Standard Deviation Explained
Standard deviation measures how spread out values are around the mean. A low standard deviation means data clusters tightly; a high value means values are widely scattered. It is the most common measure of variability in statistics.
The Two Formulas
Population standard deviation (σ) — use when you have the entire population:
Sample standard deviation (s) — use when you have a sample from a larger group:
The division by n − 1 (Bessel's correction) corrects for bias when estimating the population standard deviation from a sample. For most real-world analyses, use the sample formula.
Step-by-Step Calculation
Data: 2, 4, 4, 4, 5, 5, 7, 9 (n = 8)
- Find the mean: (2+4+4+4+5+5+7+9) / 8 = 40/8 = 5
- Subtract the mean from each value, then square:
(2−5)² = 9, (4−5)² = 1 (×3), (5−5)² = 0 (×2), (7−5)² = 4, (9−5)² = 16 - Sum the squared differences: 9+1+1+1+0+0+4+16 = 32
- Divide by N for population variance: 32/8 = 4
- Take the square root: √4 = 2
Population standard deviation = 2. (Sample: divide by 7 → √(32/7) ≈ 2.14)
The 68–95–99.7 Rule
For normally distributed data, almost all values fall within a predictable number of standard deviations from the mean:
- 68% of values fall within μ ± 1σ
- 95% fall within μ ± 2σ
- 99.7% fall within μ ± 3σ
Example: IQ scores follow a normal distribution with μ = 100 and σ = 15. About 95% of people score between 70 and 130 (100 ± 2×15).
Standard Deviation vs Variance
Variance is the average of the squared deviations from the mean. Standard deviation is the square root of variance.
Why bother with std dev? Because variance is in squared units (e.g., m², €²) which are hard to interpret. Standard deviation is in the same unit as the original data (metres, euros), making it immediately meaningful. Compute it instantly with the statistics calculator.