How to Calculate Standard Deviation from Mean

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Standard Deviation Calculator

Population (σ) Sample (s) Use "Sample" for a subset of a larger group, and "Population" if you have data for every member of the group.

Calculation Results

function calculateSD() { var rawInput = document.getElementById("dataInput").value; var calcType = document.getElementById("calcType").value; var resultsWrapper = document.getElementById("resultsWrapper"); var sdOutput = document.getElementById("sdOutput"); var errorBox = document.getElementById("errorBox"); // Reset UI resultsWrapper.style.display = "none"; errorBox.style.display = "none"; // Process Input: split by comma, space, or newline var rawArray = rawInput.split(/[,\s\n]+/); var numbers = []; for (var i = 0; i < rawArray.length; i++) { var trimmed = rawArray[i].trim(); if (trimmed !== "") { var num = parseFloat(trimmed); if (!isNaN(num)) { numbers.push(num); } } } // Validation if (numbers.length < 2) { errorBox.innerHTML = "Please enter at least two valid numbers to calculate standard deviation."; errorBox.style.display = "block"; return; } // 1. Calculate Mean var sum = 0; for (var j = 0; j < numbers.length; j++) { sum += numbers[j]; } var mean = sum / numbers.length; // 2. Sum of Squares (Differences from Mean) var sumOfSquares = 0; for (var k = 0; k < numbers.length; k++) { sumOfSquares += Math.pow(numbers[k] – mean, 2); } // 3. Variance var divisor = (calcType === "sample") ? (numbers.length – 1) : numbers.length; var variance = sumOfSquares / divisor; // 4. Standard Deviation var standardDeviation = Math.sqrt(variance); // Output formatting var html = ''; html += ''; html += ''; html += ''; html += ''; html += ''; html += '
Count (N):' + numbers.length + '
Arithmetic Mean (x̄):' + mean.toFixed(4) + '
Sum of Squares:' + sumOfSquares.toFixed(4) + '
Variance (' + (calcType === "sample" ? "s²" : "σ²") + '):' + variance.toFixed(4) + '
Standard Deviation (' + (calcType === "sample" ? "s" : "σ") + '):' + standardDeviation.toFixed(4) + '
'; sdOutput.innerHTML = html; resultsWrapper.style.display = "block"; }

How to Calculate Standard Deviation from Mean

Standard deviation is a crucial statistical measure that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range.

The Step-by-Step Process

To calculate the standard deviation manually, follow these five steps:

  1. Calculate the Mean: Find the average of your data set by adding all numbers and dividing by the count (N).
  2. Subtract the Mean: Subtract the mean from every individual data point. Some results will be positive, and some will be negative.
  3. Square the Differences: Square each of the results from the previous step. This ensures all values are positive.
  4. Find the Variance: Sum all the squared differences together. For Population standard deviation, divide by N. For Sample standard deviation, divide by N – 1.
  5. Take the Square Root: The square root of the variance is your standard deviation.

Standard Deviation Formulas

Sample Standard Deviation (s):
s = √[ Σ(xi – x̄)² / (n – 1) ]

Population Standard Deviation (σ):
σ = √[ Σ(xi – μ)² / N ]

Practical Example

Suppose you are measuring the height of 5 plants in centimeters: 10, 15, 20, 25, and 30.

  • Mean (x̄): (10+15+20+25+30) / 5 = 20
  • Squared Differences:
    • (10-20)² = 100
    • (15-20)² = 25
    • (20-20)² = 0
    • (25-20)² = 25
    • (30-20)² = 100
  • Sum of Squares: 100 + 25 + 0 + 25 + 100 = 250
  • Sample Variance: 250 / (5 – 1) = 62.5
  • Sample Standard Deviation: √62.5 ≈ 7.906

Sample vs. Population: Which one to use?

This is the most common point of confusion. Use Population Standard Deviation when your data includes everyone or everything you are interested in (e.g., test scores for a whole class). Use Sample Standard Deviation when your data is a small piece of a larger puzzle (e.g., polling 100 voters to represent a whole city).

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