Variance, correlation and sample size
Even a real edge is buried in noise. Measure the variance of a single bet, the standard error of a track record and the number of bets needed to see an edge, and learn why picks from the same match are not independent evidence.
Learning objectives
- Compute the mean, variance and standard deviation of the profit on a single bet.
- Compute the standard error of ROI and a Wilson interval for a hit rate, and interpret both.
- Estimate how many bets are needed before a given edge is distinguishable from zero.
- Quantify the correlation between two same-match picks from a score grid and explain its effect on risk.
One bet is a lopsided coin
Profit = o − 1 with probability p, −1 otherwise; Var = p(o − 1)² + (1 − p) − EV²Take a hypothetical evens price (2.00) where the true win chance is 52%. The mean profit is +0.04 per unit, but the standard deviation is 0.999: the noise on one bet is about 25 times the edge. At 2.30 with p = 48% the mean is +0.104 and the standard deviation 1.149. At a long price of 10.0 with p = 10.4% the mean is again +0.04 but the standard deviation jumps to 3.053.
Same edge, three times the noise. Because the number of bets needed grows with the square of the standard deviation, tripling the noise means roughly nine times as many bets before the edge shows: (3.053 / 0.999)² ≈ 9.3. Long-odds strategies are not riskier in expectation; they are slower to verify and far easier to fool yourself with.
Standard errors and how long it takes
SE(ROI) = sd / √n Bets needed ≈ (z × sd / edge)²| Bets | Standard error of ROI | Approximate 95% range of observed ROI |
|---|---|---|
| 100 | 9.99 points | −15.6% to +23.6% |
| 500 | 4.47 points | −4.8% to +12.8% |
| 1,000 | 3.16 points | −2.2% to +10.2% |
| 2,398 | 2.04 points | 0.0% to +8.0% |
Detecting a 4% edge at evens takes about 2,398 independent bets; a 2% edge takes about 9,601, and a 4% edge at odds of 10.0 about 22,374. Even with a genuine +4% edge at evens, the chance of being behind after 100 bets is 30.8%, and after 1,000 bets still 9.7%.
These figures assume independent bets at a single price and a single pre-registered strategy. Real records mix odds, which changes the standard deviation bet by bet, and are often the survivors of several strategies tried on the same data, which makes any one good result less surprising than its standard error suggests.
Hit rates and the Wilson interval
Wilson centre = (r + z²/2n) / (1 + z²/n); half-width = z / (1 + z²/n) × √(r(1 − r)/n + z²/4n²)Fifty-five wins from 100 picks at evens is a +10% ROI. The Wilson 95% interval for the hit rate is 45.2% to 64.4%, which comfortably contains the 50% break-even bar. A bettor with no edge at all (p = 50%) reaches 55 or more wins in 18.4% of 100-bet runs. The same 55% rate over 1,000 picks gives 51.9% to 58.1%, which does exclude 50%.
Correlation: same-match picks are not independent
Take the hypothetical score grid from lesson 1: independent Poisson goals with λ = 1.6 and μ = 1.0 . A home win has probability 51.34% and Over 2.5 goals 48.16%. If they were independent, both would land 24.73% of the time; on the grid they land together 29.95% of the time, a correlation of about 0.21. Home wins tend to come with more goals.
| Pair of picks | Correlation |
|---|---|
| Home win and Over 2.5 | +0.21 |
| BTTS Yes and Over 2.5 | +0.57 |
| Home win and BTTS Yes | −0.16 |
Stake 1 unit on the home win at a hypothetical 2.00 and 1 unit on Over 2.5 at 2.10. Under the grid, both lose 30.45% of the time, against 25.23% if the picks were independent, and the standard deviation of the combined profit is 1.594 rather than 1.449. Positive correlation adds risk and means two picks carry less than two picks' worth of evidence.
Negative correlation works the other way: a home win and BTTS Yes (−0.16) partly offset each other, so the combined result is less variable. That is a statement about risk only. Pairing picks to reduce variance does not create expected value; if each pick has negative EV, the pair does too.
Worked example
Worked example · Is 55 wins from 100 evidence of an edge?
A simulated record: 100 picks at hypothetical evens (2.00), 55 wins.
- Observed ROI
(55 × 2.00 − 100) / 100+10.0% - Wilson 95% interval for the hit rate
Wilson interval for 55 of 10045.2% to 64.4% - Compare with break-even
Break-even at evens = 1/2.00 = 50%, inside the intervalNot distinguishable from no edge - How often does luck do this?
P(≥ 55 wins of 100 at p = 0.5) = P(55+ wins | p = 50%)18.4% - Noise if the edge were real (+4%)
sd at p = 0.52, o = 2.00 is 0.999; SE over 100 bets = 0.999 / √100≈ 10.0 points - Bets needed to detect +4%
(1.96 × 0.9992 / 0.04)²≈ 2,398 bets - Correlation check
If the picks came in same-match pairs such as home win + Over 2.5 (correlation ≈ 0.21)Effective sample smaller than 100
A +10% ROI over 100 picks is well within the range of pure luck: a zero-edge bettor does at least this well almost one time in five. The record is consistent with a modest edge, no edge or a small negative one. The honest summary is “not enough evidence yet”, with roughly 2,400 independent picks needed to resolve a 4% edge at evens.
Practice · Judge a hot streak
A simulated record shows 60 wins from 100 independent picks at evens. Compute the ROI, the Wilson 95% interval for the hit rate and the probability that a zero-edge bettor wins 60 or more. Then repeat for 30 wins from 50 picks. What changes, and what extra information would you want before calling either record an edge?
Show what you should find
60/100: ROI +20%, Wilson interval 50.2% to 69.1%, zero-edge tail probability 2.84%. The interval only just excludes 50%. 30/50 has the same 60% hit rate but a Wilson interval of 46.2% to 72.4% and a tail probability of 10.1%, so it is much weaker. Before calling either an edge, ask how many strategies or filters were tried (many tries make a 2.84% tail unremarkable), whether picks were correlated within matches, and whether the prices were executable.
Common mistakes
- ✗ Judging a strategy on 50–100 bets.At evens the standard error of ROI over 100 bets is about 10 points. Use Wilson intervals and expect to need thousands of independent bets to resolve small edges.
- ✗ Counting correlated same-match picks as independent evidence.Home win and Over 2.5 correlate at about 0.21 on a typical grid; BTTS and Over 2.5 at about 0.57. Count matches as well as picks.
- ✗ Using the simple normal interval for small samples.The Wilson interval keeps sensible bounds for small n and extreme hit rates.
- ✗ Assuming long odds reach significance as fast as short odds.Per-bet variance grows with the odds: a 4% edge at 10.0 needs about 22,374 bets against about 2,398 at evens.
When it fails
Supporting research
- Winkelmann et al. (2024): Are Betting Markets Inefficient? Evidence From Simulations and Real DataJournal of Sports Economics · Peer-reviewed empiricalSignificant single-season “inefficiencies” are likely even in an efficient market, and the apparent ones in real football data did not persist across seasons or leagues.
- Clegg & Cartlidge (2025): Not feeling the buzz: Correction study of mispricing and inefficiency in online sportsbooksInternational Journal of Forecasting · Peer-reviewed replicationA published profit depended largely on one outlier bet at erroneously long odds, showing how fragile small-sample results are.
- Maher (1982): Modelling association football scoresStatistica Neerlandica · Peer-reviewed empiricalTeam-strength Poisson models give a reasonable description of football scores, which justifies using a score grid to estimate how same-match picks co-move.
Check your understanding
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