Standard Deviation Formula Table- Quick Reference Guide

What This Guide Is For

You're working on stats homework, analyzing data at your job, or preparing for an exam. You need the standard deviation formula — the right one, right now. This page gives you exactly that. No theory lectures. No history of statistics. Just the formulas you need in a table you can actually read.

The Two Standard Deviation Formulas You Need

There are two versions of standard deviation. Most people get confused about which one to use. Here's the simple rule:

• Population standard deviation — use when you have every single data point in your dataset
• Sample standard deviation — use when your data is just a sample from a larger group

Most real-world situations use sample standard deviation. Academic problems often specify which one they want.

Standard Deviation Formula Table

Formula Variables Explained

Before you panic at the Greek letters, here's what everything means:

• σ (sigma) = population standard deviation
• s or sₓ = sample standard deviation
• xᵢ = each individual data point
• μ (mu) = population mean (average)
• x̄ (x-bar) = sample mean (average)
• Σ = sum of all values
• N = total number of data points in population
• n = total number of data points in sample

The Variance Connection

Standard deviation is the square root of variance. If you see variance mentioned in your materials, just square the result of these formulas:

• Population variance = σ² = Σ(xᵢ - μ)² / N
• Sample variance = s² = Σ(xᵢ - x̄)² / (n-1)

How to Calculate Standard Deviation: Step by Step

Let's say your dataset is: 2, 4, 6, 8, 10

Step 1: Find the Mean

Add all numbers and divide by how many there are.

(2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6

Step 2: Subtract the Mean from Each Value

• 2 - 6 = -4
• 4 - 6 = -2
• 6 - 6 = 0
• 8 - 6 = 2
• 10 - 6 = 4

Step 3: Square Each Result

• (-4)² = 16
• (-2)² = 4
• (0)² = 0
• (2)² = 4
• (4)² = 16

Step 4: Sum the Squared Values

16 + 4 + 0 + 4 + 16 = 40

Step 5: Divide by N or (n-1)

For population: 40 / 5 = 8

For sample: 40 / 4 = 10

Step 6: Take the Square Root

Population: √8 = 2.83

Sample: √10 = 3.16

That's it. Six steps. The formula just automates this process.

Common Mistakes That Mess Up Your Answer

• Using N when you should use n-1 — This is the most common error. If your data is a sample, divide by n-1, not N.
• Forgetting to square the deviations — Negative and positive deviations cancel out. Squaring fixes this.
• Rounding too early — Keep full decimal precision until your final answer.
• Confusing mean types — Population mean uses μ. Sample mean uses x̄. They're calculated the same way but represent different things.

Quick Decision Guide

When Standard Deviation Is Zero

If every number in your dataset is identical, the standard deviation is 0. No variation means no spread. This is correct, not an error.

What This Doesn't Cover

This guide covers the basic formulas used in most situations. If you're working with grouped data, frequency distributions, or weighted standard deviation, you need different methods. Those situations require additional steps beyond this quick reference.

Bookmark this page. The table and formulas above are what you'll reach for every time.