How Do You Calculate Variance

Variance Calculator

Sample Variance (n – 1) Population Variance (N)

Results

Variance:

Standard Deviation:

Mean (Average):

Count (n):

Sum of Squares:

Understanding Variance: Definition and Calculation

Variance is a statistical measurement of the spread between numbers in a data set. It quantifies how far each number in the set is from the mean (average) and thus from every other number in the set. Variance is often represented by the symbol σ² (sigma squared) for populations and s² for samples.

How to Calculate Variance Manually

Calculating variance involves a few key steps that help you understand the variability of your data:

1. Find the Mean: Calculate the average of all your data points by adding them up and dividing by the total number of items.
2. Subtract the Mean: For every data point, subtract the mean you just calculated.
3. Square the Differences: Square each of the results from step 2 to ensure all values are positive.
4. Sum the Squares: Add all the squared values together. This is called the "Sum of Squares."
5. Divide:
   • For Population Variance, divide the sum by the total number of data points (N).
   • For Sample Variance, divide by the total number of data points minus one (n – 1).

Sample vs. Population Variance

Choosing the right calculation depends on the nature of your data:

• Population Variance: Use this when your data set includes every member of the group you are studying (e.g., the test scores of every student in a single small classroom).
• Sample Variance: Use this when your data is a subset of a larger population (e.g., surveying 100 random voters to estimate the behavior of an entire city). Using n – 1 (Bessel's correction) provides an unbiased estimate of the population variance.

Practical Example

Imagine you have five test scores: 70, 80, 85, 90, and 100.

1. Mean: (70+80+85+90+100) / 5 = 85.
2. Subtract/Square: (70-85)²=225, (80-85)²=25, (85-85)²=0, (90-85)²=25, (100-85)²=225.
3. Sum of Squares: 225 + 25 + 0 + 25 + 225 = 500.
4. Sample Variance: 500 / (5 – 1) = 125.
5. Population Variance: 500 / 5 = 100.

function calculateVariance() { var rawInput = document.getElementById("dataInput").value; var calcType = document.getElementById("calcType").value; // Clean input: replace commas with spaces, split by whitespace var cleanInput = rawInput.replace(/,/g, ' '); var stringArray = cleanInput.trim().split(/\s+/); var data = []; for (var i = 0; i < stringArray.length; i++) { var num = parseFloat(stringArray[i]); if (!isNaN(num)) { data.push(num); } } if (data.length < 2) { alert("Please enter at least two valid numeric data points."); return; } // Calculate Mean var sum = 0; for (var j = 0; j < data.length; j++) { sum += data[j]; } var mean = sum / data.length; // Calculate Sum of Squares var sumSquares = 0; for (var k = 0; k < data.length; k++) { sumSquares += Math.pow(data[k] – mean, 2); } // Calculate Variance var variance; var n = data.length; if (calcType === "sample") { variance = sumSquares / (n – 1); } else { variance = sumSquares / n; } var stdDev = Math.sqrt(variance); // Display Results document.getElementById("resVariance").innerText = variance.toLocaleString(undefined, {minimumFractionDigits: 2, maximumFractionDigits: 4}); document.getElementById("resStdDev").innerText = stdDev.toLocaleString(undefined, {minimumFractionDigits: 2, maximumFractionDigits: 4}); document.getElementById("resMean").innerText = mean.toLocaleString(undefined, {minimumFractionDigits: 2, maximumFractionDigits: 4}); document.getElementById("resCount").innerText = n; document.getElementById("resSS").innerText = sumSquares.toLocaleString(undefined, {minimumFractionDigits: 2, maximumFractionDigits: 4}); document.getElementById("varianceResults").style.display = "block"; }