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 you how spread out your data points are from the average (mean).
Low Standard Deviation: Indicates that the data points tend to be very close to the mean (also called the expected value) and to each other.
High Standard Deviation: Indicates that the data points are spread out over a wider range of values.
Why is Standard Deviation Important?
Standard deviation is widely used in various fields, including:
Finance: To measure the volatility of an investment. Higher standard deviation means higher risk.
Quality Control: To monitor manufacturing processes and ensure product consistency.
Science and Research: To understand the variability in experimental results.
Education: To analyze test scores and student performance.
How to Calculate Standard Deviation
There are two main types of standard deviation: population standard deviation (σ) and sample standard deviation (s). The calculator above computes the sample standard deviation, which is typically used when your data is a sample from a larger population.
Steps for 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 differences 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.
Variance (s²) = [ Σ(x - x̄)² ] / (n - 1)
Calculate the Standard Deviation: Take the square root of the variance.
Standard Deviation (s) = √Variance = √s²
Example Calculation:
Let's calculate the standard deviation for the data set: 10, 12, 15, 11, 13
The standard deviation for this sample is approximately 1.92.
function calculateStandardDeviation() {
var dataPointsInput = document.getElementById("dataPoints").value;
var errorMessageDiv = document.getElementById("errorMessage");
var resultValueDiv = document.getElementById("result-value");
errorMessageDiv.innerText = ""; // Clear previous error messages
resultValueDiv.innerText = "–"; // Reset result
if (!dataPointsInput) {
errorMessageDiv.innerText = "Please enter data points.";
return;
}
var dataPoints = dataPointsInput.split(',')
.map(function(point) {
return parseFloat(point.trim());
})
.filter(function(point) {
return !isNaN(point);
});
if (dataPoints.length < 2) {
errorMessageDiv.innerText = "Please enter at least two valid numbers to calculate standard deviation.";
return;
}
var n = dataPoints.length;
var sum = 0;
for (var i = 0; i < n; i++) {
sum += dataPoints[i];
}
var mean = sum / n;
var sumSquaredDifferences = 0;
for (var i = 0; i < n; i++) {
var difference = dataPoints[i] – mean;
sumSquaredDifferences += difference * difference;
}
// Using (n – 1) for sample standard deviation
var variance = sumSquaredDifferences / (n – 1);
var standardDeviation = Math.sqrt(variance);
resultValueDiv.innerText = standardDeviation.toFixed(4); // Display with 4 decimal places
}