Standard deviation (SD) is a fundamental statistical measure that quantifies the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (average) of the set, while a high standard deviation indicates that the values are spread out over a wider range. It is one of the most commonly used measures of variability in data.
In simpler terms, it tells you how "spread out" your numbers are. If your data points are all very similar, the SD will be small. If they are very different, the SD will be large.
How Standard Deviation is Calculated
There are two common formulas for standard deviation: one for a population (if you have data for the entire group) and one for a sample (if you have data from a subset of a larger group). The sample standard deviation is more common as it's often impractical to collect data for an entire population. This calculator uses the sample standard deviation formula.
Steps for Sample Standard Deviation:
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: For each data point (x), subtract the mean.
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 a sample.
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
Use Cases for Standard Deviation
Finance: Measuring the volatility of stock prices or investment returns. Higher SD means higher risk.
Quality Control: Monitoring variations in manufacturing processes to ensure consistency.
Science & Research: Assessing the reliability of experimental results and the spread of measurements.
Education: Analyzing test scores to understand the distribution of student performance.
Healthcare: Studying variations in patient data, like blood pressure or cholesterol levels.
A larger standard deviation means that the data points are further from the mean, indicating greater variability. A smaller standard deviation means the data points are closer to the mean, indicating less variability.
function calculateStandardDeviation() {
var dataInput = document.getElementById("dataPoints").value;
var resultDiv = document.getElementById("result");
if (!dataInput) {
resultDiv.innerHTML = 'Please enter some data points.';
return;
}
// Split the input string by comma and remove any whitespace
var dataPoints = dataInput.split(',')
.map(function(point) {
return point.trim();
})
.filter(function(point) { // Remove empty strings resulting from multiple commas
return point !== ";
});
var numbers = [];
for (var i = 0; i < dataPoints.length; i++) {
var num = parseFloat(dataPoints[i]);
if (isNaN(num)) {
resultDiv.innerHTML = 'Invalid input: Please enter valid numbers separated by commas.';
return;
}
numbers.push(num);
}
var n = numbers.length;
if (n < 2) {
resultDiv.innerHTML = 'Need at least two data points to calculate SD.';
return;
}
// 1. Calculate the Mean
var sum = 0;
for (var i = 0; i < n; i++) {
sum += numbers[i];
}
var mean = sum / n;
// 2. Calculate Deviations from the Mean, 3. Square them, 4. Sum Squared Deviations
var sumSquaredDeviations = 0;
for (var i = 0; i < n; i++) {
var deviation = numbers[i] – mean;
sumSquaredDeviations += deviation * deviation;
}
// 5. Calculate the Variance (for sample)
var variance = sumSquaredDeviations / (n – 1);
// 6. Calculate the Standard Deviation
var standardDeviation = Math.sqrt(variance);
// Display the result
resultDiv.innerHTML = 'Sample Standard Deviation: ' + standardDeviation.toFixed(4) + '';
}