Z-Score Calculator
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Z-Score Calculator
Z-Score:
This value indicates how many standard deviations your observed value is from the mean.
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. A positive Z-score indicates that the data point is above the mean, while a negative Z-score indicates that it is below the mean. A Z-score of zero means the data point is exactly at the mean.
The Z-Score Formula
The formula to calculate the Z-score is straightforward:
Z = (X – μ) / σ
Where:
- X is the observed value (the data point you are analyzing).
- μ (mu) is the mean (average) of the population or sample.
- σ (sigma) is the standard deviation of the population or sample.
How to Interpret the Z-Score
The Z-score is a powerful tool for comparing values from different datasets or understanding a value's position within a single distribution.
- Z = 0: The observed value is exactly equal to the mean.
- Z > 0: The observed value is above the mean. A higher positive Z-score means the value is further above the mean. For example, a Z-score of 2 means the value is two standard deviations above the mean.
- Z < 0: The observed value is below the mean. A lower negative Z-score (e.g., -2) means the value is further below the mean. A Z-score of -1.5 means the value is 1.5 standard deviations below the mean.
In many fields, including statistics and finance, values with Z-scores between -2 and 2 are often considered to be within a typical range. Values outside this range might be considered outliers or statistically significant.
Use Cases for Z-Scores
- Identifying Outliers: Data points with extremely high or low Z-scores (e.g., |Z| > 3) are often flagged as potential outliers.
- Comparing Distributions: You can compare Z-scores of values from different datasets with different means and standard deviations. For instance, comparing a student's score on two different tests.
- Probability Calculations: Z-scores are fundamental to probability calculations in a normal distribution (bell curve). Knowing the Z-score allows you to find the probability of observing a value less than, greater than, or between certain values.
- Quality Control: In manufacturing, Z-scores can help monitor product specifications and identify when processes deviate significantly from the norm.
- Finance: Used to assess the relative performance of investments or to identify unusual market movements.
Example Calculation
Let's say you have a dataset of exam scores with a mean (μ) of 70 and a standard deviation (σ) of 5. A student scores 85 (X).
Using the formula:
Z = (85 – 70) / 5 = 15 / 5 = 3
This Z-score of 3 means the student's score of 85 is 3 standard deviations above the mean score. This is a relatively high score compared to the average.
function calculateZScore() {
var observedValue = parseFloat(document.getElementById("observedValue").value);
var meanValue = parseFloat(document.getElementById("meanValue").value);
var stdDevValue = parseFloat(document.getElementById("stdDevValue").value);
var resultDiv = document.getElementById("result");
var zScoreResultSpan = document.getElementById("zScoreResult");
if (isNaN(observedValue) || isNaN(meanValue) || isNaN(stdDevValue)) {
alert("Please enter valid numbers for all fields.");
resultDiv.style.display = 'none';
return;
}
if (stdDevValue === 0) {
alert("Standard deviation cannot be zero.");
resultDiv.style.display = 'none';
return;
}
var zScore = (observedValue – meanValue) / stdDevValue;
zScoreResultSpan.textContent = zScore.toFixed(4); // Display with 4 decimal places
resultDiv.style.display = 'block';
}