The Normal Cumulative Distribution Function (CDF) calculates the probability that a random variable, following a normal distribution (the bell curve), will take a value less than or equal to a specific value x. Unlike the Probability Density Function (PDF), which gives the "height" of the curve at a point, the CDF gives the total area under the curve to the left of that point.
Understanding the Inputs
Mean (μ): The center of the distribution. In a standard normal distribution, the mean is 0.
Standard Deviation (σ): This measures the spread of the data. A higher value means the bell curve is wider and flatter. For a standard normal distribution, this is 1.
X Value: The specific threshold or score you are testing.
Z = (x – μ) / σ
Φ(z) = P(Z ≤ z)
How to Calculate Normal CDF
Calculating the Normal CDF manually requires complex calculus because the integral of the normal distribution function doesn't have a simple algebraic solution. Statisticians historically used "Z-tables." This calculator uses a high-precision polynomial approximation (Abramowitz and Stegun formula) to provide results instantly.
Real-World Example
Scenario: Suppose IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. What is the probability that a randomly selected person has an IQ below 130?
Enter Mean (μ) = 100
Enter Standard Deviation (σ) = 15
Enter X Value = 130
The calculator finds a Z-score of 2.0 ( (130-100)/15 ).
The result is approximately 0.9772, meaning 97.72% of people have an IQ below 130.
Important Properties
Because the normal distribution is perfectly symmetrical:
The probability of the mean (x = μ) is always 0.50 (50%).
The total area under the entire curve is always 1.0.
P(X > x) is simply 1 – P(X < x).
function calculateNormalCDF() {
var mean = parseFloat(document.getElementById('mean').value);
var stdDev = parseFloat(document.getElementById('stdDev').value);
var x = parseFloat(document.getElementById('xValue').value);
if (isNaN(mean) || isNaN(stdDev) || isNaN(x)) {
alert("Please enter valid numeric values.");
return;
}
if (stdDev 0) {
finalProb = 1.0 – prob;
} else {
finalProb = prob;
}
// Calculate tail probabilities
var lowerTail = finalProb;
var upperTail = 1.0 – finalProb;
// Display Results
document.getElementById('results').style.display = 'block';
document.getElementById('zScoreRes').innerText = z.toFixed(4);
document.getElementById('lowerTailRes').innerText = lowerTail.toFixed(6);
document.getElementById('upperTailRes').innerText = upperTail.toFixed(6);
document.getElementById('finalResult').innerText = "P(X ≤ " + x + ") = " + (lowerTail * 100).toFixed(4) + "%";
}