.probability-calculator-normal-distribution-wrapper {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
max-width: 700px;
margin: 20px auto;
padding: 25px;
border: 1px solid #e0e0e0;
border-radius: 10px;
background-color: #fdfdfd;
box-shadow: 0 4px 12px rgba(0, 0, 0, 0.08);
}
.probability-calculator-normal-distribution-wrapper h2 {
text-align: center;
color: #333;
margin-bottom: 25px;
font-size: 1.8em;
}
.probability-calculator-normal-distribution-wrapper .form-group {
margin-bottom: 18px;
display: flex;
flex-wrap: wrap;
align-items: center;
}
.probability-calculator-normal-distribution-wrapper .form-group label {
flex: 1;
min-width: 150px;
color: #555;
font-weight: bold;
margin-right: 15px;
font-size: 1em;
}
.probability-calculator-normal-distribution-wrapper .form-group input[type="number"] {
flex: 2;
min-width: 180px;
padding: 10px 12px;
border: 1px solid #ccc;
border-radius: 5px;
font-size: 1em;
box-sizing: border-box;
transition: border-color 0.3s ease;
}
.probability-calculator-normal-distribution-wrapper .form-group input[type="number"]:focus {
border-color: #007bff;
outline: none;
box-shadow: 0 0 5px rgba(0, 123, 255, 0.3);
}
.probability-calculator-normal-distribution-wrapper .form-group input[type="radio"] {
margin-right: 8px;
transform: scale(1.1);
}
.probability-calculator-normal-distribution-wrapper .form-group label[for^="prob"] {
font-weight: normal;
color: #333;
margin-right: 20px;
min-width: unset;
flex: unset;
}
.probability-calculator-normal-distribution-wrapper button {
display: block;
width: 100%;
padding: 12px 20px;
background-color: #007bff;
color: white;
border: none;
border-radius: 5px;
font-size: 1.1em;
font-weight: bold;
cursor: pointer;
transition: background-color 0.3s ease, transform 0.2s ease;
margin-top: 25px;
}
.probability-calculator-normal-distribution-wrapper button:hover {
background-color: #0056b3;
transform: translateY(-2px);
}
.probability-calculator-normal-distribution-wrapper .result-container {
margin-top: 30px;
padding: 15px;
border: 1px solid #d4edda;
background-color: #e9f7ef;
border-radius: 8px;
text-align: center;
font-size: 1.1em;
color: #155724;
}
.probability-calculator-normal-distribution-wrapper .result-container h3 {
color: #155724;
margin-top: 0;
margin-bottom: 10px;
font-size: 1.4em;
}
.probability-calculator-normal-distribution-wrapper #probabilityResult strong {
color: #0a3622;
font-size: 1.3em;
}
.probability-calculator-normal-distribution-wrapper .article-content {
margin-top: 40px;
line-height: 1.7;
color: #333;
font-size: 1em;
}
.probability-calculator-normal-distribution-wrapper .article-content h3 {
color: #007bff;
margin-top: 25px;
margin-bottom: 15px;
font-size: 1.5em;
}
.probability-calculator-normal-distribution-wrapper .article-content p {
margin-bottom: 15px;
}
.probability-calculator-normal-distribution-wrapper .article-content ul {
list-style-type: disc;
margin-left: 20px;
margin-bottom: 15px;
}
.probability-calculator-normal-distribution-wrapper .article-content ul li {
margin-bottom: 8px;
}
.probability-calculator-normal-distribution-wrapper .form-group #rangeXInputs label {
min-width: 100px;
}
@media (max-width: 600px) {
.probability-calculator-normal-distribution-wrapper .form-group {
flex-direction: column;
align-items: flex-start;
}
.probability-calculator-normal-distribution-wrapper .form-group label {
margin-bottom: 8px;
min-width: unset;
}
.probability-calculator-normal-distribution-wrapper .form-group input[type="number"] {
width: 100%;
min-width: unset;
}
.probability-calculator-normal-distribution-wrapper .form-group #rangeXInputs label {
min-width: unset;
}
}
// erf function (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;
}
// Standard Normal CDF (Cumulative Distribution Function)
function normalCDF(x, mean, stdDev) {
if (stdDev <= 0) {
return NaN; // Indicate an error or invalid input
}
var z = (x – mean) / stdDev;
return 0.5 * (1 + erf(z / Math.sqrt(2)));
}
function toggleXInputs() {
var probBetween = document.getElementById('probBetween');
var singleXInput = document.getElementById('singleXInput');
var rangeXInputs = document.getElementById('rangeXInputs');
if (probBetween.checked) {
singleXInput.style.display = 'none';
rangeXInputs.style.display = 'flex'; /* Use flex for better alignment */
} else {
singleXInput.style.display = 'flex'; /* Use flex for better alignment */
rangeXInputs.style.display = 'none';
}
}
function calculateProbability() {
var mean = parseFloat(document.getElementById('meanValue').value);
var stdDev = parseFloat(document.getElementById('stdDevValue').value);
var resultDiv = document.getElementById('probabilityResult');
// Input validation
if (isNaN(mean) || isNaN(stdDev)) {
resultDiv.innerHTML = 'Please enter valid numbers for Mean and Standard Deviation.';
return;
}
if (stdDev = x2) {
resultDiv.innerHTML = 'X1 Value must be less than X2 Value for "Between" calculation.';
return;
}
probability = normalCDF(x2, mean, stdDev) – normalCDF(x1, mean, stdDev);
}
if (isNaN(probability)) {
resultDiv.innerHTML = 'An error occurred during calculation. Please check your inputs.';
return;
}
resultDiv.innerHTML = 'The probability is: ' + (probability * 100).toFixed(4) + '%';
}
// Initialize the correct input display on page load
window.onload = function() {
toggleXInputs();
};
Normal Distribution Probability Calculator
Result:
Understanding the Normal Distribution Probability Calculator
The Normal Distribution, often referred to as the "bell curve" or Gaussian distribution, is a fundamental concept in statistics and probability theory. 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 Concepts of Normal Distribution
- Mean (μ): This is the central tendency of the distribution, representing the average value of the data. The peak of the bell curve is located at the mean.
- Standard Deviation (σ): This measures the spread or dispersion of the data points 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.
- Z-score: A Z-score (or standard score) measures how many standard deviations an element is from the mean. It's calculated as
Z = (X - μ) / σ. A positive Z-score indicates the value is above the mean, while a negative Z-score indicates it's below the mean. - Probability: In the context of a normal distribution, probability refers to the likelihood that a random variable X falls within a certain range of values. This is represented by the area under the curve.
How to Use This Calculator
This calculator allows you to determine probabilities for a given normal distribution. Follow these steps:
- Enter Mean (μ): Input the average value of your dataset.
- Enter Standard Deviation (σ): Input the measure of spread for your dataset. Remember, the standard deviation must be a positive value.
- Choose Probability Type:
- P(X < x): Select this to find the probability that a random variable X is less than a specific value 'x'.
- P(X > x): Select this to find the probability that a random variable X is greater than a specific value 'x'.
- P(x1 < X < x2): Select this to find the probability that X falls between two specific values, 'x1' and 'x2'.
- Enter X Value(s): Depending on your chosen probability type, enter either a single 'X Value' or both 'X1 Value' and 'X2 Value'. Ensure X1 is less than X2 for the "between" calculation.
- Click "Calculate Probability": The calculator will then display the probability as a percentage.
Examples
Example 1: Probability Less Than (P(X < x))
Imagine IQ scores are normally distributed with a Mean (μ) of 100 and a Standard Deviation (σ) of 15. What is the probability that a randomly selected person has an IQ score less than 115?
- Mean (μ): 100
- Standard Deviation (σ): 15
- Probability Type: P(X < x)
- X Value: 115
- Result: Approximately 84.13%
Example 2: Probability Greater Than (P(X > x))
The lifespan of a certain brand of light bulb is normally distributed with a Mean (μ) of 5000 hours and a Standard Deviation (σ) of 500 hours. What is the probability that a randomly chosen light bulb will last longer than 6000 hours?
- Mean (μ): 5000
- Standard Deviation (σ): 500
- Probability Type: P(X > x)
- X Value: 6000
- Result: Approximately 2.28%
Example 3: Probability Between Two Values (P(x1 < X < x2))
The heights of adult males in a certain population are normally distributed with a Mean (μ) of 175 cm and a Standard Deviation (σ) of 7 cm. What is the probability that a randomly selected adult male's height is between 168 cm and 182 cm?
- Mean (μ): 175
- Standard Deviation (σ): 7
- Probability Type: P(x1 < X < x2)
- X1 Value: 168
- X2 Value: 182
- Result: Approximately 68.27%