Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. In simpler terms, it tells you how spread out your numbers are from the average (mean).
Population vs. Sample: Which One to Use?
Choosing the correct calculation depends on the source of your data:
Population Standard Deviation (σ): Use this when your data set includes every member of the group you are studying (e.g., the test scores of every student in a specific classroom).
Sample Standard Deviation (s): Use this when your data is a subset of a larger population (e.g., surveying 100 people to estimate the behavior of an entire city). It uses "Bessel's correction" (n-1) to provide an unbiased estimate.
The Step-by-Step Formula
Calculate the Mean: Add all numbers and divide by the count.
Find the Deviations: Subtract the mean from each individual number.
Square the Deviations: Square each result from the previous step.
Sum of Squares: Add all the squared values together.
Variance: Divide the sum by N (Population) or N-1 (Sample).
Standard Deviation: Take the square root of the variance.
Real-World Example
Imagine you have five exam scores: 85, 90, 75, 92, and 88.
Mean: (85+90+75+92+88) / 5 = 86
Variance (Population): Calculate the squared difference from 86 for each score, sum them (164), and divide by 5 = 32.8
SD: √32.8 ≈ 5.73
A low standard deviation (like 5.73) indicates that the scores are clustered closely around the average, whereas a high standard deviation would suggest a wide range of performance.
function calculateSD() {
var input = document.getElementById('dataInput').value;
var resultDiv = document.getElementById('sdResults');
var resMean = document.getElementById('resMean');
var resCount = document.getElementById('resCount');
var resVariance = document.getElementById('resVariance');
var resSD = document.getElementById('resSD');
var stepExplanation = document.getElementById('stepExplanation');
// Clean data: split by commas or spaces, convert to numbers, filter out non-numbers
var numbers = input.split(/[,\s]+/)
.map(function(item) { return parseFloat(item.trim()); })
.filter(function(item) { return !isNaN(item); });
if (numbers.length < 2) {
alert("Please enter at least two valid numbers to calculate standard deviation.");
resultDiv.style.display = "none";
return;
}
var isPopulation = document.querySelector('input[name="sdType"]:checked').value === "population";
var n = numbers.length;
// 1. Calculate Mean
var sum = 0;
for (var i = 0; i < n; i++) {
sum += numbers[i];
}
var mean = sum / n;
// 2. Calculate Sum of Squares
var sumOfSquaredDeviations = 0;
for (var j = 0; j < n; j++) {
sumOfSquaredDeviations += Math.pow(numbers[j] – mean, 2);
}
// 3. Calculate Variance
var divisor = isPopulation ? n : n – 1;
var variance = sumOfSquaredDeviations / divisor;
// 4. Calculate SD
var sd = Math.sqrt(variance);
// Display Results
resMean.innerHTML = mean.toFixed(4).replace(/\.?0+$/, "");
resCount.innerHTML = n;
resVariance.innerHTML = variance.toFixed(4).replace(/\.?0+$/, "");
resSD.innerHTML = sd.toFixed(4).replace(/\.?0+$/, "");
var typeText = isPopulation ? "Population" : "Sample";
var formulaPart = isPopulation ? "N" : "N – 1";
stepExplanation.innerHTML = "Calculation Summary: For this " + typeText + " dataset, the sum of values is " + sum.toFixed(2) + ". The squared differences from the mean sum up to " + sumOfSquaredDeviations.toFixed(4) + ". Dividing by " + formulaPart + " (" + divisor + ") gives a variance of " + variance.toFixed(4) + ".";
resultDiv.style.display = "block";
}