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. A Z-score of 0 indicates that the data point's value is identical to the mean value. A positive Z-score indicates that the data point is above the mean, while a negative Z-score indicates that the data point is below the mean.
The Z-score is a crucial concept in statistics and is used for various purposes, including:
Comparing values from different distributions: Z-scores allow you to compare scores from different tests or datasets that may have different means and standard deviations. For example, comparing a student's score on a math test with their score on an English test.
Identifying outliers: Data points with Z-scores that are unusually high (e.g., above +3) or unusually low (e.g., below -3) can often be considered outliers.
Probability calculations: Z-scores are fundamental to calculating probabilities associated with normal distributions using Z-tables or statistical software.
Standardization: It's a key step in many statistical analyses and machine learning algorithms where data needs to be standardized to have a mean of 0 and a standard deviation of 1.
The Z-Score Formula
The formula to calculate a Z-score is straightforward:
Z = (X - μ) / σ
Where:
Z is the Z-score
X is the raw score or observation value
μ (mu) is the mean of the population or sample
σ (sigma) is the standard deviation of the population or sample
How to Use This Calculator
Observation Value (X): Enter the specific data point you want to analyze.
Mean (μ): Input the average value of the dataset from which the observation is taken.
Standard Deviation (σ): Enter the measure of the spread or dispersion of the data around the mean.
Click "Calculate Z-Score" to see how many standard deviations your observation is from the mean.
Interpreting the Results
Z = 0: Your observation is exactly at the mean.
Z > 0: Your observation is above the mean. A Z-score of 1.5 means your observation is 1.5 standard deviations above the mean.
Z < 0: Your observation is below the mean. A Z-score of -2.0 means your observation is 2.0 standard deviations below the mean.
In a normal distribution, approximately 68% of values lie within 1 standard deviation of the mean (Z-scores between -1 and 1), about 95% lie within 2 standard deviations (-2 to 2), and about 99.7% lie within 3 standard deviations (-3 to 3).
function calculateZScore() {
var observationValue = parseFloat(document.getElementById("observationValue").value);
var meanValue = parseFloat(document.getElementById("meanValue").value);
var stdDevValue = parseFloat(document.getElementById("stdDevValue").value);
var errorMessageDiv = document.getElementById("errorMessage");
errorMessageDiv.textContent = ""; // Clear previous errors
// Input validation
if (isNaN(observationValue) || isNaN(meanValue) || isNaN(stdDevValue)) {
errorMessageDiv.textContent = "Please enter valid numbers for all fields.";
document.getElementById("zScoreResult").textContent = "—";
return;
}
if (stdDevValue === 0) {
errorMessageDiv.textContent = "Standard deviation cannot be zero.";
document.getElementById("zScoreResult").textContent = "—";
return;
}
var zScore = (observationValue – meanValue) / stdDevValue;
var resultElement = document.getElementById("zScoreResult");
resultElement.textContent = zScore.toFixed(4); // Display with 4 decimal places
}