Spread to Moneyline Conversion: The Normal Model Works at 1.5 and Fails at 7
Every spread implies a moneyline. Every moneyline implies a spread. The conversion chart you find online is one model with one assumption about how NFL margins spread out. This article shows the model, runs the model on this week's board, and marks the spread where the model fails.
Short version: the normal curve prices small favorites within a few cents. The same curve underprices a 7-point favorite by 90 to 100 cents. If you convert spreads to moneylines for alt lines, teasers, or price checks, you need to know where the formula stops working.
The Formula Behind Every Conversion Chart
Hal Stern's 1991 paper in The American Statistician, "On the Probability of Winning a Football Game," treats the final NFL margin as a normal random variable. The mean equals the point spread. The standard deviation is 13.86 points, fit to the 1981, 1983, and 1984 seasons. Neil Paine replicated the work on 1978 through 2012 games for FiveThirtyEight and got 13.45. A 2022 arXiv paper on college football spreads notes recent estimates sit near 13.5.
The conversion takes three steps.
Spread to fair moneyline
1. z = spread / 13.5
2. win probability p = Φ(z), the standard normal CDF
3. favorite fair price = -100 × p / (1 - p)
underdog fair price = +100 × p / (1 - p) with the same number
Example, spread 1.5: z = 0.111, p = 54.4%
Favorite: -100 × 0.544 / 0.456 = -119
Near a pick'em, the normal curve gives each point of spread roughly 3 percentage points of win probability. The density at zero is 1 / (13.5 × 2.507), which is 2.96% per point. A 1-point favorite wins 53.0%. A 2.5-point favorite wins 57.3%. A 3.5-point favorite wins 60.2%.
The Chart at 13.5 Points
Here is the full normal-model table, computed with the formula above. Every figure recomputes from the formula.
| Spread | Win probability | Fair favorite price | Fair underdog price |
|---|---|---|---|
| 1 | 53.0% | -113 | +113 |
| 1.5 | 54.4% | -119 | +119 |
| 2.5 | 57.3% | -134 | +134 |
| 3 | 58.8% | -143 | +143 |
| 3.5 | 60.2% | -151 | +151 |
| 4.5 | 63.1% | -171 | +171 |
| 5.5 | 65.8% | -193 | +193 |
| 6.5 | 68.5% | -217 | +217 |
| 7 | 69.8% | -231 | +231 |
| 7.5 | 71.1% | -246 | +246 |
| 9.5 | 75.9% | -315 | +315 |
| 10 | 77.1% | -336 | +336 |
| 14 | 85.0% | -567 | +567 |
The fair prices here carry no vig. Real moneylines add margin to both sides, so a book posts something wider than +119 and -119.
Check One: Bears -1.5 This Week
VegasInsider's Week 7 board, updated October 19, lists Patriots at Bears with the Bears at -1.5 (-110) and moneylines of -125 and +105. Strip the vig from the moneyline pair.
No-vig the Bears moneyline
-125 implies 125 / 225 = 55.56%
+105 implies 100 / 205 = 48.78%
Sum: 104.34%. The book keeps 4.34 points.
Scale each side by 1 / 1.0434:
Bears fair: 55.56 / 104.34 = 53.25% (fair price -114)
Normal model at 1.5: 54.4% (fair price -119)
Gap: 1.2 percentage points
The market and the model agree within about 1 point. At small spreads the normal curve is a good check. A -125 price on a team the model says is worth -119 does not signal an edge. The 6-cent gap is inside the model's error and inside the vig.
Check Two: Where the Model Breaks
Run the same test on a 7-point favorite. The normal model says 69.8%, or -231. Two independent historical samples disagree.
- OpticOdds paired point spreads with results for 4,835 NFL games using the nflreadr package. Its worked example for a 7-point favorite gives a 76.8% win rate. OpticOdds also reports favorites winning over 80% of the time at 10 points or more.
- Boyd's Bets publishes a spread-to-win-probability chart built on data since 1999. Its row for a 7-point spread shows 76.2%.
Two sources, two builds, one answer: 76 to 77%. Convert those win rates to prices.
Price a 7-point favorite three ways
Normal model: 69.8% → -100 × 0.698 / 0.302 = -231
Boyd's Bets, 76.2% → -100 × 0.762 / 0.238 = -320
OpticOdds, 76.8% → -100 × 0.768 / 0.232 = -331
Model error vs the data: 6.4 to 7.0 percentage points, 89 to 100 cents
A 7-point favorite at -231 is a gift. A 7-point underdog moneyline at +231 is a trap. You pay for a 30% chance and the data says the dog wins about 23 to 24%.
The 10-point row shows the same pattern. The model gives 77.1%. OpticOdds reports favorites above 80% at 10 points or more. At both spreads checked, the model sits below the data.
Why the Normal Curve Misses
NFL margins are discrete and clumpy. ESPN reports about 14.8% of games since the extra-point change have ended on exactly 3, with 7 next at 9%. BetMGM's count from 2000 onward puts 3 at about 14.97% and 7 at about 9.09%. A 1999 through 2025 sample of 7,276 games finds 3 in 15.12% of games, 1,100 of 7,276, and 7 at 9.03%.
Test those against the bell curve. Center a game on a pick'em with a 13.5-point standard deviation. The density at 3 is 2.9% per point. A margin of exactly +3 or -3 should land in about 5.8% of games. Observed: roughly 15%. At 7 the curve gives about 5.2%. Observed: roughly 9%.
| Final margin | Normal model (pick'em game) | Observed share of games | Source |
|---|---|---|---|
| 3 | about 5.8% | 14.8% to 15.12% | ESPN, BetMGM, 1999 to 2025 sample |
| 7 | about 5.2% | 9.0% to 9.09% | ESPN, BetMGM, 1999 to 2025 sample |
The normal model assumes margins fall in a smooth bell around the spread. They do not. A 3 or 7 spread sits on the heaviest lumps in the distribution, so the win probability at those numbers differs from the smooth-curve value. One betting guide ran the same comparison and found the normal shortcut gave a 7.5-point favorite about 71% while moneylines implied roughly 75 to 77%. Our table shows the same 71.1%.
Stern's 1991 fit used games from three seasons, The formula still gives a usable answer for games inside a field goal.
Calculator: Model Price vs Market Price
Enter a spread and the standard deviation. Add a moneyline pair if you have one and the tool strips the vig for you. Above a 3-point spread, read the model as a floor for the favorite, not a fair price.
Spread to moneyline check
Calculators Disagree, So Check the Build
Action Network publishes a spread-to-moneyline converter. In its own example, a Broncos spread of 4.5 at -110 converts to roughly -224. Our normal table prices 4.5 at -171. The gap is 53 cents. The two methods differ. Books priced the same game at -205 on FanDuel, which sits between the two.
Both outputs are models. Neither is the market. Before you act on a conversion, find out which model built the number.
A Worked Alt-Line Example
Say a book hangs a 7-point favorite at -110 and offers the same team on the moneyline at -260. Compare the moneyline to the three prices above. The normal model says -231, so -260 looks expensive. The observed win rates say -320 to -331, so -260 is cheap. The same number looks like a bad price or a good price depending on which model you trust.
Do the break-even work. -260 needs 260 / 360 = 72.2% to break even. The 76.2% and 76.8% samples clear the bar by 4.0 and 4.6 points. The normal model says 69.8% and misses the bar by 2.4 points. The sample data carries the bet. The formula would have talked you out of the bet. A conversion chart built on the normal curve would have cost you the wager.
The 7-point sample also sets a limit on how much to trust the edge. The two sources use different seasons and different line sources. A 4-point cushion over break-even is real money at -260 but sits within the spread of those two samples. Size the bet for a thin edge, not a lock.
Where to Use the Conversion
- Inside 3 points: use the normal table. Error runs about 1 to 2 points of win probability.
- From 3.5 to 6.5: use the normal table as a lower bound for the favorite. Compare to a market no-vig price before you call anything an edge.
- At 7 and above: use observed win rates. Anchor at 76 to 77% for a 7-point favorite.
- For alt spreads and teasers: the model prices the move poorly across 3 and 7. See the half-point guide and the Wong teaser math for the key-number work.
- For no-vig work: strip the vig first. The Bears example shows a 4.34-point margin on a -125 and +105 pair.
The verdict: the 13.5-point normal conversion is a one-minute sanity check for spreads under 3. Above 3, the model understates the favorite by 6 to 7 points of win probability at a 7-point spread. Use observed win rates by spread, and use the market's own no-vig moneyline as the price to beat.
Sources and Method
Stern 1991 supplies the normal model and the 13.86 standard deviation. Paine's FiveThirtyEight replication supplies 13.45 on 1978 through 2012 data. The arXiv college football paper notes recent estimates near 13.5. OpticOdds and Boyd's Bets supply the observed favorite win rates. ESPN, BetMGM, and a 1999 through 2025 margin study supply the clustering at 3 and 7. VegasInsider supplies the Week 7 Bears price. Every calculation above, the table, the Bears no-vig, and the 7-point repricing, comes from the formulas shown. You recompute each in a spreadsheet with NORM.S.DIST.
Limits: the 13.5 figure averages all NFL games and ignores each game's total. The observed win rates come from closing spreads in the samples cited, and the two sources do not publish identical datasets. Treat the 76 to 77% figure as a range, not a point estimate.