Standard deviation is a fundamental statistical measure that quantifies the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean (average) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.
It is particularly useful in finance, science, engineering, and quality control to understand the consistency and predictability of a dataset. For example, in finance, it's used to measure the volatility of an investment.
How to Calculate Standard Deviation
The calculation involves several steps:
1. Calculate the Mean (Average): Sum all the data points and divide by the number of data points.
2. Calculate Deviations from the Mean: Subtract the mean from each individual data point.
3. Square the Deviations: Square each of the differences calculated in the previous step.
4. Calculate the Variance: Sum the squared deviations and divide by the number of data points (for population variance) or by (number of data points – 1) (for sample variance). This calculator computes the sample standard deviation, which is more common when dealing with a sample of data.
5. Calculate the Standard Deviation: Take the square root of the variance.
Formula for Sample Standard Deviation (s):
s = sqrt( Σ(xi - x̄)² / (n - 1) )
Where:
s is the sample standard deviation
Σ denotes summation
xi is each individual data point
x̄ (x-bar) is the sample mean
n is the number of data points
Use Cases:
Finance: Measuring investment volatility and risk.
Quality Control: Monitoring consistency in manufacturing processes.
Science: Analyzing experimental results and the spread of measurements.
Social Sciences: Understanding the distribution of survey responses or demographic data.
Education: Assessing the spread of student scores on a test.
Example Calculation:
Let's calculate the standard deviation for the data points: 10, 12, 15, 11, 13
Number of data points (n): 5
Sum of data points: 10 + 12 + 15 + 11 + 13 = 61
Mean (x̄): 61 / 5 = 12.2
Deviations from Mean (xi – x̄):
10 – 12.2 = -2.2
12 – 12.2 = -0.2
15 – 12.2 = 2.8
11 – 12.2 = -1.2
13 – 12.2 = 0.8
Squared Deviations (xi – x̄)²:
(-2.2)² = 4.84
(-0.2)² = 0.04
(2.8)² = 7.84
(-1.2)² = 1.44
(0.8)² = 0.64
Sum of Squared Deviations: 4.84 + 0.04 + 7.84 + 1.44 + 0.64 = 14.8
Variance (s²): 14.8 / (5 – 1) = 14.8 / 4 = 3.7
Standard Deviation (s): sqrt(3.7) ≈ 1.92
The standard deviation for this dataset is approximately 1.92, indicating a moderate spread of the data around the mean.
function calculateStandardDeviation() {
var dataPointsInput = document.getElementById("dataPoints").value;
var resultDiv = document.getElementById("result");
var resultValueDiv = document.getElementById("result-value");
if (!dataPointsInput) {
alert("Please enter data points.");
return;
}
var dataPoints = dataPointsInput.split(',')
.map(function(item) {
return parseFloat(item.trim());
})
.filter(function(item) {
return !isNaN(item);
});
if (dataPoints.length < 2) {
alert("Please enter at least two valid data points 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 sumSquaredDeviations = 0;
for (var i = 0; i < n; i++) {
var deviation = dataPoints[i] – mean;
sumSquaredDeviations += deviation * deviation;
}
// Calculate sample variance (divide by n-1)
var variance = sumSquaredDeviations / (n – 1);
var standardDeviation = Math.sqrt(variance);
resultValueDiv.textContent = standardDeviation.toFixed(4); // Display with 4 decimal places
resultDiv.style.display = "block";
}