Welcome to Part 9 of our series on profitable sports betting, where we delve into the critical concept of variance and its profound impact on betting outcomes. By understanding variance, bettors can better manage their expectations and strategies over both the short and long term.
We've come a long way since we started!
In sports betting, variance refers to the measure of how results differ from expectations. It quantifies the spread of your betting results around the expected value (EV). High variance means that your betting results can swing dramatically in either direction from what you expect, while low variance suggests that results will be more consistent and close to the expected value.
Variance significantly impacts bet results by introducing unpredictability into the equation. Even with a positive expected value, where the long-term prospects are profitable, short-term results can vary widely. This variability can make it challenging to predict short-term outcomes accurately, even if the long-term outlook is positive.
To illustrate how variance works, consider a series of coin flips where you bet on heads:
In the short run, even positive-EV bets can result in clusters of losses or wins due to variance. This can give bettors misleading impressions:
Over the long run, betting results tend to converge on the total amount of Closing Line Value (CLV) generated. CLV is a key indicator of a bet’s value, reflecting the quality of odds at the time of bet placement compared to the odds at game start. The more bets placed, the more likely the actual win rate will reflect the EV derived from the CLV, demonstrating the true efficacy of a betting strategy.
Understanding variance helps in setting realistic profitability targets.
The world's most successful sports bettors achieve a long-term win rate of between 55% and 60%, with an ROI of between 3% and 7%.
These figures are grounded in vast amounts of data and betting experience, illustrating that even the best strategies do not guarantee overwhelmingly high returns due to the inherent variance in sports betting.
One of the most effective strategies to mitigate the impact of variance in sports betting is to increase your betting volume. By placing a higher number of bets, you leverage the law of large numbers, which helps to stabilize your results around the expected outcomes derived from your betting strategy's true probability and Closing Line Value (CLV).
To understand how betting volume counteracts variance, let’s look at a practical example.
You make a bet at odds of -125 (55.56% chance to win) that you have determined has a positive expected value. Thus, we’d expect this bet to win 55.56% percent of the time.
But, as we now know, variance can impact the results over the short run.Let’s explore the expected win rate variance when making this bet 10, 100, 1000 and 100000 times.
Variance for a single bet in a binary outcome (win or lose) scenario can be calculated using the formula:
To find the expected range of outcomes from the true probability, we have to calculate the standard deviation of the variance:
As the number of bets increases, the law of large numbers suggests that the proportion of winning bets should converge to the expected win rate (55.56%). We can show how the variance and standard deviation evolve as we scale the number of bets.
The total variance for a number of bets (N) is scaled linearly:
The standard deviation over multiple bets provides insight into how spread out the win rate might be around the expected 55.56%. It is calculated as:
Let's calculate these for 10, 100, 1000, and 100000 bets:
Meaning, even though you expect to win between 5 and 6 of these 10 bets (55%), 2 of those bets (rounding up from 1.57) could go either way due to variance. Said differently, you’re likely going to have a result between 3 and 7 wins out of ten bets made.
Now take that to real life - If you bet $100 per wager, this means that as much as $200 of the $1000 wagered is left up to random chance. If you only won 3 out of 10 bets where you expected a win rate of 55%, you might be inclined to think, “Hey, this isn’t working!” and abandon what’s actually a winning strategy. On the other hand, if you win 7 out of 10 bets, you might think, “Hey, this is amazing!” and over-bet your bankroll on the next one.
In reality, neither of these conclusions are true. With only 10 occurrences, 20% of the results are left up to random chance.
You can’t possibly draw a conclusion about your strategy when a high percentage of results are impacted by variance.
If we increase the number of bets to 100 (a 10x) increase, note how the variance has only increased from 2 bets to 5 bets (rounding up from 4.97). If I make this bet 100 times, 5 of those bets could go either way due to variance.
Now instead of 20% of our results being random, we’re only dealing with 5% (5 of 100) of them being random. For 100 bets with a win rate of 55%, you would expect to win 55 bets. This variance calculation says that you’ll likely win somewhere between 50 bets (55 - 4.97) or 60 bets (55 + 4.97)
In terms of dollar results, using the same $100 per bet sizing, $500 out of $10,000 in total wagers is up for grabs. Much better, but still not great.
If you increase the number of bets to 1000, the expected variance has only increased from 5 bets to 16 bets (rounding up from 15.72). If you make this bet 100 times, 15 of those bets could go either way due to variance.
Instead of 5% of our results being random, you’re only dealing with 1.5% (15 of 1000) of them being random. For 1000 bets with a win rate of 55%, you would expect to win 550 bets. This variance calculation says that you’ll likely win somewhere between 534 bets (550 - 15.72) or 565 bets (550 + 15.72)
As a result of increasing our bet volume, only $1500 out of $150,000 in total wagers is up for grabs. Now you’re getting somewhere: you have reduced our variance to just 1.5% of total occurrences, giving you a much better read on the validity of your approach.
For maximum effect, let’s take this scenario and play it out over 100,000 bets.
Even though we’ve increased the number of bets we’ve made by 100x from the 1000 bet level, the variance of wins/losses has only increased to 157 bets. That’s just .16% of all bets made.
If you make 100,000 bets with an expected win rate of 55%, you’d expect to win 55,000 bets and lose 45,000 bets. Since we’ve crushed variance with volume, that win rate could fluctuate between 54,843 and 55,157.
At $100 a bet, you’re talking about dollar variance of $15,700 on total wagers of $10,000,000.
This graph shows the relationship between volume and variance, between 10 and 1,000,000 bets.
Want to reduce the impact of "luck” on your betting results? Get more bets down.
Before we move into a discussion of Closing Line Value and volume, it's important to note that the above calculation are based on one standard deviation (aka "one sigma variance").
One sigma of variance refers to the expected dispersion of results around the mean (average), denoted by the Greek letter σ (sigma). One sigma (±1σ) from the mean encompasses approximately 68% of the data points in any distribution. This means that about 68% of the time you're going to get the ranges calculated above, for each volume of bets.
Two sigma variance refers to a range within two standard deviations (2σ) of the mean in a normal distribution. Two sigma variance means that about 95% of the data points in a normally distributed dataset are expected to fall within that range. Multiple 1σ x 2, and you'll get an expected range of wins that will occur 95% of the time.
Three sigma covers approximately 99.7% of all results, or 3 x 1σ. Graphed, the distribution of betting results would look like this:
To wrap up this math-heavy discussion on standard deviation, what this means is that to understand where your betting results will fall 99.7% of the time for bets that average a 55% win rate, you take the standard deviation we calculated for 1σ (one standard deviation), which was 15.72 bets, and multiply it by 3 (let's call that 48 bets).
99.7% of the time, your expected range of wins on 1,000 bets placed that have a 55% win rate (-125 odds) will be between 502 wins and 598 wins. At 502 wins you're barely breaking even. At 598 wins you feel like a genius. The truth is that you're going to get a number in this range 99.7% of the time, and it's dumb luck where it lands. That's Variance 101.
Accelerating Toward True CLV with Higher Volume
We just demonstrated how, in this simple but very relevant example, increasing volume from 10 bets to 100,000 bets reduced variance from 20% of results to just .16% of results using one sigma of standard deviation.
Hopefully you now understand a few bedrock truths about bet volume. As you place more bets:
Based on our experience working with our users at Optimal, we’ve noted that users start to see variance really smooth out at around 2000 total wagers made. That might seem like a lot, but you can get 2000 bets down in around three months if you’re making between 20-25 bets a day.
And even though we used a $100 bet size as an example, the variance math holds at any wager size. Whether you’re a $5 bettor or a $1000 bettor, more bets equals more consistent profits, assuming the bets you’re making all carry positive expected value.
The principle behind this approach is straightforward: the greater the number of bets, the quicker you can reach a volume where the results reflect the true probabilities.
This concept is similar to the coin flip example we discussed earlier. In scenarios with only a handful of bets, the results can vary wildly from expected outcomes due to high variance. However, as the number of bets increases into the hundreds or thousands, the average of the results tends to converge closely to the expected value, effectively reducing the impact of variance.
This convergence is crucial because it demonstrates the effectiveness of your betting strategy over time. With enough volume, the true skill or edge in your strategy is revealed, distinguishing real gains from those influenced by short-term luck or variance.
For most bettors, manually maintaining the discipline and oversight required to manage 20-50 bets per day would be overwhelming, if not impossible. This is where a sophisticated tool like Optimal becomes invaluable. Optimal automates the process of identifying valuable betting opportunities and calculating the ideal bet size relative to the expected value of each bet, based on meticulously designed algorithms and betting models.
To truly counteract the effects of variance and confirm the reliability of your betting strategy, a high volume of bets is essential. Utilizing a tool like Optimal to manage and execute a large number of bets daily allows you to reach the statistical significance needed to validate your strategy faster.
This high volume approach ensures that you move quickly towards realizing your strategy’s true CLV, paving the way for sustainable long-term profitability in your sports betting.
