P(X < x)
P(X > x)
P(x1 < X < x2)
P(X is outside x1, x2)
Calculation Results
Understanding the Normal Distribution
The Normal Distribution, often referred to as the Bell Curve or Gaussian Distribution, is the most important probability distribution in statistics. It describes many natural phenomena, from heights and weights to test scores and manufacturing errors. This calculator helps you determine the probability of an event occurring within a specific range of values based on the distribution's mean and standard deviation.
Key Parameters of Normal Probability
Mean (μ): The average value where the peak of the curve is centered.
Standard Deviation (σ): A measure of how spread out the data points are. A smaller standard deviation creates a narrow, tall curve, while a larger one creates a wide, flat curve.
Z-Score: The number of standard deviations a specific data point is away from the mean. The formula is: z = (x – μ) / σ.
How to Use This Calculator
Follow these steps to find the probability of your specific data set:
Enter the Mean: Input the central value of your data set.
Enter the Standard Deviation: Input the spread of your data.
Select the Range: Choose whether you want the probability "less than", "greater than", "between two values", or "outside two values".
Input your x-values: Provide the specific numbers you are testing.
Interpret results: The calculator provides the Z-score and the total probability as both a decimal (0 to 1) and a percentage (0% to 100%).
Real-World Example: Standardized Testing
Suppose an exam (like the SAT) has a mean score of 1050 and a standard deviation of 200. If you want to know what percentage of students score above 1300:
Mean (μ) = 1050
Standard Deviation (σ) = 200
Type = P(X > x)
x = 1300
The calculator will compute a Z-score of 1.25. The area to the right of this Z-score represents roughly 10.56% of students who scored above 1300.
Common Probability Thresholds
In statistics, researchers often look for the Empirical Rule (68-95-99.7 rule):
68.27% of the data falls within 1 standard deviation of the mean.
95.45% of the data falls within 2 standard deviations of the mean.
99.73% of the data falls within 3 standard deviations of the mean.
function toggleInputs() {
var type = document.getElementById("probType").value;
var x2Container = document.getElementById("x2-container");
var x1Label = document.getElementById("x1-label");
if (type === "between" || type === "outside") {
x2Container.style.display = "block";
x1Label.innerText = "Lower Value (x1)";
} else {
x2Container.style.display = "none";
x1Label.innerText = "Value (x)";
}
}
function erf(x) {
var a1 = 0.254829592;
var a2 = -0.284496736;
var a3 = 1.421413741;
var a4 = -1.453152027;
var a5 = 1.061405429;
var p = 0.3275911;
var sign = 1;
if (x < 0) sign = -1;
x = Math.abs(x);
var t = 1.0 / (1.0 + p * x);
var y = 1.0 – (((((a5 * t + a4) * t) + a3) * t + a2) * t + a1) * t * Math.exp(-x * x);
return sign * y;
}
function normalCDF(x, mean, stdDev) {
return 0.5 * (1 + erf((x – mean) / (stdDev * Math.sqrt(2))));
}
function calculateNormalProb() {
var mu = parseFloat(document.getElementById("meanVal").value);
var sigma = parseFloat(document.getElementById("stdDevVal").value);
var type = document.getElementById("probType").value;
var x1 = parseFloat(document.getElementById("x1Val").value);
var x2 = parseFloat(document.getElementById("x2Val").value);
if (isNaN(mu) || isNaN(sigma) || isNaN(x1)) {
alert("Please enter valid numeric values.");
return;
}
if (sigma <= 0) {
alert("Standard deviation must be greater than zero.");
return;
}
var prob = 0;
var resultText = "";
var zText = "";
if (type === "less") {
prob = normalCDF(x1, mu, sigma);
var z = (x1 – mu) / sigma;
zText = "Z-Score: " + z.toFixed(4);
resultText = "P(X " + x1 + ")";
} else if (type === "between") {
if (isNaN(x2)) { alert("Please enter x2"); return; }
var p1 = normalCDF(x1, mu, sigma);
var p2 = normalCDF(x2, mu, sigma);
prob = Math.abs(p2 – p1);
var z1 = (x1 – mu) / sigma;
var z2 = (x2 – mu) / sigma;
zText = "Z1: " + z1.toFixed(4) + " | Z2: " + z2.toFixed(4);
resultText = "P(" + Math.min(x1, x2) + " < X < " + Math.max(x1, x2) + ")";
} else if (type === "outside") {
if (isNaN(x2)) { alert("Please enter x2"); return; }
var p1 = normalCDF(x1, mu, sigma);
var p2 = normalCDF(x2, mu, sigma);
prob = 1 – Math.abs(p2 – p1);
var z1 = (x1 – mu) / sigma;
var z2 = (x2 – mu) / sigma;
zText = "Z1: " + z1.toFixed(4) + " | Z2: " + z2.toFixed(4);
resultText = "P(X is outside " + Math.min(x1, x2) + " and " + Math.max(x1, x2) + ")";
}
document.getElementById("resultsArea").style.display = "block";
document.getElementById("zScoreRes").innerText = zText;
document.getElementById("decimalRes").innerText = resultText + " = " + prob.toFixed(6);
document.getElementById("percentRes").innerText = "Probability: " + (prob * 100).toFixed(4) + "%";
document.getElementById("resultsArea").scrollIntoView({ behavior: 'smooth', block: 'nearest' });
}