Enter your data points (numbers) separated by commas.
Results
Calculation Steps
Understanding Standard Deviation
Standard deviation is a fundamental statistical measure that quantifies the amount of variation or dispersion of a set of data values. In simpler terms, it tells us how spread out the numbers are from their average (mean). A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation suggests that the data points are spread out over a wider range of values.
Why is Standard Deviation Important?
Standard deviation is widely used across various fields:
Finance: To measure the volatility or risk of an investment. A higher standard deviation for an asset's returns implies greater risk.
Quality Control: To monitor the consistency of a manufacturing process. Low standard deviation indicates consistent output.
Science and Research: To assess the reliability and variability of experimental results.
Economics: To understand income inequality or the dispersion of economic indicators.
Education: To analyze the spread of test scores among students.
The Math Behind Standard Deviation
Calculating standard deviation involves several steps. There are two common types: population standard deviation (σ) and sample standard deviation (s). This calculator computes the sample standard deviation, which is more common when you're working with a subset of a larger population.
Steps for Calculating Sample Standard Deviation (s):
Calculate the Mean (Average): Sum all the data points and divide by the number of data points (n).
Mean (x̄) = (Σx) / n
Calculate Deviations from the Mean: Subtract the mean from each individual data point.
Deviation = x - x̄
Square the Deviations: Square each of the deviations calculated in the previous step.
Squared Deviation = (x - x̄)²
Sum the Squared Deviations: Add up all the squared deviations.
Sum of Squared Deviations = Σ(x - x̄)²
Calculate the Variance: Divide the sum of squared deviations by (n-1) for sample variance.
Sample Variance (s²) = [ Σ(x - x̄)² ] / (n - 1)
Calculate the Standard Deviation: Take the square root of the variance.
Sample Standard Deviation (s) = √s²
Example Calculation
Let's say we have the following data points representing the daily sales of a small shop over five days: 100, 120, 110, 130, 140.
n = 5 (number of data points)
Step 1: Calculate the Mean (x̄)
Sum = 100 + 120 + 110 + 130 + 140 = 600
Mean = 600 / 5 = 120
Step 6: Calculate Standard Deviation (s)
Standard Deviation = √250 ≈ 15.81
So, the sample standard deviation of the daily sales is approximately 15.81. This indicates the typical variation in daily sales around the average of 120.
function calculateStandardDeviation() {
var dataInput = document.getElementById("dataPoints").value;
var errorMessageDiv = document.getElementById("error-message");
var resultsDiv = document.getElementById("results");
var stepsList = document.getElementById("stepsList");
var varianceResultDiv = document.getElementById("varianceResult");
var standardDeviationResultDiv = document.getElementById("standardDeviationResult");
// Clear previous results and errors
errorMessageDiv.style.display = "none";
resultsDiv.style.display = "none";
stepsList.innerHTML = "";
varianceResultDiv.innerHTML = "";
standardDeviationResultDiv.innerHTML = "";
// Parse data points
var dataPointsArray = dataInput.split(',')
.map(function(item) {
return parseFloat(item.trim());
})
.filter(function(item) {
return !isNaN(item);
});
var n = dataPointsArray.length;
if (n < 2) {
errorMessageDiv.innerHTML = "Please enter at least two valid numbers to calculate standard deviation.";
errorMessageDiv.style.display = "block";
return;
}
// — Calculation Steps —
var mean = 0;
var sum = 0;
for (var i = 0; i < n; i++) {
sum += dataPointsArray[i];
}
mean = sum / n;
var squaredDeviations = [];
var sumSquaredDeviations = 0;
for (var i = 0; i < n; i++) {
var deviation = dataPointsArray[i] – mean;
var squaredDeviation = deviation * deviation;
squaredDeviations.push(squaredDeviation);
sumSquaredDeviations += squaredDeviation;
}
var variance = sumSquaredDeviations / (n – 1);
var standardDeviation = Math.sqrt(variance);
// — Display Results and Steps —
// Step 1: Mean
var step1 = document.createElement('li');
step1.innerHTML = "Calculated the mean (average) of the data points: " + mean.toFixed(4) + ".";
stepsList.appendChild(step1);
// Step 2 & 3: Deviations and Squared Deviations
var step2_3 = document.createElement('li');
var deviationsListHTML = "
";
var squaredDeviationsListHTML = "
";
for (var i = 0; i < n; i++) {
var deviation = dataPointsArray[i] – mean;
var squaredDeviation = deviation * deviation;
deviationsListHTML += "
";
step2_3.innerHTML = "Calculated the deviation of each data point from the mean and then squared these deviations:Deviations: " + deviationsListHTML + "Squared Deviations: " + squaredDeviationsListHTML;
stepsList.appendChild(step2_3);
// Step 4: Sum of Squared Deviations
var step4 = document.createElement('li');
step4.innerHTML = "Summed the squared deviations: " + sumSquaredDeviations.toFixed(4) + ".";
stepsList.appendChild(step4);
// Step 5: Variance
var step5 = document.createElement('li');
step5.innerHTML = "Calculated the sample variance by dividing the sum of squared deviations by (n-1): " + sumSquaredDeviations.toFixed(4) + " / (" + n + " – 1) = " + variance.toFixed(4) + ".";
stepsList.appendChild(step5);
// Step 6: Standard Deviation
var step6 = document.createElement('li');
step6.innerHTML = "Calculated the sample standard deviation by taking the square root of the variance: " + standardDeviation.toFixed(4) + ".";
stepsList.appendChild(step6);
varianceResultDiv.innerHTML = "Variance (s²): " + variance.toFixed(4);
standardDeviationResultDiv.innerHTML = "Standard Deviation (s): " + standardDeviation.toFixed(4);
resultsDiv.style.display = "block";
}