01 · The forecast
A spread of −3.5 is a forecast: the favorite by 3.5 points.
A point spread is a forecast of the final margin, written from the favorite's side with a minus sign. A closing spread of −3.5 says the favorite is expected to win by 3.5 points. The chart measures the favorite's final margin: right of zero the favorite wins, left of zero the underdog wins, and the marker sits at 3.5. The spread says where results should center, not how often the favorite wins.
02 · The scatter
Across 7,017 games, results scatter around the spread with a standard deviation of 13.2 points.
Each bar counts games by how far the real margin landed from the closing spread, for 7,017 NFL games from 2000 to 2025 in nflverse data, drawn around the 3.5 marker. The misses average +0.07 points, so the spread is close to unbiased, but they are wide: their standard deviation is 13.2 points. Only 7 games missed by more than 45 points, the edge of the chart.
03 · The curve
A normal curve centered on 3.5 with a standard deviation of 13.2 follows the bars.
A normal curve is fixed by two numbers: its center and its standard deviation. Centered on the spread, 3.5, with the measured 13.2 points, it traces the bars closely. Hal Stern fitted the same model in 1991 and reported 13.86 points, drawn here as the dashed line; the scatter of NFL results has barely changed since. The steps below use 13.2, the figure measured on these 7,017 games.
04 · The area
The area above zero is the favorite's win probability: 60.5 percent.
The favorite wins when the margin ends above zero, so the win probability is the share of the curve's area right of zero. As a formula it is Φ(−spread ÷ σ), where Φ is the standard normal distribution function: Φ(3.5 ÷ 13.2) = Φ(0.265) = 60.5 percent. The remaining 39.5 percent, left of zero, belongs to the underdog. With Stern's 13.86 the same spread gives 60.0 percent.
05 · The price
Written as a price with no margin, 60.5 percent is −153; the other side is +153.
American prices express a probability on a base of 100. For a probability p of at least one half, the fair price is −100p ÷ (1 − p); below one half it is +100(1 − p) ÷ p. With p = 0.6046 that is −100 × 0.6046 ÷ 0.3954 = −152.9, rounded to −153, and the underdog's fair price is +152.9, or +153. Fair means the two probabilities add up to exactly 100 percent.
06 · The margin
Two prices of −110 imply 52.38 percent each: 104.76 together, 4.76 over 100.
Turn a price back into a probability and the arithmetic runs the other way: −110 implies 110 ÷ (110 + 100) = 52.38 percent. When both sides of a game are priced at −110, the two implied probabilities add up to 104.76 percent, although exactly one side can win. The 4.76 points above 100 are the margin built into the pair. Dividing each side by 104.76 percent removes it and returns 50.00 percent each.
07 · Against the record · the check
Favorites of 3.5 won 64.0 percent of 655 games; the curve says 60.5. At 3: 57.9 versus 59.0.
The check uses the same nflverse games. Favorites at a closing spread of exactly 3 won 648 of 1,119 games, 57.9 percent, against 59.0 from the curve. Favorites at 3.5 won 419 of 655, 64.0 percent, against 60.5. The standard errors are 1.5 and 1.9 points, so both gaps are within two of them, and all 1,774 games together come out at 60.1 percent against 59.5.
08 · Lumpy margins · what breaks without it
Margins are lumpy: 15.1 percent of games end by exactly 3; the smooth curve says 5.3.
Football scores in chunks of 3 and 7, so final margins pile up on a few numbers. Of these 7,017 games, 15.1 percent ended with one team exactly 3 points ahead and 9.0 percent exactly 7 ahead. The smooth curve, applied to every game's own spread, gives the band from 2.5 to 3.5 points either way only 5.3 percent, and the band around 7 only 4.9. It tells who wins well, and exact margins poorly.
09 · Any spread
The same curve prices any spread: −3 is 59.0 percent, −7 is 70.2, −14 is 85.6.
Moving the spread slides the whole curve, and the shaded area follows it. A 1-point favorite is at 53.0 percent, a 3-point favorite at 59.0, then 70.2 at 7 points, 77.6 at 10 and 85.6 at 14, with fair prices of −113, −144, −236, −346 and −592. The spread converter runs the same arithmetic for any spread and any standard deviation, and the table below sets every common spread against its real record.
The arithmetic
Three formulas, one worked example
With the spread written from the favorite's side as a negative number and σ the standard deviation of results around the spread:
P(favorite wins) = Φ(−spread ÷ σ) = Φ(3.5 ÷ 13.2) = 0.6046 fair price, p ≥ 0.5: −100·p ÷ (1 − p) = −100 × 0.6046 ÷ 0.3954 = −152.9 fair price, p < 0.5: +100·(1 − p) ÷ p = +100 × 0.6046 ÷ 0.3954 = +152.9 implied by a price of −110: 110 ÷ (110 + 100) = 52.38%, both sides 104.76%
Φ is computed in the page from the error function, and a separate check recomputes every probability here with Python's standard library. The spread converter takes any spread, standard deviation or pair of prices; the win probability calculator uses the pregame spread as its measure of team strength and then follows the game clock. The wider background, from key numbers to how the built-in margin works, is in the point spread guide.
The data · 18 spreads
Every common closing spread: the curve against the record
Each row is one closing spread that occurred in at least 100 of the 7,017 games from 2000–2025. "Curve" is Φ(spread ÷ 13.2); "fair price" is that probability as an American price with no margin; "won" is the share of those games the favorite won outright, with ties counted as not won.
| Spread | Games | Curve | Fair price | Won |
|---|---|---|---|---|
| −1 | 534 | 53.0% | −113 | 51.1% |
| −1.5 | 194 | 54.5% | −120 | 52.6% |
| −2 | 164 | 56.0% | −127 | 56.7% |
| −2.5 | 513 | 57.5% | −135 | 53.0% |
| −3 | 1,119 | 59.0% | −144 | 57.9% |
| −3.5 | 655 | 60.5% | −153 | 64.0% |
| −4 | 331 | 61.9% | −163 | 65.9% |
| −4.5 | 261 | 63.3% | −173 | 66.7% |
| −5 | 159 | 64.8% | −184 | 70.4% |
| −5.5 | 276 | 66.2% | −195 | 65.2% |
| −6 | 340 | 67.5% | −208 | 68.2% |
| −6.5 | 329 | 68.9% | −221 | 71.4% |
| −7 | 465 | 70.2% | −236 | 75.1% |
| −7.5 | 286 | 71.5% | −251 | 75.9% |
| −9 | 145 | 75.2% | −304 | 74.5% |
| −9.5 | 155 | 76.4% | −324 | 81.3% |
| −10 | 208 | 77.6% | −346 | 80.8% |
| −10.5 | 132 | 78.7% | −369 | 83.3% |
Single rows move a few points either way: with these sample sizes the standard error of a row runs from 1.5 to 3.9 points.
The data · final margins
How often each final margin happens, and what the smooth curve expects
"Real" is the share of all 7,017 games that ended with one team ahead by exactly that many points. "Curve" is the share the normal model assigns to the band half a point either side of that margin, on either side of zero, computed for every game from its own closing spread with σ = 13.2 and then averaged. The curve spreads its probability evenly; real margins collect on 3 and 7.
| Margin | Real | Curve |
|---|---|---|
| 0 | 0.2% | 2.7% |
| 1 | 4.1% | 5.4% |
| 2 | 4.0% | 5.4% |
| 3 | 15.1% | 5.3% |
| 4 | 4.9% | 5.2% |
| 5 | 3.6% | 5.1% |
| 6 | 6.0% | 5.0% |
| 7 | 9.0% | 4.9% |
| 8 | 3.7% | 4.7% |
| 9 | 1.6% | 4.5% |
| 10 | 5.5% | 4.3% |
| 11 | 2.4% | 4.1% |
| 12 | 1.7% | 3.9% |
| 13 | 2.8% | 3.7% |
| 14 | 4.9% | 3.5% |
| 15 | 1.5% | 3.2% |
| 16 | 2.1% | 3.0% |
| 17 | 3.5% | 2.8% |
| 18 | 2.4% | 2.6% |
| 19 | 1.1% | 2.4% |
| 20 | 2.2% | 2.2% |
| 21 | 2.7% | 2.0% |
Frequently asked questions
How is a point spread converted into a win probability?
The final margin is treated as normally distributed around the spread, with a standard deviation σ measured from past games. The favorite's win probability is Φ(−spread ÷ σ), where Φ is the standard normal distribution function. Across 7,017 NFL games from 2000 to 2025, σ is 13.2 points, so a spread of −3.5 gives Φ(0.265) = 60.5 percent for the favorite and 39.5 percent for the underdog.
Where does the standard deviation of 13.2 points come from?
It is measured, not assumed. Across 7,017 games in nflverse data the real margin landed on average +0.07 points from the closing spread, with a standard deviation of 13.2 points. Hal Stern fitted the same normal model in 1991 and reported 13.86 points, so the scatter of NFL results has barely changed. Using Stern's figure moves the probability for a 3.5-point favorite only from 60.5 to 60.0 percent.
What win probability does a 3-point or 7-point favorite have?
With σ = 13.2, a 1-point favorite is at 53.0 percent, a 3-point favorite at 59.0, a 7-point favorite at 70.2, a 10-point favorite at 77.6 and a 14-point favorite at 85.6. Against the record, favorites at exactly 3 won 648 of 1,119 games, 57.9 percent, and favorites at 3.5 won 419 of 655, 64.0 percent. Both gaps to the curve are within two standard errors.
What probability does an American price of −110 imply?
For a negative American price, the implied probability is the price without its sign divided by that number plus 100, so −110 implies 110 ÷ (110 + 100) = 52.38 percent. When both sides of a game carry −110, the two implied probabilities add up to 104.76 percent, although exactly one side can win. The 4.76 points above 100 are the margin built into the pair; dividing each side by 104.76 percent returns 50.00 percent each.
Does the normal curve predict exact final margins?
Poorly. Football scores come in chunks of 3 and 7, so final margins pile up on a few numbers. Of the 7,017 games, 15.1 percent ended with one team exactly 3 points ahead and 9.0 percent exactly 7 ahead, while the smooth curve gives those bands only 5.3 and 4.9 percent. The curve estimates who wins well, but it spreads probability evenly across margins that real games do not reach evenly.