Sample variance is a fundamental statistical measure used to quantify the spread or dispersion of a set of data points relative to their mean. It is an estimate of the population variance, calculated from a sample of data. A higher variance indicates that the data points are spread further from the mean, while a lower variance suggests they are clustered more closely around the mean.
$x_i$ represents each individual data point in the sample.
$\bar{x}$ (x-bar) represents the sample mean (the average of all data points).
$n$ is the number of data points in the sample.
$n-1$ is used in the denominator because it provides a less biased estimate of the population variance (Bessel's correction).
How to Calculate Sample Variance (Step-by-Step)
Collect Your Data: Gather all the individual data points for your sample.
Calculate the Sample Mean ($\bar{x}$): Sum all the data points and divide by the number of data points ($n$).
Mean = (Sum of all data points) / n
Calculate Deviations from the Mean: For each data point ($x_i$), subtract the sample mean ($\bar{x}$).
Deviation = x_i - $\bar{x}$
Square the Deviations: Square each of the deviations calculated in the previous step.
Squared Deviation = (x_i - $\bar{x}$)^2
Sum the Squared Deviations: Add up all the squared deviations.
Sum of Squared Deviations = $\sum (x_i - \bar{x})^2$
Divide by (n-1): Divide the sum of squared deviations by the number of data points minus one ($n-1$). This is your sample variance.
Sample Variance ($s^2$) = (Sum of Squared Deviations) / (n-1)
Why Use Sample Variance?
Sample variance is crucial in inferential statistics. When you cannot survey an entire population, you take a sample. The sample variance helps you understand the variability within that sample, which then allows you to make inferences or predictions about the variability of the entire population from which the sample was drawn. It is a key component in calculating other statistics like standard deviation, confidence intervals, and is used in hypothesis testing.
Example Calculation
Let's calculate the sample variance for the following data points: 2, 4, 6, 8, 10
Therefore, the sample variance for the data set {2, 4, 6, 8, 10} is 10.
function calculateSampleVariance() {
var dataInput = document.getElementById("dataPoints").value;
var resultDisplay = document.getElementById("result-value");
if (!dataInput) {
resultDisplay.textContent = "Enter data points";
return;
}
var dataPointsArray = dataInput.split(',')
.map(function(item) {
return parseFloat(item.trim());
})
.filter(function(item) {
return !isNaN(item);
});
var n = dataPointsArray.length;
if (n < 2) {
resultDisplay.textContent = "Need at least 2 data points";
return;
}
// Calculate Mean
var sum = dataPointsArray.reduce(function(acc, val) {
return acc + val;
}, 0);
var mean = sum / n;
// Calculate Sum of Squared Deviations
var sumSquaredDeviations = dataPointsArray.reduce(function(acc, val) {
var deviation = val – mean;
return acc + (deviation * deviation);
}, 0);
// Calculate Sample Variance
var sampleVariance = sumSquaredDeviations / (n – 1);
resultDisplay.textContent = sampleVariance.toFixed(4); // Display with 4 decimal places
}