Variance and Standard Deviation Calculator
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Variance and Standard Deviation Calculator
Results
Variance: N/A
Standard Deviation: N/A
Understanding Variance and Standard Deviation
Variance and Standard Deviation are fundamental statistical measures used to understand the spread or dispersion of a dataset. They tell us how much individual data points deviate from the average (mean) of the dataset.
What is Variance?
Variance measures the average of the squared differences from the mean. In simpler terms, it quantifies how far apart the data points are from each other and from the mean. A higher variance indicates that data points are further from the mean and from each other, suggesting greater variability. A lower variance indicates that data points are clustered closely around the mean.
The formula for population variance (σ²) is:
σ² = Σ(xᵢ - μ)² / N
Where:
σ² is the population variancexᵢ represents each individual data pointμ (mu) is the population meanN is the total number of data points
For a sample variance (s²), we divide by (n-1) instead of n to provide a less biased estimate of the population variance:
s² = Σ(xᵢ - x̄)² / (n - 1)
Where:
s² is the sample variancexᵢ represents each individual data point in the samplex̄ (x-bar) is the sample meann is the number of data points in the sample
This calculator uses the sample variance formula, which is more common when working with a subset of a larger population.
What is Standard Deviation?
Standard Deviation is simply the square root of the variance. It is often preferred because it is expressed in the same units as the original data, making it easier to interpret the spread of the data.
The formula for population standard deviation (σ) is:
σ = √σ²
The formula for sample standard deviation (s) is:
s = √s²
A small standard deviation indicates that the data points tend to be close to the mean, while a large standard deviation indicates that the data points are spread out over a wider range of values.
Example Calculation
Let's say we have the following data points representing the daily sales (in dollars) for a small business over 5 days: 100, 120, 110, 130, 140.
Step 1: Calculate the Mean (Average)
Sum = 100 + 120 + 110 + 130 + 140 = 600
Mean (x̄) = Sum / N = 600 / 5 = 120
Step 2: Calculate the Deviations from the Mean
- 100 – 120 = -20
- 120 – 120 = 0
- 110 – 120 = -10
- 130 – 120 = 10
- 140 – 120 = 20
Step 3: Square the Deviations
- (-20)² = 400
- (0)² = 0
- (-10)² = 100
- (10)² = 100
- (20)² = 400
Step 4: Sum the Squared Deviations
Sum of Squared Deviations = 400 + 0 + 100 + 100 + 400 = 1000
Step 5: Calculate the Variance (Sample)
Variance (s²) = Sum of Squared Deviations / (N – 1) = 1000 / (5 – 1) = 1000 / 4 = 250
Step 6: Calculate the Standard Deviation (Sample)
Standard Deviation (s) = √Variance = √250 ≈ 15.81
So, the variance is 250, and the standard deviation is approximately 15.81 dollars. This indicates that, on average, the daily sales figures deviate from the mean of $120 by about $15.81.
When to Use Variance and Standard Deviation
- Finance: Measuring market volatility, risk assessment of investments.
- Quality Control: Monitoring variations in manufacturing processes to ensure consistency.
- Science and Research: Analyzing experimental data to understand the reliability and spread of results.
- Social Sciences: Studying the distribution of opinions, behaviors, or demographic data.
- Performance Analysis: Evaluating the consistency of an athlete's performance, a stock's returns, or a machine's output.
function calculateVarianceAndStdDev() {
var dataInput = document.getElementById("dataPoints").value;
var resultsDiv = document.getElementById("result");
var varianceResultP = document.getElementById("varianceResult");
var stdDevResultP = document.getElementById("stdDevResult");
// Clear previous results
varianceResultP.innerHTML = "Variance: N/A";
stdDevResultP.innerHTML = "Standard Deviation: N/A";
if (!dataInput) {
alert("Please enter some data points.");
return;
}
var dataPoints = dataInput.split(',')
.map(function(item) { return parseFloat(item.trim()); })
.filter(function(item) { return !isNaN(item); }); // Filter out non-numeric values
var n = dataPoints.length;
if (n < 2) {
alert("Please enter at least two valid data points to calculate variance and standard deviation.");
return;
}
// Calculate the mean
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 sumSquaredDifferences = 0;
for (var i = 0; i < n; i++) {
var difference = dataPoints[i] – mean;
sumSquaredDifferences += difference * difference;
}
// Calculate the sample variance (divide by n-1)
var variance = sumSquaredDifferences / (n – 1);
// Calculate the sample standard deviation
var standardDeviation = Math.sqrt(variance);
// Display the results
varianceResultP.innerHTML = "Variance: " + variance.toFixed(4);
stdDevResultP.innerHTML = "Standard Deviation: " + standardDeviation.toFixed(4);
}