Sample size in betting is the number of bets behind a result, and it decides how much of that result could be luck. With level stakes, luck shrinks only with the square root of the number of bets. At a price of $3.00, 200 bets leave about 10 percentage points of noise either side of a punter's true yield, and it takes 800 bets to halve that.
So the useful question about any profit, yours or a tipster's, is how far above zero it sits compared with the swing luck alone produces at those prices. The same arithmetic applies to racing, sport and any betting strategy you want to test, and every figure used to show it is illustrative.
How much luck moves a betting result
The standard error is the typical gap luck puts between a result and the punter's true average. For a run of level-stake bets, a close approximation is:
standard error of yield = square root of (average decimal odds - 1) / square root of the number of bets
It comes from a single bet. A $1 bet at decimal odds o either wins o - 1 or loses 1, and if the price is fair, the standard deviation of that result is the square root of (o - 1). Averaging n bets divides it by the square root of n.
Each bet adds variance, the square of that standard deviation, in proportion to its price minus 1, so with level stakes the plain average of your decimal prices is all the formula needs. With stakes that vary, it does not hold, so restate a record in level units first.
One standard error, in percentage points of yield, at illustrative average prices:
| Average price | 100 bets | 500 bets | 2,000 bets |
|---|---|---|---|
| $1.90 | 9.5 | 4.2 | 2.1 |
| $3.00 | 14.1 | 6.3 | 3.2 |
| $6.00 | 22.4 | 10.0 | 5.0 |
| $11.00 | 31.6 | 14.1 | 7.1 |
About two results in three land within one standard error of the true yield, and about 19 in 20 within two. That rule comes from the normal curve, and it is rough for long prices over small samples, where two or three winners decide everything. How wide the swings get over a long run, and what they do to a bankroll, is covered in variance in betting.
Worked example: 200 bets at $3.00
Example: Illustrative figures: 200 bets of $10, every one at $3.00, by a punter with no edge, so each bet's true chance is 1 in 3.
| Line | Working | Result |
|---|---|---|
| Turnover | 200 x $10 | $2,000 |
| Winners expected | 200 / 3 | 66.7 |
| Spread of the winner count | Square root of (200 x 1/3 x 2/3) | 6.67 |
| Profit swing per extra winner | A $20 win instead of a $10 loss | $30 |
| Standard error of profit | 6.67 x $30 | $200 |
| Standard error of yield | $200 / $2,000, or square root of 2 / square root of 200 | 10 points |
| Range luck allows, winners | 66.7 minus and plus 2 x 6.67 | About 53 to 80 |
| Range luck allows, profit | 2 x $200 either side of zero | About -$400 to +$400 |
| Range luck allows, yield | 2 x 10 points either side of zero | About -20% to +20% |
A punter with no edge at all, betting 200 times at $3.00, finishes somewhere between about $400 down and $400 up roughly 19 times in 20. Counting every way the 200 bets can land (the exact binomial count) puts 96% of results between 53 and 80 winners. A record inside that band says nothing about skill either way.
Say this punter finishes with 77 winners: $310 profit, a 15.5% yield, 1.55 standard errors above zero. Counted exactly, a punter with no edge gets 77 or more winners about 7% of the time, roughly 1 time in 14. That is a result worth tracking, not evidence of an edge.
At real bookmaker prices the band also sits lower. A punter with no skill pays the margin on every bet, so the middle of the range is below zero by about the size of that margin, and the whole band moves down with it.
Statistical significance in plain words
A result is significant when luck alone would rarely produce one as good. The working is one division, z = yield / standard error, and the normal curve then says how often a punter with no edge does at least that well:
| Standard errors above zero (z) | Luck alone does at least this well | Plain reading |
|---|---|---|
| 1 | About 1 time in 6 | Ordinary luck |
| 2 | About 1 time in 44 | Worth a closer look, not proof |
| 3 | About 1 time in 740 | Hard to put down to luck, if it was the only record tested |
The usual convention draws the line at 1 time in 20 on a two-sided test, which is 1.96 standard errors either way, so about 2 is a fair working rule.
Two limits come with it. Significance describes bets already placed, not the next ones. And a result that falls short of the line is not proof of no edge: it means the sample cannot tell yet.
How many bets to know if you are profitable
Turn the standard error around and it gives the bets needed. An edge, written as a decimal, at average decimal odds reaches two standard errors above zero, on average, after about:
bets needed = 4 x (odds - 1) / (edge x edge)
At $3.00 with a 5% edge that is 4 x 2 / (0.05 x 0.05) = 3,200 bets. Rounded, at illustrative edges:
| Average price | 2% edge | 5% edge | 10% edge |
|---|---|---|---|
| $1.90 | 9,000 | 1,440 | 360 |
| $3.00 | 20,000 | 3,200 | 800 |
| $6.00 | 50,000 | 8,000 | 2,000 |
| $11.00 | 100,000 | 16,000 | 4,000 |
These are the counts at which a real edge of that size would, on average, just reach the line, so half the time it would still fall short. To clear it about 4 times in 5, roughly double them. A 2% edge at $6.00 needs about 50,000 bets on this rule: at, say, 50 bets a week, that is about 19 years.
Luck vs skill as the bets add up
Skill and luck grow at different speeds. With level $10 stakes at $3.00 and a true 5% edge (illustrative), the edge's expected profit rises in step with the bets, while one standard error of luck rises only with their square root:
| Bets | Expected profit from the edge | One standard error of luck | Edge divided by luck |
|---|---|---|---|
| 100 | $50 | $141 | 0.35 |
| 1,000 | $500 | $447 | 1.12 |
| 10,000 | $5,000 | $1,414 | 3.54 |
The working: expected profit = 0.05 x bets x $10, and one standard error = square root of (bets x 2) x $10. Luck swamps the first hundred bets, and the edge only pulls clear in the thousands, which is why a short record, good or bad, says so little.
Over 200 bets the same edge expects a 5% yield against a standard error of 10 points, only 0.5 standard errors above zero, so the normal curve puts the result below zero about 3 times in 10 (an exact count gives 30%). The law of large numbers is why the average settles in the end, and it never makes a losing run due to reverse.
Why closing line value needs fewer bets
Closing line value (CLV) compares your price with the final price before the start: CLV = your odds / closing odds - 1. Its advantage is noise. At $3.00 a single bet's result swings between losing the whole stake and winning twice it, while its CLV is the gap between two prices for the same outcome, which is usually far smaller. Less noise per bet means fewer bets to reach the same confidence.
Example: Illustrative: the same 200 bets at $3.00 average +4% CLV against a closing price with the margin taken out, and the bet-to-bet spread of those CLV figures is 12 percentage points. The standard error of the average is 12 / square root of 200 = 0.85 points, so +4% sits 4.7 standard errors above zero. By profit alone, a 4% edge at $3.00 needs about 4 x 2 / (0.04 x 0.04) = 5,000 bets to reach two standard errors. By CLV it needs about 4 x 0.12 x 0.12 / (0.04 x 0.04) = 36.
Measure the spread of your own CLV figures rather than borrowing the 12 points used here. The shortcut rests on one assumption: that the margin-free close sits near the true chance, in which case your average CLV stands in for the yield you can expect.
That assumption holds less well in thin markets, and for racing the exchange starting price can serve as the close, as CLV for horse racing explains. CLV also says nothing about how long a bookmaker will keep taking your bets.
Applying sample size to a tipster record
The same arithmetic tests any record you are shown. Take an illustrative tipster with 150 tips at an average of $5.00 and a 30% yield at level stakes. The standard error is the square root of 4 / the square root of 150 = 16.3 points, so 30% sits 1.84 standard errors above zero. By the normal-curve rule, a tipster with no edge does at least that well about 1 time in 30; counted exactly with every tip at $5.00, it is nearer 1 time in 22, because the curve is rough at long prices.
That sounds rare until you ask how the record was found. If 30 tipsters with no edge each publish 150 tips like these, the chance that at least one shows a record this good is 1 - (29 / 30) to the power of 30 = about 64% on the normal-curve figure, and 1 - (21 / 22) to the power of 30 = about 75% on the exact count. The record at the top of a leaderboard is the luckiest of many, which is survivorship bias at work.
Before a tipster record counts as evidence, restate it at one stake and at prices you could actually have taken, as how to check a tipster record explains step by step. Then:
- Count the settled tips and work out the standard error at their average price.
- Ask how many records you were choosing from, because the best of many is far likelier to be luck than one picked before the tips were posted.
- Ask for the CLV of the tips, which needs far fewer of them to show anything.
Risk: Betting involves risk. Clearing two standard errors makes luck less likely, not impossible, and an edge found in past bets can fade. Past results are no guarantee of future results. See responsible gambling for limits and support.
Mistakes that make a small sample look convincing
| Mistake | What it costs |
|---|---|
| Judging the record at its high point | Checking after every bet and declaring a method proven the moment the record looks good lets ordinary luck pass for a method far more often than 1 time in 20. Stopping betting is always fine: the mistake is reading the high point as proof |
| Testing many systems and keeping the best | Of 20 systems with no edge, expect one to pass a 1 in 20 test by luck alone |
| Judging by strike rate | An illustrative 60% winners at $1.50 is a yield of 0.6 x 1.50 - 1 = -10%, as strike rate betting shows |
| Counting linked bets as separate | Bets on the same runner or side, such as a win and a place bet on one horse or a head to head and a line bet on one team, and multis that share legs, tend to win and lose together, so the real sample is smaller than the count |
| Trusting a hot streak | Extreme runs tend to be followed by ordinary ones, the pattern called regression to the mean |
| Ignoring the prices | 100 bets at $11.00 carry more than three times the luck of 100 bets at $1.90 |