Standard Score (Z-score) Calculator
Use this calculator to determine the Z-score for a specific data point. A Z-score, also known as a standard score, indicates how many standard deviations an element is from the mean. It's a useful statistical measure for comparing data points from different distributions or identifying outliers.
Understanding the Standard Score (Z-score)
The Z-score is a fundamental concept in statistics that quantifies the relationship between a data point and the mean of a group of data. It tells you how many standard deviations away from the mean your individual score lies. A positive Z-score indicates the data point is above the mean, while a negative Z-score indicates it's below the mean.
Why is the Z-score Important?
- Comparison Across Different Distributions: Z-scores allow you to compare scores from different normal distributions. For example, you can compare a student's performance on a math test with a class average of 70 and a standard deviation of 10, to their performance on a science test with an average of 60 and a standard deviation of 5.
- Identifying Outliers: Data points with very high or very low Z-scores (typically beyond +2 or -2, or +3 or -3 depending on the context) are often considered outliers, meaning they are unusually far from the average.
- Probability and Percentiles: In a normal distribution, Z-scores can be used with a Z-table to find the probability of a score occurring or to determine the percentile rank of a score.
How to Interpret Z-scores
- Z = 0: The individual score is exactly equal to the population mean.
- Z = 1: The individual score is one standard deviation above the population mean.
- Z = -1: The individual score is one standard deviation below the population mean.
- Larger Absolute Z-score: The further away the score is from the mean.
The Z-score Formula
The formula used by this calculator is:
Z = (X - μ) / σ
- X: The individual score or data point.
- μ (Mu): The population mean (average).
- σ (Sigma): The population standard deviation.
Example Calculation
Let's say a student scores 85 on a test. The class average (population mean) was 70, and the standard deviation for the class was 10.
- Individual Score (X) = 85
- Population Mean (μ) = 70
- Population Standard Deviation (σ) = 10
Using the formula:
Z = (85 - 70) / 10
Z = 15 / 10
Z = 1.5
This means the student's score of 85 is 1.5 standard deviations above the class average.
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function calculateZScore() {
var individualScore = parseFloat(document.getElementById("individualScore").value);
var populationMean = parseFloat(document.getElementById("populationMean").value);
var populationStdDev = parseFloat(document.getElementById("populationStdDev").value);
var resultDiv = document.getElementById("result");
if (isNaN(individualScore) || isNaN(populationMean) || isNaN(populationStdDev)) {
resultDiv.innerHTML = "
Please enter valid numbers for all fields.";
return;
}
if (populationStdDev <= 0) {
resultDiv.innerHTML = "
Population Standard Deviation must be greater than zero.";
return;
}
var zScore = (individualScore – populationMean) / populationStdDev;
resultDiv.innerHTML = "Your Z-score is:
" + zScore.toFixed(4) + "";
}