Score Distributions — Bets Learn

For continuous outcomes, use the bell curve

Win/loss is binary — we use Beta (Lesson 2). But spreads, totals, and margins are CONTINUOUS. The right tool is the Normal (Gaussian) distribution.

f(x) = (1 / (σ√(2π))) × exp(−½ × ((x − μ) / σ)²)

μ = mean (expected margin) · σ = standard deviation

Translation: "most games land near the spread, with fewer landing in the extremes."

🎮 Interactive: Spread Cover Probability

Set the spread (μ) and your view of the standard deviation, see the probability of covering.

Spread (favorite favored by): 4.5 points

Standard deviation (typical game volatility): 12 points

Bet line (cover threshold): 4.5 points

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Probability favorite covers

📊 Typical Standard Deviations by Sport

Sport	σ (margin std dev)	Why
NBA	~11-12 points	High scoring, many possessions, late-game garbage time
NFL	~13.5 points	Few possessions, big single-play swings
NCAAB	~10-11 points	Similar to NBA but slightly tighter
MLB	~3.5 runs	Lower-scoring, more single-event randomness
NHL	~2 goals	Lowest scoring, lots of close games

📖 The 68/95/99.7 Rule

In a Normal distribution:

68% of outcomes land within 1σ of the mean
95% within 2σ
99.7% within 3σ

For an NBA game with spread 4.5 and σ=12: 68% of margins land between −7.5 and +16.5. 95% land between −19.5 and +28.5. A 30-point blowout is rare (less than 2.5% of games).

Practical use: If a model says "expected margin 6.5, σ=12" and the line is 4.5, you can compute the probability the favorite covers as the area under the curve from 4.5 to infinity. That's roughly 56-57%. Compare to break-even at -110 (52.4%) → small +EV edge.