Calculate probabilities (area under the curve) for a standard normal distribution (Z-distribution).
P(Z < z) – Area to the Left
P(Z > z) – Area to the Right
P(z1 < Z < z2) – Area Between Two Z-Scores
Understanding the Standard Normal Distribution
The Standard Normal Distribution, often denoted by the letter Z, is a special case of the normal distribution. It has a mean ($\mu$) of 0 and a standard deviation ($\sigma$) of 1. This distribution is fundamental in statistics because any normal distribution can be transformed into a standard normal distribution using a technique called standardization. This allows us to use a single standard table (the Z-table) or calculator to find probabilities for any normally distributed variable.
The probability associated with a Z-score represents the area under the standard normal curve up to, beyond, or between certain Z-score values. The total area under the curve is always 1.
The Z-Score Formula
A Z-score measures how many standard deviations a particular data point (X) is away from the mean ($\mu$) of its distribution. The formula for calculating a Z-score is:
Z = (X - μ) / σ
Where:
Z is the Z-score
X is the raw score or data point
μ (mu) is the mean of the population distribution
σ (sigma) is the standard deviation of the population distribution
When we talk about the "Standard Normal Distribution Calculator", we are generally working with pre-calculated Z-scores or using this calculator to find probabilities directly from Z-scores, assuming a mean of 0 and a standard deviation of 1.
How this Calculator Works
This calculator utilizes the Cumulative Distribution Function (CDF) of the standard normal distribution. The CDF, often denoted as $\Phi(z)$, gives the probability that a standard normal random variable Z is less than or equal to a specific value z, i.e., P(Z ≤ z).
Area to the Left (P(Z < z)): This is the direct output of the CDF, $\Phi(z)$. It represents the proportion of the distribution that falls below the given Z-score.
Area to the Right (P(Z > z)): Since the total area under the curve is 1, the area to the right of a Z-score is calculated as 1 - P(Z < z), or 1 - Φ(z).
Area Between Two Z-Scores (P(z1 < Z < z2)): This is calculated by finding the area to the left of the second Z-score (z2) and subtracting the area to the left of the first Z-score (z1). The formula is P(Z < z2) - P(Z < z1), or Φ(z2) - Φ(z1).
Implementing the CDF accurately often involves approximations or numerical integration methods, as there isn't a simple closed-form elementary function for it. This calculator uses a common and accurate approximation method.
Use Cases
The standard normal distribution and Z-scores are used extensively in:
Hypothesis Testing: Determining the statistical significance of results.
Confidence Intervals: Estimating a range of plausible values for a population parameter.
Quality Control: Monitoring processes and identifying deviations.
Risk Management: Assessing the probability of certain financial outcomes.
Data Analysis: Understanding the distribution and variability of data.
By calculating these probabilities, we can make informed decisions and draw meaningful conclusions from data.
// Based on the approximation of the standard normal CDF (Φ(x))
// This is a common approximation using the error function (erf)
// Φ(x) = 0.5 * (1 + erf(x / sqrt(2)))
// Source for erf approximation: Abramowitz and Stegun, Handbook of Mathematical Functions, Eq. 7.1.26
function erf(x) {
var a1 = 0.254829592;
var a2 = -0.284496736;
var a3 = 1.421413741;
var a4 = -1.453152027;
var a5 = 1.061405429;
var p = 0.3275911;
var sign = 1;
if (x < 0) {
sign = -1;
x = -x;
}
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 standardNormalCdf(z) {
if (isNaN(z)) return NaN;
return 0.5 * (1 + erf(z / Math.SQRT2));
}
function calculateProbability() {
var zScoreInput = document.getElementById("zScore");
var zScore2Input = document.getElementById("zScore2");
var probabilityTypeSelect = document.getElementById("probabilityType");
var resultDiv = document.getElementById("result");
var z1 = parseFloat(zScoreInput.value);
var z2 = parseFloat(zScore2Input.value); // May be NaN if not visible/filled
var type = probabilityTypeSelect.value;
// Clear previous result
resultDiv.innerHTML = '';
// Input validation
if (isNaN(z1)) {
resultDiv.innerHTML = "Please enter a valid Z-Score.";
resultDiv.style.backgroundColor = "#dc3545"; // Error color
return;
}
var prob;
var formattedProb;
if (type === "left") {
prob = standardNormalCdf(z1);
} else if (type === "right") {
prob = 1 – standardNormalCdf(z1);
} else if (type === "between") {
if (isNaN(z2)) {
resultDiv.innerHTML = "Please enter a valid second Z-Score (z1) for 'between' calculation.";
resultDiv.style.backgroundColor = "#dc3545"; // Error color
return;
}
// Ensure z1 is the smaller value for logical consistency, though math works either way
var lowerZ = Math.min(z1, z2);
var upperZ = Math.max(z1, z2);
prob = standardNormalCdf(upperZ) – standardNormalCdf(lowerZ);
}
// Check for calculation errors
if (isNaN(prob) || prob 1) {
resultDiv.innerHTML = "Calculation error. Please check inputs.";
resultDiv.style.backgroundColor = "#dc3545"; // Error color
} else {
// Format probability to a reasonable number of decimal places
formattedProb = prob.toFixed(6); // Adjust decimals as needed
resultDiv.innerHTML = 'P = ' + formattedProb + '';
resultDiv.style.backgroundColor = "var(–success-green)"; // Reset to success color
}
}
// Function to toggle visibility of the second Z-score input
function toggleZ2Input() {
var selectElement = document.getElementById("probabilityType");
var z2InputGroup = document.getElementById("z2InputGroup");
if (selectElement.value === "between") {
z2InputGroup.style.display = "flex"; // Use flex to match parent alignment
} else {
z2InputGroup.style.display = "none";
}
}
// Add event listener to the select element
document.getElementById("probabilityType").addEventListener("change", toggleZ2Input);
// Initial call to set correct visibility on page load
toggleZ2Input();