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
The mathematical formula used by this calculator to find the Z-score is:
z = (x – μ) / σ
z is the Z-score.
x is the raw value (observed data point).
μ (mu) is the population mean.
σ (sigma) is the population standard deviation.
How to Interpret Z-Score Results
Understanding the result is crucial for statistical analysis:
Positive Z-Score: The score is above the average (mean).
Negative Z-Score: The score is below the average (mean).
Zero Z-Score: The score is exactly average.
High Z-Score (above 2 or below -2): The score is considered unusual or an outlier in many distributions.
Practical Example
Imagine you took a test where the mean score (μ) was 75 and the standard deviation (σ) was 10. If your raw score (x) was 85:
z = (85 – 75) / 10 = 10 / 10 = 1.00
This means your score was exactly 1 standard deviation above the mean.
function calculateZScore() {
var x = parseFloat(document.getElementById('rawScore').value);
var mu = parseFloat(document.getElementById('populationMean').value);
var sigma = parseFloat(document.getElementById('standardDeviation').value);
var resultContainer = document.getElementById('zResultContainer');
var zDisplay = document.getElementById('zScoreDisplay');
var interpretation = document.getElementById('interpretation');
if (isNaN(x) || isNaN(mu) || isNaN(sigma)) {
alert("Please enter valid numeric values for all fields.");
return;
}
if (sigma === 0) {
alert("Standard deviation cannot be zero. Division by zero is undefined.");
return;
}
var zScore = (x – mu) / sigma;
var roundedZ = zScore.toFixed(4);
zDisplay.innerText = roundedZ;
resultContainer.style.display = 'block';
var message = "";
if (zScore === 0) {
message = "The raw score is exactly at the mean.";
} else if (zScore > 0) {
message = "The raw score is " + Math.abs(roundedZ) + " standard deviations above the mean.";
} else {
message = "The raw score is " + Math.abs(roundedZ) + " standard deviations below the mean.";
}
if (Math.abs(zScore) > 3) {
message += " This is considered a significant outlier in a normal distribution.";
}
interpretation.innerHTML = message;
window.scrollTo({
top: resultContainer.offsetTop – 50,
behavior: 'smooth'
});
}