Variance and standard deviation

Measure dispersion around the mean and keep squared units distinct.

12 minutes · intermediate · Original Poker4Quant lesson

What you will learn

Calculate population variance and distinguish scale, location and tail shape.

Prerequisites: Expected value as a weighted average

Square deviations from the mean

For a fully specified finite payoff distribution with mean μ, variance is E[(X−μ)²]. Squaring avoids positive and negative deviations canceling. Standard deviation is the nonnegative square root of variance and returns to the payoff’s original units.

A constant payoff has zero variance. A constant fee changes the mean but not the deviations about it, so it leaves variance unchanged. Multiplying all payoffs by a multiplies variance by a² and standard deviation by |a|.

A spread measure does not describe every tail

An equal-probability +30/−10 payoff has mean 10. Each outcome lies 20 from that mean, so variance is 400 squared chips and standard deviation is 20 chips. The mean and standard deviation alone do not identify the full payoff distribution.

A 99% chance of +1 and a 1% chance of −99 has mean zero and variance 99. Most trials win, yet the rare loss matters greatly. Probability of loss, worst-case loss, variance and expected payoff are different summaries. These are exact model-distribution calculations, not sample-variance estimators from observed data.

Worked example

A fair binary payoff is +10 or −10. What happens to its variance if every payoff is doubled and then a fixed 3-chip fee is subtracted?

The original mean is 0 and variance is 100.
Doubling makes outcomes +20 and −20, so variance becomes 400.
Subtracting 3 shifts outcomes to +17 and −23 with mean −3.
Both deviations from −3 are still 20 in magnitude.

Final variance remains 400 squared chips and standard deviation is 20 chips. The fee changes the mean, not the variance.

Common mistakes

Averaging signed deviations, which always gives zero.
Reporting variance in chips instead of squared chips.
Assuming a high win rate guarantees small downside.

Keep these ideas

Variance is a weighted mean of squared deviations.
Location shifts leave variance unchanged; scaling squares.
Dispersion is not a complete description of tail risk.