Use this calculator to find probabilities and probability density for a Normal (Gaussian) distribution given its mean and standard deviation.
Calculate Probability Density (PDF) and Cumulative Probability (CDF) for a specific value (x):
Calculate Cumulative Probability (CDF) between two values (x1 and x2):
Results:
.probability-distribution-calculator {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
max-width: 600px;
margin: 20px auto;
padding: 25px;
border: 1px solid #e0e0e0;
border-radius: 10px;
background-color: #f9f9f9;
box-shadow: 0 4px 12px rgba(0, 0, 0, 0.08);
}
.probability-distribution-calculator h2,
.probability-distribution-calculator h3 {
color: #333;
text-align: center;
margin-bottom: 20px;
}
.probability-distribution-calculator p {
color: #555;
line-height: 1.6;
margin-bottom: 15px;
text-align: center;
}
.calc-input-group {
margin-bottom: 15px;
display: flex;
flex-direction: column;
}
.calc-input-group label {
margin-bottom: 8px;
color: #444;
font-weight: bold;
}
.calc-input-group input[type="number"] {
padding: 10px 12px;
border: 1px solid #ccc;
border-radius: 5px;
font-size: 16px;
width: 100%;
box-sizing: border-box;
}
.calc-input-group input[type="number"]:focus {
border-color: #007bff;
outline: none;
box-shadow: 0 0 0 2px rgba(0, 123, 255, 0.25);
}
.probability-distribution-calculator button {
display: block;
width: 100%;
padding: 12px 20px;
background-color: #007bff;
color: white;
border: none;
border-radius: 5px;
font-size: 18px;
font-weight: bold;
cursor: pointer;
transition: background-color 0.3s ease, transform 0.2s ease;
margin-top: 20px;
}
.probability-distribution-calculator button:hover {
background-color: #0056b3;
transform: translateY(-2px);
}
.probability-distribution-calculator button:active {
transform: translateY(0);
}
.calc-results {
margin-top: 30px;
padding-top: 20px;
border-top: 1px solid #eee;
}
.calc-results div {
background-color: #e9f7ff;
border: 1px solid #cce5ff;
padding: 10px 15px;
margin-bottom: 10px;
border-radius: 5px;
color: #004085;
font-size: 16px;
}
.calc-results div:last-child {
margin-bottom: 0;
}
#errorMessage {
background-color: #f8d7da;
border-color: #f5c6cb;
color: #721c24;
}
// Function to approximate the error function (erf)
// This is a common approximation from Abramowitz and Stegun
function erf(x) {
// constants
var a1 = 0.254829592;
var a2 = -0.284496736;
var a3 = 1.421413741;
var a4 = -1.453152027;
var a5 = 1.061405429;
var p = 0.3275911;
// Save the sign of x
var sign = 1;
if (x < 0) {
sign = -1;
x = -x;
}
// A&S formula 7.1.26
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 to calculate the Cumulative Distribution Function (CDF) for a standard normal distribution (Z-score)
function standardNormalCDF(z) {
return 0.5 * (1 + erf(z / Math.SQRT2));
}
function calculateNormalDistribution() {
var mean = parseFloat(document.getElementById("meanValue").value);
var stdDev = parseFloat(document.getElementById("stdDevValue").value);
var x = parseFloat(document.getElementById("xValue").value);
var x1 = parseFloat(document.getElementById("x1Value").value);
var x2 = parseFloat(document.getElementById("x2Value").value);
var pdfResultDiv = document.getElementById("pdfResult");
var cdfLessThanResultDiv = document.getElementById("cdfLessThanResult");
var cdfGreaterThanResultDiv = document.getElementById("cdfGreaterThanResult");
var cdfBetweenResultDiv = document.getElementById("cdfBetweenResult");
var errorMessageDiv = document.getElementById("errorMessage");
pdfResultDiv.innerHTML = "";
cdfLessThanResultDiv.innerHTML = "";
cdfGreaterThanResultDiv.innerHTML = "";
cdfBetweenResultDiv.innerHTML = "";
errorMessageDiv.innerHTML = "";
if (isNaN(mean) || isNaN(stdDev)) {
errorMessageDiv.innerHTML = "Please enter valid numbers for Mean and Standard Deviation.";
return;
}
if (stdDev <= 0) {
errorMessageDiv.innerHTML = "Standard Deviation must be a positive number.";
return;
}
var PI = Math.PI;
var SQRT2PI = Math.sqrt(2 * PI);
// Calculate PDF
if (!isNaN(x)) {
var exponent = -Math.pow((x – mean), 2) / (2 * Math.pow(stdDev, 2));
var pdf = (1 / (stdDev * SQRT2PI)) * Math.exp(exponent);
pdfResultDiv.innerHTML = "Probability Density Function (PDF) at x = " + x + ": " + pdf.toFixed(6);
} else {
pdfResultDiv.innerHTML = "Probability Density Function (PDF) at x: N/A (Specific Value 'x' not provided)";
}
// Calculate CDF (P(X < x))
if (!isNaN(x)) {
var z_x = (x – mean) / stdDev;
var cdfLessThan = standardNormalCDF(z_x);
cdfLessThanResultDiv.innerHTML = "Cumulative Probability P(X < " + x + "): " + (cdfLessThan * 100).toFixed(4) + "%";
// Calculate P(X > x)
var cdfGreaterThan = 1 – cdfLessThan;
cdfGreaterThanResultDiv.innerHTML = "Cumulative Probability P(X > " + x + "): " + (cdfGreaterThan * 100).toFixed(4) + "%";
} else {
cdfLessThanResultDiv.innerHTML = "Cumulative Probability P(X < x): N/A (Specific Value 'x' not provided)";
cdfGreaterThanResultDiv.innerHTML = "Cumulative Probability P(X > x): N/A (Specific Value 'x' not provided)";
}
// Calculate CDF between x1 and x2
if (!isNaN(x1) && !isNaN(x2)) {
if (x1 >= x2) {
errorMessageDiv.innerHTML += "For P(x1 < X < x2), Lower Bound (x1) must be less than Upper Bound (x2).";
cdfBetweenResultDiv.innerHTML = "Cumulative Probability P(" + x1 + " < X < " + x2 + "): Invalid input";
} else {
var z_x1 = (x1 – mean) / stdDev;
var z_x2 = (x2 – mean) / stdDev;
var cdf_x1 = standardNormalCDF(z_x1);
var cdf_x2 = standardNormalCDF(z_x2);
var cdfBetween = cdf_x2 – cdf_x1;
cdfBetweenResultDiv.innerHTML = "Cumulative Probability P(" + x1 + " < X < " + x2 + "): " + (cdfBetween * 100).toFixed(4) + "%";
}
} else {
cdfBetweenResultDiv.innerHTML = "Cumulative Probability P(x1 < X < x2): N/A (Both x1 and x2 must be provided)";
}
}
Understanding the Normal Distribution and Its Probabilities
The Normal Distribution, often referred to as the Gaussian distribution or the "bell curve," is one of the most fundamental and widely used probability distributions in statistics and natural sciences. It describes how the values of a variable are distributed, with most values clustering around a central mean and tapering off symmetrically as they move away from the mean.
Key Characteristics of the Normal Distribution:
Symmetry: The distribution is perfectly symmetrical around its mean. This means that the left side of the curve is a mirror image of the right side.
Mean, Median, Mode: In a perfect normal distribution, the mean, median, and mode are all equal and located at the center of the curve.
Bell Shape: The graph of a normal distribution is a distinctive bell shape.
Asymptotic: The tails of the curve approach the x-axis but never actually touch it, meaning that theoretically, extreme values are possible, though highly improbable.
Defined by Two Parameters: A normal distribution is completely characterized by two parameters:
Mean (μ): This is the central value of the distribution, representing the average of all values. It determines the location of the peak of the bell curve.
Standard Deviation (σ): This measures the spread or dispersion of the data around the mean. A smaller standard deviation indicates that data points are clustered closely around the mean, resulting in a tall, narrow curve. A larger standard deviation means data points are more spread out, leading to a flatter, wider curve.
Probability Density Function (PDF)
The Probability Density Function (PDF) for a continuous distribution like the normal distribution does not give the probability of a specific value occurring. Instead, it describes the likelihood of a random variable falling within a given range of values. The value of the PDF at a specific point 'x' indicates the relative likelihood of the variable being close to 'x'. The formula for the normal distribution PDF is:
The area under the entire PDF curve is always equal to 1 (or 100%), representing the total probability of all possible outcomes.
Cumulative Distribution Function (CDF)
The Cumulative Distribution Function (CDF) gives the probability that a random variable takes a value less than or equal to a specific value 'x'. In other words, it calculates the area under the PDF curve from negative infinity up to 'x'.
P(X < x): This is the probability that a randomly selected value from the distribution will be less than 'x'.
P(X > x): This is the probability that a randomly selected value will be greater than 'x'. It is calculated as 1 - P(X < x).
P(x1 < X < x2): This is the probability that a randomly selected value will fall between two specific values, 'x1' and 'x2'. It is calculated as P(X < x2) - P(X < x1).
The CDF is crucial for understanding the likelihood of events occurring within certain ranges.
How to Use the Calculator:
Enter Mean (μ): Input the average value of your dataset. For example, if you're looking at IQ scores, the mean is typically 100.
Enter Standard Deviation (σ): Input the measure of spread. For IQ scores, the standard deviation is often 15.
Enter Specific Value (x): If you want to find the probability density at a particular point or the cumulative probability up to that point, enter 'x'.
Enter Lower Bound (x1) and Upper Bound (x2): If you want to find the probability of a value falling within a specific range, enter 'x1' and 'x2'.
Click "Calculate Probabilities": The calculator will then display the PDF at 'x', the CDF for P(X < x), P(X > x), and P(x1 < X < x2) based on your inputs.
Example Scenarios:
Imagine a standardized test where scores are normally distributed with a mean (μ) of 500 and a standard deviation (σ) of 100.
What is the probability density at a score of 600?
Mean: 500
Standard Deviation: 100
Specific Value (x): 600
Result: The PDF value will indicate the relative likelihood of a score around 600.
What is the probability of a student scoring less than 650?
Mean: 500
Standard Deviation: 100
Specific Value (x): 650
Result: P(X < 650) will be displayed, likely around 93.32%.
What is the probability of a student scoring between 450 and 550?
Mean: 500
Standard Deviation: 100
Lower Bound (x1): 450
Upper Bound (x2): 550
Result: P(450 < X < 550) will be displayed, likely around 38.29%.
This calculator provides a practical tool for exploring the probabilities associated with normally distributed data, making complex statistical concepts more accessible.