Two classes average 70% on a test. In one, everyone scored 68–72. In the other, scores ran 40–100. Same mean — totally different stories. Standard deviation is the number that tells them apart.
Standard deviation (σ) measures how spread out data is around the mean. Small σ = values cluster tight; large σ = values scatter wide.
Finance (volatility/risk), quality control, grading curves, scientific error bars, machine learning — σ is the universal 'how variable is this?' measure.
Subtract the mean from each value, square it, average those, take the square root. (Use n−1 for a sample estimate.)
Data: 2, 4, 4, 4, 5, 5, 7, 9. Mean is 5. Roughly what's the standard deviation?
Why do investors call standard deviation 'risk'?
A stock with high σ swings wildly — big gains *and* big losses. Low σ means steady, predictable returns. σ quantifies how bumpy the ride is.
Variance vs standard deviation. Variance is σ² — it's in *squared* units (squared dollars, squared cm), which is hard to interpret. Take the square root to get σ back in the original units.
For a normal distribution: ~68% of values lie within ±1σ of the mean, ~95% within ±2σ, ~99.7% within ±3σ. The '68-95-99.7 rule'.
- σ measures spread around the mean.
- σ = √(variance) — square root brings it back to original units.
- In a normal distribution, 68-95-99.7% fall within ±1, ±2, ±3 σ.