Calculate the value from a standard normal distribution given a cumulative probability.
Result: —
Understanding the Inverse Normal CDF
The Inverse Normal Cumulative Distribution Function (CDF), often referred to as the quantile function or probit function, is a fundamental concept in statistics and probability. While the standard normal CDF (often denoted as Φ(x)) tells you the probability that a standard normal random variable is less than or equal to a given value x (i.e., P(Z ≤ x)), the inverse normal CDF answers the reverse question: given a probability, what is the corresponding value x?
Mathematically, if P = Φ(x), then the inverse normal CDF is the function that returns x for a given P. We can denote this as x = Φ⁻¹(P).
The Standard Normal Distribution
The standard normal distribution is a special case of the normal distribution with a mean (μ) of 0 and a standard deviation (σ) of 1. Its probability density function (PDF) is given by:
f(z) = (1 / √(2π)) * e^(-z²/2)
The CDF, Φ(z), is the integral of the PDF from negative infinity to z.
The General Normal Distribution
For a general normal distribution with mean μ and standard deviation σ, the relationship between a value X and the standard normal variable Z is given by the z-score formula:
Z = (X - μ) / σ
Therefore, if we have a cumulative probability P for a general normal distribution, we first find the corresponding z-score (z) using the inverse standard normal CDF, and then we can find the value X using:
X = μ + z * σ
How This Calculator Works
This calculator takes the following inputs:
Cumulative Probability (P): The probability value (between 0 and 1) for which you want to find the corresponding value in the normal distribution.
Mean (μ): The mean of the normal distribution. For the standard normal distribution, this is 0.
Standard Deviation (σ): The standard deviation of the normal distribution. For the standard normal distribution, this is 1.
The calculator first finds the z-score (z) corresponding to the input probability P using an approximation of the inverse standard normal CDF. Then, it uses the formula X = μ + z * σ to calculate the value for the specified general normal distribution.
Use Cases
The inverse normal CDF has numerous applications in statistics, finance, engineering, and data science:
Hypothesis Testing: Determining critical values for hypothesis tests. For example, finding the z-score corresponding to a 95% confidence level (which is approximately 1.96).
Confidence Intervals: Calculating the boundaries for confidence intervals.
Simulations: Generating random numbers from a normal distribution using techniques like the inverse transform sampling method.
Quality Control: Setting tolerance limits based on probability.
Risk Management: Calculating Value at Risk (VaR) or other risk metrics.
Example Calculation
Let's find the value that corresponds to the 97.5th percentile (0.975 cumulative probability) for a standard normal distribution (mean = 0, std dev = 1).
Cumulative Probability (P): 0.975
Mean (μ): 0
Standard Deviation (σ): 1
The calculator will find the z-score for P = 0.975. This value is approximately 1.96.
Using the formula: X = 0 + 1.96 * 1 = 1.96.
This means that 97.5% of the values in a standard normal distribution are less than 1.96.
Now, let's consider a normal distribution with a mean of 100 and a standard deviation of 15 (like an IQ test score distribution).
Cumulative Probability (P): 0.975
Mean (μ): 100
Standard Deviation (σ): 15
The z-score for P = 0.975 is still approximately 1.96.
Using the formula: X = 100 + 1.96 * 15 = 100 + 29.4 = 129.4.
This indicates that a score of approximately 129.4 represents the 97.5th percentile in this specific IQ distribution.
// Function to approximate the inverse of the standard normal CDF (probit function)
// This is a common approximation polynomial. More complex ones exist for higher accuracy.
// Source: Adapted from Abramowitz and Stegun, Handbook of Mathematical Functions.
// Note: This approximation is generally good for probabilities between 0.0001 and 0.9999.
function standardNormalInverseCdf(p) {
if (p = 1) return Infinity;
if (p === 0.5) return 0;
var q = p;
if (q > 0.5) q = 1 – q;
var a = [-3.3440454, -1.3312723, -0.19332795, 0.4496477];
var b = [-2.1857587, -0.19855093, 0.19197421];
var c = [-2.7827701, -0.34776101, 0.10654055, 0.00689853];
var d = [-1.3626762, -0.08976755, 0.12499712, -0.00817696];
var r = Math.min(q, 1 – q);
var t = Math.sqrt(-2 * Math.log(r));
var qp = 0.5 – (a[0] + t * (a[1] + t * (a[2] + t * a[3]))) / (1 + t * (b[0] + t * (b[1] + t * b[2])));
var res = qp + (c[0] + t * (c[1] + t * (c[2] + t * c[3]))) / (1 + t * (d[0] + t * (d[1] + t * (d[2] + t * d[3]))));
if (p < 0.5) {
res = -res;
}
return res;
}
function calculateInverseNormalCdf() {
var p = parseFloat(document.getElementById("probability").value);
var mean = parseFloat(document.getElementById("mean").value);
var stddev = parseFloat(document.getElementById("stddev").value);
var resultValueElement = document.getElementById("resultValue");
// Input validation
if (isNaN(p) || isNaN(mean) || isNaN(stddev)) {
resultValueElement.textContent = "Invalid input. Please enter numbers.";
return;
}
if (p 1) {
resultValueElement.textContent = "Probability must be between 0 and 1.";
return;
}
if (stddev <= 0) {
resultValueElement.textContent = "Standard deviation must be positive.";
return;
}
var zScore = standardNormalInverseCdf(p);
// Check if zScore calculation resulted in +/- Infinity (for p=0 or p=1)
if (!isFinite(zScore)) {
resultValueElement.textContent = zScore;
return;
}
var finalValue = mean + zScore * stddev;
resultValueElement.textContent = finalValue.toFixed(6); // Display with reasonable precision
}