Standard deviation measures how spread out numbers are from their average. This comprehensive guide will teach you how to calculate it step by step, whether you're working with a population or sample data.

Understanding the Basics

Before diving into calculations, it's essential to understand what standard deviation represents. It tells us how much individual data points deviate from the mean (average) of the dataset.

Key Terms to Know

Mean (μ or x̄): The average of all data points
Variance (σ² or s²): The average of squared differences from the mean
Standard Deviation (σ or s): The square root of variance
Population: The entire group you want to study
Sample: A subset of the population

Step 1: Calculate the Mean

First, find the average of your dataset by adding all values and dividing by the number of values.

Mean = (Sum of all values) ÷ (Number of values)

Example with Dataset: [2, 4, 6, 8, 10]

Mean = (2 + 4 + 6 + 8 + 10) ÷ 5 = 30 ÷ 5 = 6

Step 2: Find Deviations from the Mean

Subtract the mean from each data point to find how far each value is from the average.

Data Point	Deviation from Mean
2	2 - 6 = -4
4	4 - 6 = -2
6	6 - 6 = 0
8	8 - 6 = 2
10	10 - 6 = 4

Step 3: Square Each Deviation

Square each deviation to eliminate negative values and emphasize larger differences.

Deviation	Squared Deviation
-4	(-4)² = 16
-2	(-2)² = 4
0	(0)² = 0
2	(2)² = 4
4	(4)² = 16

Step 4: Calculate Variance

Find the average of the squared deviations. The formula differs for population and sample:

Population Variance (σ²)

σ² = Σ(x - μ)² / N

Where N is the total number of data points

Our example:

σ² = (16 + 4 + 0 + 4 + 16) / 5 = 40 / 5 = 8

Sample Variance (s²)

s² = Σ(x - x̄)² / (n - 1)

Where n is the sample size (use n-1 for Bessel's correction)

Our example:

s² = (16 + 4 + 0 + 4 + 16) / (5 - 1) = 40 / 4 = 10

Step 5: Calculate Standard Deviation

Take the square root of the variance to get the standard deviation:

Population Standard Deviation (σ)

σ = √(σ²) = √8 ≈ 2.83

Sample Standard Deviation (s)

s = √(s²) = √10 ≈ 3.16

Quick Reference Formulas

Population

Mean: μ = Σx / N

Variance: σ² = Σ(x - μ)² / N

Std Dev: σ = √(σ²)

Sample

Mean: x̄ = Σx / n

Variance: s² = Σ(x - x̄)² / (n - 1)

Std Dev: s = √(s²)

When to Use Population vs Sample Formulas

Use Population Formula when you have data for the entire group you're studying (e.g., all students in a class, all products in inventory)

Use Sample Formula when you have data from only a subset of the larger group (e.g., survey responses from 100 people representing thousands)

Practice Tips

Always double-check your mean calculation first
Keep track of positive and negative deviations
Remember to square all deviations before summing
Use the correct denominator (N for population, n-1 for sample)
Don't forget to take the square root at the end

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Step-by-Step Guide to Calculating Standard Deviation