While a mysterious concept to many, variance is a highly real and important part of the game. From upswings to downswings, it is a source of both excitement and utter frustration. In this article, we want to answer some of the most common questions surrounding the concept of variance and how it affects us. Chances are you’ll want to read on.
What is it?
In statistics, variance is defined as the expectation of the squared deviation of a random variable from its mean. For those interested, the mathematical formula for the variance of a random variable X with outcomes x_1, x_2, …, x_n, each with probabilities p_1, p_2, …, p_n, looks like this:
In plain english, however, variance is a measure of how far a set of numbers are spread out from their average.
The numbers can be anything, for example heights in a given group of people. Let’s say you have two groups of people, both with an average height of 180 cm. In one group, everyone have approximately equal height, so if you were to choose one at random, you’d be fairly sure that the person you selected will be roughly 180 cm. The other group, however, consist of only very short and very tall people. The average is the same, but if you were to choose one from this group at random, your results would fluctuate more than the first time. In this case, the variance for the second group is greater than for the first.
If you prefer to watch a video that explains variance, you can do so below:
Why is it important?
The above example can be translated into investing and sports trading. Substitute people’s heights with different profits, and you’ll see where we’re going. As an example, consider the following investment options and the probability of different outcomes:
As you can easily calculate, both options yield an expected value of 100. Variance, however, is much larger in the first case, making it a far riskier option.
High return and low risk is what you want from any investment instrument, whether it is in financial or sports markets. The lesson is simple: Variance = risk. You want to reduce it.
What does the number represent?
Variance is basically a measurement of how much you can expect any outcome to vary from what you expect it to be. In order to make any conclusions from the number itself, take the root of it and obtain the standard deviation:
Now, this new number can be used to determine the probability of profits staying within different ranges. Assuming the outcomes are normally distributed, there’s a 68 % chance that the value will stay within one standard deviation, 95 % chance it will stay within two, and 99.7 % within three:
In sports trading, the curve would represent different outcomes after a certain number of trades, and their probability. It would be symmetrical around its expected value(rather than 0 on the figure), illustrating that your profits are equally likely to end up above and below what is expected.
The point is once again that reducing variance means reducing risk. That’s valuable when sports trading.
Why is the number of trades important?
If you haven’t read our article on the law of large numbers, we recommend that you do so. Anyways, one of its major points is that the outcome of an individual bet is negligible in the long run. What matters is that the average of all results converge towards the expectation.
Consider a coin flip with 50 % probability of each outcome. You do a series of trials and register heads as a success(1) and tails as a failure(0). As you should, you expect the mean of your trials to equal 0.5 after a while, since the numbers of ones and zeros should be roughly the same. However, the question remains: How close to 0.5 can you expect it to be?
As it turns out, it depends on how many trials you conduct. After only a few coin flips, everything can happen. For example, there’s still a 6.25 % chance you’ll lose all of the first four, making the mean 0. If you increase the number of trials, however, the mean can be approximated as a normally distributed random variable. In other words, its probability distribution would look like the bell-shaped curve illustrated above (symmetrical around 0.5)
In fact, we can calculate how much the mean is expected to vary by utilizing the following formula:
You don’t need to know exactly what the above formula means, but note that the mean variance is inversely proportional to sample size. In other words, when sample size n increases, variance of the mean decreases.
Allow us to illustrate. The following three graphs each show the probability distribution of the mean after a series of coin flips, but with a different number of trials. The purple one corresponds with 10 trials, the red with 100 trials, and the blue with 1000:
The point here is that as you increase the number of coin flips, the mean is increasingly likely to be close to 0.5. If the number of trials is too small, variance will dominate results, making them unstable.
Therefore, it’s important for any value bettor to place a large amount of trades. If you combine valuable bets with volume, you’ll be profitable in the long run. If you combine valuable bets (+ Expected Value) with a large volume of trades you will significantly increase your chances of being profitable in the long run. As an example, this is the probability distribution of different profits after betting on 1000 coin flips with a 5 % edge (odds 2.10):
How to reduce it?
Hopefully, you’ve gained some insight on what variance is and why you want to minimize it. As to how you might do that, we’ve made an article on the subject already. Check it out here.