A Z-score, also known as a standard score, is a statistical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean score. A Z-score of 1.0 would indicate a value that is one standard deviation from the mean.
The Z-Score Formula
z = (x – μ) / σ
x: The individual value (Raw Score)
μ: The population mean
σ: The population standard deviation
Real-World Example
Imagine you took a standardized test. The average score (mean) for all students was 75, and the standard deviation was 5. If your score was 85, your Z-score would be:
z = (85 – 75) / 5 = 2.0
This means your score was 2 standard deviations above the mean, placing you in approximately the 97.7th percentile of all test-takers.
Interpreting the Results
Z-Score Range
Meaning
Positive (> 0)
The score is above the average (mean).
Negative (< 0)
The score is below the average (mean).
Zero (0)
The score is exactly the average.
function calculateZScore() {
var rawScore = parseFloat(document.getElementById('rawScore').value);
var popMean = parseFloat(document.getElementById('popMean').value);
var stdDev = parseFloat(document.getElementById('stdDev').value);
var resultWrapper = document.getElementById('zResultWrapper');
var output = document.getElementById('zScoreOutput');
var interpretation = document.getElementById('zInterpretation');
var percentileEl = document.getElementById('zPercentile');
if (isNaN(rawScore) || isNaN(popMean) || isNaN(stdDev)) {
alert("Please enter valid numerical values for all fields.");
return;
}
if (stdDev 0) {
interText = "Your score is " + Math.abs(zScore).toFixed(2) + " standard deviations ABOVE the mean.";
} else if (zScore < 0) {
interText = "Your score is " + Math.abs(zScore).toFixed(2) + " standard deviations BELOW the mean.";
} else {
interText = "Your score is exactly equal to the mean.";
}
interpretation.innerText = interText;
// Standard Normal Cumulative Distribution Function approximation
var percentile = GetZPercent(zScore);
percentileEl.innerText = "Percentile: " + (percentile * 100).toFixed(2) + "%";
resultWrapper.style.display = "block";
resultWrapper.scrollIntoView({ behavior: 'smooth', block: 'nearest' });
}
function GetZPercent(z) {
// Check for extreme values
if (z 6.5) return 1.0;
var factK = 1;
var sum = 0;
var term = 1;
var k = 0;
var loopStop = Math.exp(-0.5 * z * z) * Math.abs(z) / Math.sqrt(2 * Math.PI);
// Series approximation
while (Math.abs(term) > 0.00000001) {
term = Math.pow(z * z, k) / (2 * k + 1);
for (var i = 1; i 100) break;
}
var prob = 0.5 + (z > 0 ? 1 : -1) * loopStop * sum;
// Fallback for simple z logic
if (z > 0 && prob < 0.5) prob = 1 – prob;
if (z 0.5) prob = 1 – prob;
return prob;
}