Standard deviation (σ or s) measures how spread out values in a dataset are from the mean. A low standard deviation means data points cluster near the average; a high standard deviation means they are spread widely.
How Standard Deviation Works
The calculation involves: (1) find the mean, (2) subtract the mean from each value and square the result, (3) average those squared differences (variance), (4) take the square root. Use population SD (σ) when you have all data points, or sample SD (s) when working with a subset (divide by n−1 instead of n).
The 68-95-99.7 Rule
For normally distributed data: 68% of values fall within ±1 SD of the mean, 95% within ±2 SD, and 99.7% within ±3 SD. Values beyond 3 SD are considered outliers.
Real-World Example
Test scores: 72, 78, 80, 82, 88. Mean: 80. Deviations: -8, -2, 0, +2, +8. Squared: 64, 4, 0, 4, 64. Variance: 136 ÷ 5 = 27.2. Standard deviation: √27.2 = 5.22. Most scores fall within 80 ± 5.22, or between 74.8 and 85.2.
Frequently Asked Questions
When should I use population vs sample standard deviation?
Use population SD (σ, divide by n) when your dataset includes every member of the group. Use sample SD (s, divide by n−1) when analyzing a subset of a larger population — which is the case in most real-world statistical analysis.
What does a standard deviation of 0 mean?
A standard deviation of 0 means all values in the dataset are identical — there is no variation at all. Every data point equals the mean.
How is standard deviation used in finance?
In investing, standard deviation measures volatility — how much returns fluctuate around the average. A stock with 20% annual SD is more volatile (risky) than one with 10% SD. It is a key component of risk assessment and portfolio optimization.