Standard Deviation: What Statisticians Mean by 'Spread'

Ask ten experts and you will get twelve answers. The underlying math, thankfully, only has one. Mean tells you the centre. Standard deviation tells you how messy or tidy the data is around it.

The intuition

Small SD = tight cluster. Large SD = scattered. Two classes with identical mean test scores can have very different SDs.

Calculation steps

Subtract mean from each value → square → average those → square root. Sample SD divides by n − 1.

68-95-99.7 rule

For normal data: 68% within ±1 SD, 95% within ±2 SD, 99.7% within ±3 SD. Used for confidence intervals.

Where it shows up

Quality control, finance (volatility), polling margins, medical reference ranges.

Pro tips

• Sample SD (n−1) vs population SD (n). Use sample for almost everything.
• Coefficient of variation (SD / mean) lets you compare spread across different units.

Frequently asked questions

Why divide by n − 1? Sample SD slightly underestimates without it (Bessel's correction). Pure math thing.

What's variance? SD squared. Useful in algebra; SD is more interpretable for humans.

Run your own numbers before you trust a rule of thumb. Two minutes saves a lot of regret.