Result will appear here.
Enter values above to see the cumulative probability P(X ≤ value).
Understanding the Normal Distribution Calculator
This calculator helps you understand probabilities associated with a normal (or Gaussian) distribution. The normal distribution is a fundamental concept in statistics and probability theory, often visualized as a bell-shaped curve. It describes many natural phenomena, from heights of people to measurement errors.
A normal distribution is defined by two parameters:
Mean (μ): This represents the center of the distribution, the peak of the bell curve. It's the average value.
Standard Deviation (σ): This measures the spread or dispersion of the data around the mean. A smaller standard deviation means the data is tightly clustered around the mean, while a larger one indicates more spread.
How the Calculator Works
This calculator computes the cumulative probability P(X ≤ x) for a given value x. This is the probability that a random variable from the specified normal distribution will take a value less than or equal to x.
The calculation involves standardizing the value x into a z-score and then using the cumulative distribution function (CDF) of the standard normal distribution (mean 0, standard deviation 1).
The steps are:
Calculate the z-score: The formula for the z-score is:
z = (X - μ) / σ
Where:
X is the specific value you are interested in.
μ (mu) is the mean of the distribution.
σ (sigma) is the standard deviation of the distribution.
Find the Cumulative Probability: The z-score tells us how many standard deviations X is away from the mean. The calculator then uses a statistical function (often approximated or based on lookup tables/algorithms for the standard normal CDF) to find the probability associated with this z-score. This probability represents P(Z ≤ z), which is equivalent to P(X ≤ x).
Use Cases
Normal distribution calculators are invaluable in various fields:
Social Sciences: Analyzing survey results, understanding population characteristics.
Example:
Let's say a certain type of light bulb has a lifespan that follows a normal distribution with a mean (μ) of 1000 hours and a standard deviation (σ) of 150 hours. We want to find the probability that a randomly selected bulb will last up to 1200 hours.
Find P(Z ≤ 1.33). Based on standard normal distribution tables or functions, this value is approximately 0.9082.
Therefore, the probability that a light bulb will last 1200 hours or less is about 90.82%.
function calculateNormalDistribution() {
var mean = parseFloat(document.getElementById("mean").value);
var stdDev = parseFloat(document.getElementById("stdDev").value);
var valueX = parseFloat(document.getElementById("valueX").value);
var resultDiv = document.getElementById("result");
// Clear previous results and error messages
resultDiv.innerHTML = 'Result will appear here.Enter values above to see the cumulative probability P(X ≤ value).';
resultDiv.style.backgroundColor = "#28a745"; // Default to success green
// Input validation
if (isNaN(mean) || isNaN(stdDev) || isNaN(valueX)) {
resultDiv.innerHTML = "Please enter valid numbers for all fields.";
resultDiv.style.backgroundColor = "#dc3545"; // Red for error
return;
}
if (stdDev <= 0) {
resultDiv.innerHTML = "Standard deviation must be a positive number.";
resultDiv.style.backgroundColor = "#dc3545"; // Red for error
return;
}
// Calculate z-score
var zScore = (valueX – mean) / stdDev;
// — Calculate Cumulative Probability P(Z <= zScore) —
// This is a simplified approximation. For production, a more robust statistical library
// or an accurate approximation like the error function (erf) should be used.
// For this example, we'll use a common approximation.
// Reference for approximation: https://www.itl.nist.gov/frequently_asked_questions.html#15
// More accurate formula for erf(x) for positive x:
// erf(x) = 1 – (a1*t + a2*t^2 + a3*t^3 + a4*t^4 + a5*t^5) * exp(-x^2)
// where t = 1 / (1 + p*x), p = 0.3275911, a1=0.254829592, a2=-0.284496736, a3=1.421413741, a4=-1.453152027, a5=1.061405429
var p = 0.3275911;
var a1 = 0.254829592;
var a2 = -0.284496736;
var a3 = 1.421413741;
var a4 = -1.453152027;
var a5 = 1.061405429;
var erf;
var sign = 1;
if (zScore < 0) {
sign = -1;
zScore = -zScore;
}
var t = 1.0 / (1.0 + p * zScore);
var erf_approx = 1.0 – (((((a5 * t + a4) * t) + a3) * t + a2) * t + a1) * t * Math.exp(-zScore * zScore);
erf = sign * erf_approx;
// The CDF of standard normal distribution is 0.5 * (1 + erf(z / sqrt(2)))
// However, the common approximation directly calculates erf(x) where the relation to CDF is:
// CDF(z) = 0.5 * [1 + erf(z / sqrt(2))]
// A more direct approximation related to the provided constants often leads to:
// P(X <= x) = 0.5 * (1 + erf(z / sqrt(2)))
// Let's use a commonly cited approximation for P(Z <= z) which relates to erf function
// P(Z <= z) ≈ 0.5 * [1 + erf(z / sqrt(2))]
// The constants above are for erf(x). Let's adjust for z / sqrt(2).
// var y = zScore / Math.sqrt(2);
// The approximation above is for erf(x), not P(Z <= z) directly.
// A common, simpler direct approximation for P(Z 6) { // Handle very large positive z-scores
cdf = 1.0;
} else if (zScore < -6) { // Handle very large negative z-scores
cdf = 0.0;
} else {
// Using the approximation for erf, then converting to CDF
// CDF(z) = 0.5 * (1 + erf(z / sqrt(2)))
var z_over_sqrt2 = zScore / Math.sqrt(2);
var sign_erf = 1;
if (z_over_sqrt2 < 0) {
sign_erf = -1;
z_over_sqrt2 = -z_over_sqrt2;
}
var t_erf = 1.0 / (1.0 + p * z_over_sqrt2);
var erf_val = 1.0 – (((((a5 * t_erf + a4) * t_erf) + a3) * t_erf + a2) * t_erf + a1) * t_erf * Math.exp(-z_over_sqrt2 * z_over_sqrt2);
erf_val = sign_erf * erf_val;
cdf = 0.5 * (1 + erf_val);
}
var probability = cdf;
// Display result
resultDiv.innerHTML = probability.toFixed(5) + "P(X ≤ " + valueX + ")";
}