Standard deviation tells you how spread out data is around the average. A small standard deviation means data clusters tightly; a large one means it's widely scattered.
Why Standard Deviation Matters
Two classes both average 75% on a test. But in Class A, scores range from 70–80%. In Class B, scores range from 40–100%. The average hides important information — standard deviation reveals it.
The Formula
For a population (all data):
σ = √[ Σ(x - μ)² / N ]
For a sample (subset of data):
s = √[ Σ(x - x̄)² / (n-1) ]
Where:
- σ (sigma) = population standard deviation
- s = sample standard deviation
- x = each value
- μ or x̄ = mean
- N = population size, n = sample size
The sample formula divides by n-1 (not n) to correct for bias when estimating from a subset.
Step-by-Step Example
Data: 4, 7, 13, 2, 9 (sample of 5 values)
Step 1: Calculate the mean:
Mean = (4 + 7 + 13 + 2 + 9) / 5 = 35 / 5 = 7
Step 2: Subtract mean from each value and square:
| x | x - mean | (x - mean)² | |---|----------|-------------| | 4 | -3 | 9 | | 7 | 0 | 0 | | 13 | 6 | 36 | | 2 | -5 | 25 | | 9 | 2 | 4 |
Step 3: Sum the squared differences: 9 + 0 + 36 + 25 + 4 = 74
Step 4: Divide by n-1 = 4: 74 / 4 = 18.5
Step 5: Take the square root: √18.5 ≈ 4.30
Standard deviation = 4.30
The 68-95-99.7 Rule
For normally distributed data:
- 68% of values fall within ±1 standard deviation of the mean
- 95% fall within ±2 standard deviations
- 99.7% fall within ±3 standard deviations
Example: Heights with mean 170 cm, SD 10 cm:
- 68% are between 160–180 cm
- 95% are between 150–190 cm
Real-World Applications
- Finance: Measures investment volatility (risk)
- Manufacturing: Quality control — products outside ±3σ are defects
- Medicine: Identifying abnormal test results
- Education: Grading on a curve
Use our Standard Deviation Calculator to calculate mean, median, variance, and standard deviation for any dataset.