Normal Distribution Calculator Probability

Normal Distribution Probability Calculator

Mean (μ):

Standard Deviation (σ):

Probability Type:
P(X < x)
P(X > x)
P(x1 < X < x2)

Value X (or x1):

Value X2:

Result:

// Function to toggle visibility of xValue2 input function toggleX2() { var probType = document.querySelector('input[name="probType"]:checked').value; var xValue2Group = document.getElementById('xValue2Group'); if (probType === 'between') { xValue2Group.style.display = 'block'; } else { xValue2Group.style.display = 'none'; } } // Error function (erf) approximation // From Abramowitz and Stegun, Handbook of Mathematical Functions, 7.1.26 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 Cumulative Distribution Function (CDF) function normalCDF(x) { return 0.5 * (1 + erf(x / Math.sqrt(2))); } function calculateNormalProbability() { var mean = parseFloat(document.getElementById('mean').value); var stdDev = parseFloat(document.getElementById('stdDev').value); var xValue1 = parseFloat(document.getElementById('xValue1').value); var xValue2 = parseFloat(document.getElementById('xValue2').value); var probType = document.querySelector('input[name="probType"]:checked').value; var resultDiv = document.getElementById('result'); if (isNaN(mean) || isNaN(stdDev) || isNaN(xValue1)) { resultDiv.innerHTML = "Please enter valid numbers for Mean, Standard Deviation, and X Value(s)."; return; } if (stdDev <= 0) { resultDiv.innerHTML = "Standard Deviation must be a positive number."; return; } var probability = 0; var z1, z2; switch (probType) { case 'lessThan': z1 = (xValue1 – mean) / stdDev; probability = normalCDF(z1); resultDiv.innerHTML = "P(X " + xValue1 + ") = " + (probability * 100).toFixed(4) + "%"; break; case 'between': if (isNaN(xValue2)) { resultDiv.innerHTML = "Please enter a valid number for X2."; return; } if (xValue1 >= xValue2) { resultDiv.innerHTML = "For P(x1 < X < x2), x1 must be less than x2."; return; } z1 = (xValue1 – mean) / stdDev; z2 = (xValue2 – mean) / stdDev; probability = normalCDF(z2) – normalCDF(z1); resultDiv.innerHTML = "P(" + xValue1 + " < X < " + xValue2 + ") = " + (probability * 100).toFixed(4) + "%"; break; } } // Initialize xValue2 visibility on page load window.onload = toggleX2; .calculator-container { background-color: #f9f9f9; border: 1px solid #ddd; padding: 20px; border-radius: 8px; max-width: 500px; margin: 20px auto; font-family: Arial, sans-serif; } .calculator-container h2 { text-align: center; color: #333; margin-bottom: 20px; } .calc-input-group { margin-bottom: 15px; } .calc-input-group label { display: block; margin-bottom: 5px; color: #555; font-weight: bold; } .calc-input-group input[type="number"], .calc-input-group select { width: calc(100% – 12px); padding: 8px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; } .calc-input-group input[type="radio"] { margin-right: 5px; } .calc-input-group input[type="radio"] + label { display: inline-block; font-weight: normal; margin-right: 15px; } button { background-color: #007bff; color: white; padding: 10px 15px; border: none; border-radius: 4px; cursor: pointer; font-size: 16px; width: 100%; box-sizing: border-box; margin-top: 10px; } button:hover { background-color: #0056b3; } .calc-result-group { margin-top: 20px; padding: 15px; background-color: #e9ecef; border: 1px solid #ced4da; border-radius: 4px; } .calc-result-group h3 { color: #333; margin-top: 0; margin-bottom: 10px; } #result { font-size: 1.1em; color: #000; font-weight: bold; }

Understanding the Normal Distribution and Its Probabilities

The normal distribution, often called the Gaussian distribution or "bell curve," is a fundamental concept in statistics and probability theory. It describes how the values of a variable are distributed, with most observations clustering around a central peak and probabilities tapering off symmetrically in both directions.

Key Characteristics of the Normal Distribution:

Symmetry: The distribution is perfectly symmetrical around its mean.
Mean, Median, Mode: For a normal distribution, the mean, median, and mode are all equal and located at the center of the curve.
Asymptotic: The tails of the curve approach the x-axis but never quite touch it, meaning there's always a tiny probability for extreme values.
Defined by Two Parameters: The entire shape of a normal distribution is determined by just two values: the mean and the standard deviation.

The Two Defining Parameters:

Mean (μ): This is the central tendency of the distribution. It represents the average value of the data and is the peak of the bell curve. A change in the mean shifts the entire curve along the x-axis.
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 tightly 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.

What is a Z-Score?

A Z-score (also known as a standard score) measures how many standard deviations an element is from the mean. It's a way to standardize data from different normal distributions, allowing for comparison. The formula for a Z-score is:

Z = (X - μ) / σ

Where:

X is the individual data point
μ (mu) is the mean of the distribution
σ (sigma) is the standard deviation of the distribution

A positive Z-score indicates the data point is above the mean, while a negative Z-score indicates it's below the mean.

Calculating Probabilities with the Normal Distribution

The area under the normal distribution curve represents probability. The total area under the curve is always equal to 1 (or 100%). Our calculator helps you find the probability of a random variable X falling within a certain range.

1. Probability Less Than a Value (P(X < x)):

This calculates the probability that a randomly selected value from the distribution will be less than a specific value 'x'. It corresponds to the area under the curve to the left of 'x'.

Example: If the average height of adult males is 175 cm (μ=175) with a standard deviation of 7 cm (σ=7), what is the probability that a randomly selected male is shorter than 170 cm (x=170)?

Z = (170 – 175) / 7 = -0.714
Using the calculator for P(X < 170) with μ=175, σ=7, you would get approximately 23.76%.

2. Probability Greater Than a Value (P(X > x)):

This calculates the probability that a randomly selected value will be greater than a specific value 'x'. It corresponds to the area under the curve to the right of 'x'.

Example: Using the same height distribution (μ=175, σ=7), what is the probability that a randomly selected male is taller than 185 cm (x=185)?

Z = (185 – 175) / 7 = 1.429
Using the calculator for P(X > 185) with μ=175, σ=7, you would get approximately 7.65%.

3. Probability Between Two Values (P(x1 < X < x2)):

This calculates the probability that a randomly selected value will fall between two specific values, 'x1' and 'x2'. It corresponds to the area under the curve between 'x1' and 'x2'.

Example: For the same height distribution (μ=175, σ=7), what is the probability that a randomly selected male's height is between 170 cm (x1=170) and 180 cm (x2=180)?

Z1 = (170 – 175) / 7 = -0.714
Z2 = (180 – 175) / 7 = 0.714
Using the calculator for P(170 < X < 180) with μ=175, σ=7, you would get approximately 52.49%.

How to Use the Calculator:

Enter Mean (μ): Input the average value of your dataset.
Enter Standard Deviation (σ): Input the measure of spread for your dataset.
Select Probability Type: Choose whether you want to find the probability of a value being less than, greater than, or between specific points.
Enter X Value(s):
- For "less than" or "greater than," enter a single 'X' value.
- For "between," enter both 'X1' (lower bound) and 'X2' (upper bound).
Click "Calculate Probability": The calculator will display the probability as a percentage.

This calculator is a valuable tool for students, researchers, and professionals in fields ranging from finance to engineering, helping to quickly understand and apply the principles of normal distribution.