Sample variance is a statistical measure that describes how spread out the numbers in a data sample are from their average value. It's a crucial concept in inferential statistics, allowing us to estimate the variability of a population based on a subset (sample) of that population.
Unlike population variance, sample variance uses a denominator of n-1 instead of n (where n is the number of data points) to provide a less biased estimate of the population variance. This adjustment, known as Bessel's correction, is particularly important when dealing with small sample sizes.
How to Calculate Sample Variance:
The formula for sample variance (denoted as s²) is:
s² = Σ(xi - x̄)² / (n - 1)
Where:
s² is the sample variance.
Σ represents the summation (adding up).
xi is each individual data point in the sample.
x̄ (x-bar) is the sample mean (average).
n is the number of data points in the sample.
Steps to Calculate:
Collect Your Data Points: Gather all the numerical values for your sample.
Calculate the Sample Mean (x̄): Sum all the data points and divide by the number of data points (n).
Calculate Deviations: For each data point (xi), subtract the sample mean (x̄).
Square the Deviations: Square each of the results from step 3.
Sum the Squared Deviations: Add up all the squared deviations calculated in step 4.
Divide by (n-1): Divide the sum from step 5 by the number of data points minus one.
Use Cases:
Sample variance is used extensively in:
Hypothesis Testing: To determine if observed differences between samples or between a sample and a population are statistically significant.
Confidence Intervals: To estimate a range within which a population parameter (like the mean) is likely to fall.
Quality Control: To monitor the consistency and variability of products or processes.
Research: In fields like biology, economics, psychology, and engineering to understand the spread of data in experiments and surveys.
Example:
Let's calculate the sample variance for the data points: 5, 8, 12, 15, 10.
So, the sample variance for this data set is 14.5.
function calculateSampleVariance() {
var dataPointsInput = document.getElementById("dataPoints").value;
var resultValueElement = document.getElementById("result-value");
if (!dataPointsInput) {
resultValueElement.textContent = "Error: Please enter data points.";
return;
}
var dataPoints = dataPointsInput.split(',')
.map(function(item) {
return parseFloat(item.trim());
})
.filter(function(item) {
return !isNaN(item);
});
if (dataPoints.length < 2) {
resultValueElement.textContent = "Error: Need at least 2 valid data points.";
return;
}
var n = dataPoints.length;
var sum = 0;
for (var i = 0; i < n; i++) {
sum += dataPoints[i];
}
var mean = sum / n;
var sumSquaredDeviations = 0;
for (var i = 0; i < n; i++) {
var deviation = dataPoints[i] – mean;
sumSquaredDeviations += deviation * deviation;
}
var sampleVariance = sumSquaredDeviations / (n – 1);
// Format the output to a reasonable number of decimal places
resultValueElement.textContent = sampleVariance.toFixed(4);
}