Variance and Standard Deviation

How spread out numbers are, and why the average alone misleads.

Also known as: variance, standard deviation, spread, dispersion

Variance measures how far numbers spread around their average: the mean of squared deviations from the mean. Its square root, the standard deviation, puts the spread back in original units — “average order $40, typically ±$35” tells you the business lives on variability that the average alone hides.

team A deals: 38, 40, 42  → mean 40, sd ≈ 2   (predictable)
team B deals:  5, 40, 75  → mean 40, sd ≈ 35  (same average, different business)

Same mean, different worlds: one team to forecast, the other to investigate. Whenever two groups share an average, the spread is where the story went.

The classic mistakes:

Reporting the centre without the spread. A mean forecast with no variability band is a promise the data never made. Quote both, always.
Treating standard deviation as the error of the mean. Spread of the data (sd) and precision of the average (standard error) shrink differently with sample size — confusing them overstates what big samples prove about individuals.
Assuming symmetry. “Mean ± 2 sd covers ~95%” holds for bell-shaped data only. On skewed data (incomes, latencies, deal sizes) it misstates both tails — check the shape (distribution shapes) before quoting bands.
Comparing variances across different scales. A sd of $35 means opposite things for $40 and $4,000 averages. Compare relative spread (sd ÷ mean) across scales, absolute within one.
Letting outliers set the spread. A few extreme values inflate variance enormously since deviations are squared. Inspect the tails before trusting the number.

The habit: never let an average travel alone. Centre plus spread plus shape is the minimum honest description of any batch of numbers — see confidence intervals for what the spread implies about estimates.