insideout Get Pro

07 · Variance and risk

Pro

Probability, lesson 7

Variance and risk

Same average, different ride: why spread matters as much as the mean.

The last lesson boiled a bet down to one number, its expected value. Two bets can share that number and still play nothing alike. This lesson measures how far results spread around the average. You need expected value: each outcome times its chance, added up.

By the end, you can:

  1. Measure a bet's spread with its variance and standard deviation.
  2. Work out a die's variance, by the gaps and by a shortcut.
  3. Add variances over independent rounds, and see why an edge outgrows the noise.

Part 1. How far results spread

Measure a bet's spread with its variance and standard deviation.

Two bets on a fair coin. Bet A wins 2 or loses 1. Bet B wins 10 or loses 9. Both are worth (2 − 1) / 2 = (10 − 9) / 2 = 0.5 a round. Yet nobody would call them the same bet: B swings much further each round. You need a number for how far.

Example 1.1

Two bets with one average

Bet A wins 2 or loses 1 on a fair coin, and bet B wins 10 or loses 9. Both average 0.5 a round. How far from that average does each one's result typically land?

  1. Measure A's gaps2 − 0.5 = 1.5 and −1 − 0.5 = −1.5

    Each result's distance from the average, with a sign for above or below.

  2. Square them and average(1.5² + 1.5²) / 2 = 2.25

    Squaring makes every gap positive. Each result has chance 1/2.

  3. Take the square root√2.25 = 1.5

    Back from squared chips to chips.

  4. Do the same for B√((9.5² + 9.5²) / 2) = 9.5

    Both of B's results sit 9.5 from the average.

Answer1.5 for bet A and 9.5 for bet B: one average, and more than six times the spread.

Why square the gaps? The plain gaps always average to zero, because the average is the balance point from the last lesson: gaps above it cancel gaps below. Squaring stops the cancelling, and it counts a big miss for more than a small one. The square root at the end brings the answer back to the units of the bet.

Bet A: win 2 or lose 1 −1 2 −10 −5 0 5 10 σ = 1.5: both results 1.5 from 0.5 Bet B: win 10 or lose 9 −9 10 −10 −5 0 5 10 σ = 9.5: both results 9.5 from 0.5
Bet A: win 2 or lose 1 −1 2 −10 −5 0 5 10 σ = 1.5: both results 1.5 from 0.5 Bet B: win 10 or lose 9 −9 10 −10 −5 0 5 10 σ = 9.5: both results 9.5 from 0.5
Figure 1Both bets balance at 0.5. Bet A's two results sit 1.5 either side of that point, and bet B's sit 9.5 either side. The standard deviation measures that distance.

Variance and standard deviation

The variance of a random amount X is the average squared gap from its mean μ (read "mu"). Its square root is the standard deviation, σ (read "sigma"), in the same units as X. Bet A has variance 2.25 and σ = 1.5.

Var(X) = E[(X − μ)²]

Spread matters most when your money is limited. Play each bet 100 times. A bet A player ends behind about 1 time in 2,300. A bet B player ends behind about 31% of the time, with exactly the same edge. Part 3 shows where the difference comes from.

Your turn

Bet C pays +100 or −99 on a fair coin. Compared with bet A, its expected value and its standard deviation are:

Mean: (100 − 99) / 2 = 0.5, the same as A. Both results are 99.5 from that mean, so σ = 99.5, about 66 times A's 1.5. Expected value can't tell bets A and C apart. The standard deviation can.

Part 2. The spread of a die

Work out a die's variance, by the gaps and by a shortcut.

Keep going with Pro

You've done the first part. Pro opens the rest of this lesson, and every other one.

Still to come
  1. The spread of a die
  2. Spread over many rounds
See Pro plansI have Pro: sign in
Next lessonDistributions you meet dailyBinomial, Poisson and normal, and where each one shows up. Start lesson 8

Probability overviewSpotted a mistake? Tell us.