Standard Deviation Formula
Calculate standard deviation with σ = √(Σ(x - μ)² / N).
Measure how spread out values are from the mean in any data set.
The Formula
Sample: s = √(Σ(x - x̄)² / (n - 1))
Standard deviation measures how spread out data values are from the mean. A small standard deviation means values are clustered close to the average. A large standard deviation means values are more spread out.
Variables
| Symbol | Meaning |
|---|---|
| σ (sigma) | Population standard deviation |
| s | Sample standard deviation |
| x | Each individual data value |
| μ (mu) | Population mean |
| x̄ | Sample mean |
| N | Population size (total number of values) |
| n - 1 | Degrees of freedom (used for sample data) |
Example 1 — Population Standard Deviation
Find the standard deviation of: 4, 8, 6, 5, 3 (entire population)
Step 1: Find the mean — μ = (4 + 8 + 6 + 5 + 3) / 5 = 26 / 5 = 5.2
Step 2: Find squared differences from the mean:
(4 - 5.2)² = 1.44, (8 - 5.2)² = 7.84, (6 - 5.2)² = 0.64, (5 - 5.2)² = 0.04, (3 - 5.2)² = 4.84
Step 3: Sum = 1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.80
Step 4: Divide by N — 14.80 / 5 = 2.96
Step 5: Take the square root — √2.96 = 1.72
σ = 1.72 — Values typically fall within about 1.72 of the mean.
Example 2 — Sample Standard Deviation
Test scores from a sample of students: 70, 80, 90, 85, 75
Step 1: Find the mean — x̄ = (70 + 80 + 90 + 85 + 75) / 5 = 400 / 5 = 80
Step 2: Squared differences:
(70 - 80)² = 100, (80 - 80)² = 0, (90 - 80)² = 100, (85 - 80)² = 25, (75 - 80)² = 25
Step 3: Sum = 100 + 0 + 100 + 25 + 25 = 250
Step 4: Divide by (n - 1) — 250 / 4 = 62.5
Step 5: Take the square root — √62.5 = 7.91
s = 7.91 — The sample scores vary by about 7.91 points from the mean.
When to Use It
Use standard deviation when:
- You need to understand the spread or variability in your data
- Comparing consistency between two data sets
- Determining whether a data point is unusually high or low
- Building confidence intervals or performing hypothesis tests