ between 80.78 and 119.22
Tail Area Types
Left-tailed
P(X < x) = p
Area to the left of x
Right-tailed
P(X > x) = p
Area to the right of x
Two-tailed
P(|X - μ| > x) = p
Area in both tails
Central Area
P(|X - μ| < x) = p
Area around the mean
Quick Tips
Probability must be between 0 and 1
Standard deviation must be positive
Z-score = (x - μ) / σ
Use μ=0, σ=1 for standard normal
Two-tailed splits probability equally
Understanding Inverse Normal Distribution
What is Inverse Normal Distribution?
The inverse normal distribution (invnorm) function calculates the x-value from a given probability p, mean μ, and standard deviation σ. It works backward from the area under the normal curve to find the critical value corresponding to that area.
Key Applications
- •Finding percentiles and quartiles
- •Determining critical values for hypothesis testing
- •Calculating confidence interval boundaries
- •Quality control and process capability analysis
Mathematical Foundation
If F(x) = p, then F⁻¹(p) = x
x = μ + Z × σ
- F⁻¹(p): Inverse cumulative distribution function
- p: Probability (area under curve)
- x: Critical value we're solving for
- μ: Mean of the distribution
- σ: Standard deviation
- Z: Standard normal quantile
Remember: Normal distribution calculates probability from x-values, while inverse normal calculates x-values from probabilities.
Normal vs. Inverse Normal
Normal Distribution
- • Input: x-value
- • Output: Probability (area)
- • Question: "What's the probability?"
- • Use: P(X < x) = ?
Inverse Normal
- • Input: Probability (area)
- • Output: x-value
- • Question: "What's the critical value?"
- • Use: P(X < ?) = p