Calculate the Z-score for a given data point, mean, and standard deviation.
— Z-Score Will Appear Here —
Understanding the Z-Score
The 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, measured in terms of standard deviations from the mean. In simpler terms, it tells you how far a particular data point is from the average of its dataset, and in which direction (above or below).
The Formula
The formula for calculating a Z-score is straightforward:
Z = (X - μ) / σ
Z: The Z-score itself.
X: The individual data point you want to analyze.
μ (Mu): The mean (average) of the population or sample.
σ (Sigma): The standard deviation of the population or sample.
How to Interpret a Z-Score
Z = 0: The data point is exactly equal to the mean.
Z > 0: The data point is above the mean.
Z < 0: The data point is below the mean.
Typically:
A Z-score between -1 and 1 indicates the data point is within one standard deviation of the mean.
A Z-score between -2 and 2 indicates the data point is within two standard deviations of the mean.
A Z-score between -3 and 3 indicates the data point is within three standard deviations of the mean.
Values outside of 3 standard deviations (Z > 3 or Z < -3) are often considered outliers or statistically unusual.
Use Cases for Z-Scores
Z-scores are incredibly versatile and used in many fields:
Comparing scores from different tests: You can compare a student's score on a math test (mean 70, std dev 5) to their score on an English test (mean 80, std dev 10) by converting both to Z-scores.
Identifying outliers: Data points with very high or low Z-scores can signal unusual observations that warrant further investigation.
Quality control: In manufacturing, Z-scores can help determine if a product's measurement falls within acceptable tolerances.
Standardizing data: Z-scores transform data to a standard scale, making it easier to perform further statistical analyses or build predictive models.
Probability calculations: Z-scores are fundamental to understanding probability distributions (like the normal distribution) and finding the likelihood of certain outcomes.
Example Calculation
Let's say we have a dataset of exam scores with a mean (μ) of 75 and a standard deviation (σ) of 4. We want to find the Z-score for a student who scored 82 (X).
Using the formula:
Z = (82 - 75) / 4 = 7 / 4 = 1.75
This means the student's score of 82 is 1.75 standard deviations above the mean.
Consider another student who scored 70:
Z = (70 - 75) / 4 = -5 / 4 = -1.25
This indicates that a score of 70 is 1.25 standard deviations below the mean.
function calculateZScore() {
var dataPoint = parseFloat(document.getElementById("dataPoint").value);
var mean = parseFloat(document.getElementById("mean").value);
var stdDev = parseFloat(document.getElementById("stdDev").value);
var resultDiv = document.getElementById("result");
if (isNaN(dataPoint) || isNaN(mean) || isNaN(stdDev)) {
resultDiv.textContent = "Please enter valid numbers.";
return;
}
if (stdDev === 0) {
resultDiv.textContent = "Standard deviation cannot be zero.";
return;
}
var zScore = (dataPoint – mean) / stdDev;
resultDiv.textContent = "Z-Score: " + zScore.toFixed(4); // Display with 4 decimal places
}