Standard Deviation Formula Explained

This reference explains the Standard Deviation formula used by our free Standard Deviation Calculator. Use it to understand variables, units, and common mistakes before you calculate.

The formula

s = √(variance)

Variables and units

• σ — Population standard deviation (same as data)
• s — Sample standard deviation (same as data)
• x̄ — Mean (same as data)

Worked example

Inputs: Sample: 2, 4, 4, 4, 5, 5, 7, 9 (n = 8, mean = 5)

1. Deviations: −3, −1, −1, −1, 0, 0, 2, 4
2. Squared: 9, 1, 1, 1, 0, 0, 4, 16 → sum = 32
3. Sample variance s² = 32 / (8−1) = 4.571
4. s = √4.571 ≈ 2.138

Result: Sample standard deviation ≈ 2.14

When to use this formula

Use this formula when you need a quick standard deviation estimate with values you already know. Our Standard Deviation Calculator applies the same relationship automatically so you can verify handwritten work.

Common mistakes

• Using population formula (÷ n) instead of sample (÷ n−1)
• Forgetting to square root the variance
• Calculating SD before checking for outliers

Frequently asked questions