Variance Calculator
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Variance Calculator
Enter your data points to calculate the sample variance.
Understanding Variance
Variance is a fundamental statistical measure that quantifies the degree of spread or dispersion of a set of data points around their mean (average). In simpler terms, it tells us how much the individual data points tend to deviate from the average value of the dataset.
Why is Variance Important?
- Risk Assessment: In finance, higher variance in asset returns suggests higher volatility and thus higher risk.
- Quality Control: In manufacturing, low variance indicates consistency and high quality in production.
- Data Interpretation: It helps understand the reliability and variability of data. A low variance indicates that the data points are close to the mean, while a high variance indicates they are spread out.
- Statistical Modeling: Variance is a key component in many statistical tests and models, such as ANOVA (Analysis of Variance).
Types of Variance:
There are two main types of variance:
- Population Variance ($\sigma^2$): This is calculated when you have data for the entire population. The formula involves dividing the sum of squared differences from the mean by the total number of data points (N).
- Sample Variance ($s^2$): This is calculated when you have data from a sample of a larger population. It's used to estimate the population variance. The formula involves dividing the sum of squared differences from the mean by the number of data points minus one (n-1). This is known as Bessel's correction and provides a less biased estimate of the population variance.
This calculator computes the Sample Variance, which is the most common scenario in practical data analysis.
How to Calculate Sample Variance
The formula for sample variance ($s^2$) is:
s² = Σ(xᵢ - μ)² / (n - 1)
Where:
s² is the sample variance.Σ denotes the sum.xᵢ represents each individual data point in your sample.μ (mu) is the sample mean (average) of your data points.n is the number of data points in your sample.
Steps:
- Calculate the Mean ($\mu$): Sum all the data points and divide by the total number of data points (n).
- Calculate Deviations: Subtract the mean from each individual data point (xᵢ – μ).
- Square the Deviations: Square each of the differences calculated in the previous step (xᵢ – μ)².
- Sum the Squared Deviations: Add up all the squared differences from step 3.
- Divide by (n – 1): Divide the sum from step 4 by the number of data points minus one.
Example Calculation:
Let's calculate the variance for the data points: 10, 15, 12, 18, 20
- Data Points: 10, 15, 12, 18, 20
- Number of data points (n): 5
- Calculate the Mean ($\mu$):
(10 + 15 + 12 + 18 + 20) / 5 = 75 / 5 = 15
- Calculate Deviations (xᵢ – $\mu$):
10 – 15 = -5
15 – 15 = 0
12 – 15 = -3
18 – 15 = 3
20 – 15 = 5
- Square the Deviations (xᵢ – $\mu$)²:
(-5)² = 25
(0)² = 0
(-3)² = 9
(3)² = 9
(5)² = 25
- Sum the Squared Deviations:
25 + 0 + 9 + 9 + 25 = 68
- Divide by (n – 1):
68 / (5 – 1) = 68 / 4 = 17
Therefore, the sample variance for this dataset is 17.
function calculateVariance() {
var dataPointsInput = document.getElementById("dataPoints").value;
var resultDiv = document.getElementById("result");
var varianceResultSpan = document.getElementById("varianceResult");
var resultMessage = document.getElementById("resultMessage");
// Clear previous results
varianceResultSpan.innerText = "–";
resultMessage.innerText = "";
resultDiv.style.display = 'none';
if (!dataPointsInput) {
resultMessage.innerText = "Please enter your data points.";
return;
}
var dataPoints = dataPointsInput.split(',')
.map(function(item) { return parseFloat(item.trim()); })
.filter(function(item) { return !isNaN(item); });
if (dataPoints.length < 2) {
resultMessage.innerText = "At least two valid data points are required to calculate variance.";
return;
}
var n = dataPoints.length;
var sum = 0;
for (var i = 0; i < n; i++) {
sum += dataPoints[i];
}
var mean = sum / n;
var sumSquaredDifferences = 0;
for (var i = 0; i < n; i++) {
var difference = dataPoints[i] – mean;
sumSquaredDifferences += difference * difference;
}
// Calculate sample variance (n-1 in the denominator)
var variance = sumSquaredDifferences / (n – 1);
varianceResultSpan.innerText = variance.toFixed(4); // Display with 4 decimal places
resultMessage.innerText = "Sample Variance ($s^2$) calculated using " + n + " data points.";
resultDiv.style.display = 'block';
}