Enter your data points, separated by commas or spaces. For example: 10, 15, 22, 18, 25 or 5 8 12 9 11.
Results:
Variance:
Standard Deviation:
Understanding Standard Deviation and Variance
In statistics, variance and standard deviation are two fundamental measures of dispersion or spread. They tell us how much the individual data points in a dataset deviate from the mean (average) of that dataset. A low variance and standard deviation indicate that the data points tend to be very close to the mean, while a high variance and standard deviation indicate that the data points are spread out over a wider range of values.
How to Calculate Variance
The formula for population variance (σ²) is:
σ² = Σ(xi – μ)² / N
Where:
Σ represents the sum
xi is each individual data point
μ (mu) is the population mean
N is the total number of data points in the population
For sample variance (s²), which is more commonly used when analyzing a subset of a larger population, the formula is slightly different:
s² = Σ(xi – x̄)² / (n – 1)
Where:
Σ represents the sum
xi is each individual data point
x̄ (x-bar) is the sample mean
n is the total number of data points in the sample
The use of (n - 1) in the denominator for sample variance is known as Bessel's correction, which provides a less biased estimate of the population variance. Our calculator uses the sample variance formula by default.
How to Calculate Standard Deviation
The standard deviation is simply the square root of the variance. It is often preferred because it is in the same units as the original data, making it easier to interpret.
For population standard deviation (σ):
σ = √σ²
For sample standard deviation (s):
s = √s²
When to Use These Measures
Quality Control: To monitor consistency in manufacturing processes.
Finance: To assess the risk of an investment (higher standard deviation implies higher risk).
Research: To understand the variability in experimental results.
Data Analysis: To describe the spread of data in reports and analyses.
This calculator helps you quickly compute these important statistical measures for any set of numerical data.
function calculateStandardDeviation() {
var dataInput = document.getElementById("dataInput").value;
var resultsDiv = document.getElementById("result");
var varianceResultSpan = document.getElementById("varianceResult");
var stdDevResultSpan = document.getElementById("stdDevResult");
// Parse the input string into an array of numbers
var dataPoints = dataInput.split(/[\s,]+/).map(function(item) {
return parseFloat(item);
}).filter(function(item) {
// Remove any non-numeric entries or empty strings resulting from split
return !isNaN(item);
});
if (dataPoints.length < 2) {
alert("Please enter at least two data points to calculate variance and standard deviation.");
resultsDiv.style.display = 'none';
return;
}
var n = dataPoints.length;
// Calculate the mean (average)
var sum = 0;
for (var i = 0; i < n; i++) {
sum += dataPoints[i];
}
var mean = sum / n;
// Calculate the sum of squared differences from the mean
var sumSquaredDiffs = 0;
for (var i = 0; i < n; i++) {
sumSquaredDiffs += Math.pow(dataPoints[i] – mean, 2);
}
// Calculate variance (using n-1 for sample variance)
var variance = sumSquaredDiffs / (n – 1);
// Calculate standard deviation
var stdDev = Math.sqrt(variance);
// Display the results
varianceResultSpan.textContent = variance.toFixed(4); // Display with 4 decimal places
stdDevResultSpan.textContent = stdDev.toFixed(4); // Display with 4 decimal places
resultsDiv.style.display = 'block';
}