Enter your data set separated by commas, spaces, or new lines.
Please enter at least two valid numbers for a calculation.
Calculated Results
Standard Deviation (σ or s):–
Variance (σ² or s²):–
Mean (μ or x̄):–
Sum (Σx):–
Count (n):–
Sum of Squares (SS):–
Understanding Standard Deviation
Standard deviation is a fundamental statistical metric that quantifies the amount of variation or dispersion in a set of data values. A low standard deviation indicates that the data points tend to be close to the mean (average), while a high standard deviation indicates that the data points are spread out over a wider range of values.
Population vs. Sample Standard Deviation
It is crucial to distinguish between whether your data represents an entire population or just a sample from it, as the mathematical formula changes slightly.
Metric
Population (N)
Sample (n-1)
Use Case
When you have every single data point in the group.
When you have a subset used to estimate the whole group.
Symbol
Sigma (σ)
s
Denominator
Divide by N
Divide by n – 1 (Bessel's correction)
Step-by-Step Calculation Example
Let's calculate the Sample Standard Deviation for the following numbers: 4, 8, 6
Calculate the Mean: (4 + 8 + 6) / 3 = 6.
Subtract the Mean and Square the Result:
(4 – 6)² = (-2)² = 4
(8 – 6)² = (2)² = 4
(6 – 6)² = (0)² = 0
Calculate the Sum of Squares: 4 + 4 + 0 = 8.
Divide by (n – 1): 8 / (3 – 1) = 4 (This is the Variance).
Take the Square Root: √4 = 2.
The standard deviation is 2.
Why Use Standard Deviation?
In finance, standard deviation is used as a measure of volatility or risk. In manufacturing, it helps ensure quality control by measuring how much products deviate from the standard specification. In education, it helps teachers understand the range of scores in a classroom, identifying if the whole class is performing similarly or if there is a massive gap between high and low achievers.
function calculateStandardDeviation() {
var rawInput = document.getElementById('dataInput').value;
var errorDiv = document.getElementById('sdError');
var resultsDiv = document.getElementById('sdResults');
var type = document.querySelector('input[name="sdType"]:checked').value;
// Clean input and convert to numbers
var numbers = rawInput.split(/[,\s\n]+/)
.map(function(item) { return item.trim(); })
.filter(function(item) { return item !== "" && !isNaN(item); })
.map(Number);
if (numbers.length < 2) {
errorDiv.style.display = 'block';
resultsDiv.style.display = 'none';
return;
} else {
errorDiv.style.display = 'none';
resultsDiv.style.display = 'block';
}
// Step 1: Count
var n = numbers.length;
// Step 2: Sum
var sum = 0;
for (var i = 0; i < n; i++) {
sum += numbers[i];
}
// Step 3: Mean
var mean = sum / n;
// Step 4: Sum of Squares
var sumSqDiff = 0;
for (var j = 0; j < n; j++) {
sumSqDiff += Math.pow(numbers[j] – mean, 2);
}
// Step 5: Variance
var variance;
if (type === 'population') {
variance = sumSqDiff / n;
} else {
variance = sumSqDiff / (n – 1);
}
// Step 6: Standard Deviation
var stdDev = Math.sqrt(variance);
// Update UI
document.getElementById('resCount').innerText = n;
document.getElementById('resSum').innerText = sum.toLocaleString(undefined, {maximumFractionDigits: 4});
document.getElementById('resMean').innerText = mean.toLocaleString(undefined, {maximumFractionDigits: 4});
document.getElementById('resSS').innerText = sumSqDiff.toLocaleString(undefined, {maximumFractionDigits: 4});
document.getElementById('resVariance').innerText = variance.toLocaleString(undefined, {maximumFractionDigits: 6});
document.getElementById('resSD').innerText = stdDev.toLocaleString(undefined, {maximumFractionDigits: 6});
}