Normal Distribution Calculator
body {
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background-color: #f8f9fa;
color: #333;
line-height: 1.6;
margin: 0;
padding: 20px;
}
.loan-calc-container {
max-width: 800px;
margin: 30px auto;
background-color: #ffffff;
padding: 30px;
border-radius: 8px;
box-shadow: 0 4px 15px rgba(0, 0, 0, 0.1);
border: 1px solid #e0e0e0;
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h1, h2 {
color: #004a99;
text-align: center;
margin-bottom: 25px;
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.input-group {
margin-bottom: 20px;
display: flex;
align-items: center;
flex-wrap: wrap; /* Allow wrapping on smaller screens */
}
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flex: 0 0 180px; /* Fixed width for labels */
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min-width: 150px; /* Minimum width for inputs */
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margin-top: 30px;
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margin-bottom: 8px;
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.input-group input[type="number"],
.input-group input[type="text"],
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.loan-calc-container {
padding: 20px;
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}
Normal Distribution Calculator
Understanding Normal Distribution
The normal distribution, also known as the Gaussian distribution or bell curve, is one of the most important probability distributions in statistics. It describes a continuous random variable whose distribution is symmetric about the mean. The mean ($\mu$) and standard deviation ($\sigma$) are the two key parameters that define a normal distribution.
Key Concepts:
- Mean ($\mu$): The average value of the data set. It's the center of the distribution.
- Standard Deviation ($\sigma$): A measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.
- Bell Curve: The characteristic shape of the normal distribution, symmetrical around the mean.
- Empirical Rule (68-95-99.7 Rule): For a normal distribution, approximately 68% of data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.
Calculating Probabilities:
To calculate probabilities for a normal distribution, we often convert the value of interest (x) into a z-score. The z-score tells us how many standard deviations a particular value is away from the mean. The formula for a z-score is:
z = (x - μ) / σ
Once we have the z-score, we can use a standard normal distribution table (z-table) or statistical software/calculators to find the probability associated with that z-score. The probabilities represent the area under the bell curve.
Calculator Functionality:
This calculator helps you compute different probabilities based on the mean ($\mu$), standard deviation ($\sigma$), and a specific value (x) or range of values (a and b) from a normal distribution.
- P(X < x): Calculates the probability that a random variable X will be less than a given value x. This corresponds to the area under the curve to the left of x.
- P(X > x): Calculates the probability that a random variable X will be greater than a given value x. This corresponds to the area under the curve to the right of x.
- P(a < X < b): Calculates the probability that a random variable X will fall between two values, a and b. This corresponds to the area under the curve between a and b.
Use Cases:
Normal distributions are ubiquitous in real-world phenomena, including:
- Natural Sciences: Heights of people, measurement errors, growth patterns.
- Finance: Stock price movements (often modeled as log-normal, but related), credit default probabilities.
- Quality Control: Variations in manufactured product dimensions.
- Social Sciences: Test scores, IQ scores.
- Medicine: Blood pressure readings, cholesterol levels.
Understanding and calculating probabilities from a normal distribution is crucial for making informed decisions, risk assessment, and statistical inference in various fields.
// Function to calculate the cumulative distribution function (CDF) for a normal distribution
// This is a simplified approximation using the error function (erf)
// For higher precision, a more complex algorithm or a lookup table would be needed.
// This implementation uses a common approximation found online.
function normalCDF(z) {
var t = 1.0 / (1.0 + 0.3275911 * Math.abs(z));
var cdf = 1.0 – 0.254829592 * t – 0.284496736 * Math.pow(t, 2) – 1.421413741 * Math.pow(t, 3) – 1.453152027 * Math.pow(t, 4) + 1.061405429 * Math.pow(t, 5);
if (z < 0) {
cdf = 1.0 – cdf;
}
return cdf;
}
function calculateNormalDistribution() {
var mean = parseFloat(document.getElementById("mean").value);
var stdDev = parseFloat(document.getElementById("stdDev").value);
var valueX = parseFloat(document.getElementById("valueX").value);
var calculationType = document.getElementById("calculationType").value;
var valueA = parseFloat(document.getElementById("valueA").value);
var valueB = parseFloat(document.getElementById("valueB").value);
var resultValueElement = document.getElementById("result-value");
var resultUnitElement = document.getElementById("result-unit");
resultValueElement.textContent = "–";
resultUnitElement.textContent = "";
// — Input Validation —
if (isNaN(mean) || isNaN(stdDev) || isNaN(valueX)) {
resultValueElement.textContent = "Invalid Input";
resultUnitElement.textContent = "Please enter valid numbers for Mean, Standard Deviation, and Value(s).";
return;
}
if (stdDev <= 0) {
resultValueElement.textContent = "Invalid Std. Dev.";
resultUnitElement.textContent = "Standard Deviation must be a positive number.";
return;
}
var probability = 0;
var displayResult = "";
if (calculationType === "probability_less_than") {
var z = (valueX – mean) / stdDev;
probability = normalCDF(z);
displayResult = probability.toFixed(6);
resultUnitElement.textContent = "Probability P(X " + valueX + ")";
} else if (calculationType === "probability_between") {
if (isNaN(valueA) || isNaN(valueB)) {
resultValueElement.textContent = "Invalid Input";
resultUnitElement.textContent = "Please enter valid numbers for both Lower Bound (a) and Upper Bound (b).";
return;
}
if (valueA >= valueB) {
resultValueElement.textContent = "Invalid Range";
resultUnitElement.textContent = "Lower Bound (a) must be less than Upper Bound (b).";
return;
}
var zA = (valueA – mean) / stdDev;
var zB = (valueB – mean) / stdDev;
probability = normalCDF(zB) – normalCDF(zA);
displayResult = probability.toFixed(6);
resultUnitElement.textContent = "Probability P(" + valueA + " < X < " + valueB + ")";
}
resultValueElement.textContent = displayResult;
}
// Show/hide input fields for range based on selection
var calculationTypeSelect = document.getElementById("calculationType");
var valueAInputGroup = document.getElementById("valueA-group");
var valueBInputGroup = document.getElementById("valueB-group");
calculationTypeSelect.onchange = function() {
if (this.value === "probability_between") {
valueAInputGroup.style.display = "flex";
valueBInputGroup.style.display = "flex";
} else {
valueAInputGroup.style.display = "none";
valueBInputGroup.style.display = "none";
}
};
// Initial check on load
if (calculationTypeSelect.value === "probability_between") {
valueAInputGroup.style.display = "flex";
valueBInputGroup.style.display = "flex";
} else {
valueAInputGroup.style.display = "none";
valueBInputGroup.style.display = "none";
}