Calculate probabilities and areas under the curve for a Z-score.
P(Z < z) [Left Tail]:0
P(Z > z) [Right Tail]:0
P(-|z| < Z < |z|) [Two-Sided]:0
Percentile:0%
What is the Standard Normal Distribution?
The Standard Normal Distribution, often referred to as the Z-distribution, is a special case of the normal distribution where the mean (μ) is 0 and the standard deviation (σ) is 1. It is the foundation of many statistical tests and allows researchers to compare different data sets by converting raw scores into "Z-scores."
A Z-score represents how many standard deviations a data point is from the mean. For example, a Z-score of 1.5 means the value is 1.5 standard deviations above the average.
Practical Example:
Suppose an IQ test has a mean of 100 and a standard deviation of 15. If a student scores 130, their Z-score is (130 – 100) / 15 = 2.0. Using this calculator, a Z-score of 2.0 shows that the student scored higher than approximately 97.72% of the population.
How to Read Results
Left Tail P(Z < z): The probability that a value is less than the given Z-score. This corresponds to the area under the curve to the left of the score.
Right Tail P(Z > z): The probability that a value is greater than the Z-score.
Two-Sided: The probability that a value falls between -z and +z. This is frequently used in confidence intervals (e.g., 1.96 for a 95% confidence interval).
Common Z-score Benchmarks
In statistics, certain Z-scores appear frequently due to their relationship with confidence levels:
z = 1.645: 90% Confidence (Two-tailed) / 95th Percentile
z = 1.960: 95% Confidence (Two-tailed) / 97.5th Percentile
z = 2.576: 99% Confidence (Two-tailed) / 99.5th Percentile
function calculateZScore() {
var zInput = document.getElementById("zScoreInput").value;
if (zInput === "" || isNaN(zInput)) {
alert("Please enter a valid numeric Z-score.");
return;
}
var z = parseFloat(zInput);
var prob = cumulativeDistribution(z);
var leftTail = prob;
var rightTail = 1 – prob;
var absZ = Math.abs(z);
var twoSided = cumulativeDistribution(absZ) – cumulativeDistribution(-absZ);
var percentile = prob * 100;
document.getElementById("resZ1").innerHTML = z.toFixed(3);
document.getElementById("resZ2").innerHTML = z.toFixed(3);
document.getElementById("leftTail").innerHTML = leftTail.toFixed(5);
document.getElementById("rightTail").innerHTML = rightTail.toFixed(5);
document.getElementById("twoSided").innerHTML = (twoSided * 100).toFixed(2) + "%";
document.getElementById("percentile").innerHTML = percentile.toFixed(2) + "th percentile";
document.getElementById("sncResults").style.display = "block";
}
// Abramowitz & Stegun formula for the Cumulative Distribution Function of the Standard Normal Distribution
function cumulativeDistribution(z) {
var t = 1 / (1 + 0.2316419 * Math.abs(z));
var d = 0.3989423 * Math.exp(-z * z / 2);
var p = d * t * (0.31938153 + t * (-0.356563782 + t * (1.781477937 + t * (-1.821255978 + 1.330274429 * t))));
if (z >= 0) {
return 1 – p;
} else {
return p;
}
}