Standard Deviation Calculator
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Standard Deviation Calculator
Standard Deviation
—
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 signifies that the data points are spread out over a wider range of values.
How is Standard Deviation Calculated?
The calculation involves several steps:
- Calculate the Mean: Sum all the data points and divide by the total number of data points (N). This gives you the average value.
Mean (μ) = (Σx) / N
- Calculate the Variance: For each data point (x), subtract the mean (μ) and square the result (x – μ)². Sum all these squared differences. Then, divide this sum by the total number of data points (N) for a population standard deviation, or by (N-1) for a sample standard deviation. Our calculator computes the sample standard deviation, which is more common when dealing with a subset of data.
Sample Variance (s²) = Σ(x - μ)² / (N - 1)
- Calculate the Standard Deviation: Take the square root of the variance.
Sample Standard Deviation (s) = √[ Σ(x - μ)² / (N - 1) ]
Why is Standard Deviation Important?
Standard deviation is widely used across various fields:
- Finance: To measure the volatility of an investment or the risk associated with a portfolio. A higher standard deviation typically implies higher risk.
- Quality Control: To monitor consistency in manufacturing processes. If product measurements have a low standard deviation, it indicates consistency.
- Science and Research: To assess the reliability and variability of experimental results.
- Education: To understand the distribution of test scores among students.
- General Data Analysis: To describe the spread of any dataset, helping to understand its characteristics.
Example Calculation
Let's calculate the standard deviation for the data set: 5, 10, 15, 20, 25
- Mean: (5 + 10 + 15 + 20 + 25) / 5 = 75 / 5 = 15
- Squared Differences from Mean:
- (5 – 15)² = (-10)² = 100
- (10 – 15)² = (-5)² = 25
- (15 – 15)² = (0)² = 0
- (20 – 15)² = (5)² = 25
- (25 – 15)² = (10)² = 100
- Sum of Squared Differences: 100 + 25 + 0 + 25 + 100 = 250
- Sample Variance: 250 / (5 – 1) = 250 / 4 = 62.5
- Sample Standard Deviation: √62.5 ≈ 7.91
Therefore, the standard deviation for this sample data set is approximately 7.91. This indicates a moderate spread of the data around the mean of 15.
function calculateStandardDeviation() {
var dataInput = document.getElementById("dataPoints").value;
var errorMessageDiv = document.getElementById("errorMessage");
var resultSpan = document.getElementById("standardDeviationResult");
// Clear previous error messages and results
errorMessageDiv.innerText = "";
resultSpan.innerText = "–";
if (dataInput.trim() === "") {
errorMessageDiv.innerText = "Please enter some data points.";
return;
}
// Split the input string by commas and convert to numbers
var dataPoints = dataInput.split(',')
.map(function(item) { return parseFloat(item.trim()); })
.filter(function(item) { return !isNaN(item); }); // Filter out any non-numeric entries
if (dataPoints.length < 2) {
errorMessageDiv.innerText = "Please enter at least two valid data points.";
return;
}
var n = dataPoints.length;
// 1. Calculate the Mean
var sum = 0;
for (var i = 0; i < n; i++) {
sum += dataPoints[i];
}
var mean = sum / n;
// 2. Calculate the Sum of Squared Differences from the Mean
var sumSquaredDifferences = 0;
for (var i = 0; i < n; i++) {
sumSquaredDifferences += Math.pow(dataPoints[i] – mean, 2);
}
// 3. Calculate the Sample Variance (divide by n-1)
var variance = sumSquaredDifferences / (n – 1);
// 4. Calculate the Sample Standard Deviation
var standardDeviation = Math.sqrt(variance);
// Display the result, rounded to a reasonable number of decimal places
resultSpan.innerText = standardDeviation.toFixed(4);
}