What Do You Expect the Mean to be equal to?
If you’ve ever done some fundamental statistics, you have most definitely heard of the mean, median and mode. Each of these statistics are all great measures of central tendency (central/typical value)when implemented under the right circumstances, but in this blog I’ll just be focusing on the mean.
Now just to keep you interested through all the boring stuff, the goal of this post is to show you one potential way of using the “mean” whenever you want to decide the best action to take.
Now the mean is the also known as the average. Like said above, it is a form of measure for the central tendency, but more specifically it is where the point is the very center of your data. Since it is a form of central tendency, it can also be the median or mode, but the mean is special in that it may or may not be in the data set. Whether we are looking at the heights of a sample of 10 people in a room or everyone in the world, there will be a mean that would measure what height is at the center.
The most common way to calculate the mean is by taking the sum of all your data points and divide by the number of elements. If given a sample set of numbers like (1,1,3,4,6,) you would simply add up the outcomes and divide by the number of elements shown below.
The result we get is 3 so we say the sample mean of the set (1,1,3,4,6) is 3. Simple. That is the most common way to look at it, but what if we went about it another way and by using the value of the outcome and the relative frequency of the outcome.
Surprise, surprise it’s the same! We can simply even see that we can do this just by splitting the fraction up into 5 parts to be 1/5, 1/5, 3/5, 4/5, and 6/5. What we are doing here is actually very similar to finding the sample expected value, which is the defined by the sum of the expected outcome multiplied by the probability of it occurring.
Expected Value
Finding the expected value of that sample doesn’t make much sense, but we can apply this sample to a very simple dice game. In this game each outcome of the sample is a dice roll, but if you roll a number less than or equal to 3 you lose that amount, and if you roll a number greater than 3 then you gain that amount. The expected value with the relative frequency is as follows:
This results means if we play this game can expect to win $2, or to be even more correct, if we play this game a number of times, we would expect our winnings be $2. If we kept on playing this game a number of times, we would say on average, we would win $2. Now if I were to ask if you would play this simple game given all these rules, would you play it? For me, despite the chance of losing $3, there’s a higher chance of me gaining more so I would. This is the most basic example of how I would use the “mean” to make a decision of to play or not to play.
Simple Poker Example
We can apply this to popular card games like Texas Hold ’em. For those unfamiliar with poker the goal is to make the best 5 card combination with the 2 cards you have and the 5 cards in the community pool. For educational purposes let’s just do a simple look at a Texas Hold ’em game where we know every card community card in play, only narrow the conditions to whoever has the highest 2 card combination (meaning pair or single card wins), and can either bet $100 or fold. In your hand are Ace&10 and your current highest 2-combination is Pair 10s. The other player decided to bet $100, so now you will either bet or fold, and the results of betting would be win, lose, or tie. From your own perspective (Player 1), the only combinations that could beat you would be any pair higher than your pair like Queens, Jacks, else you would tie if they had a 10, and win any other time. But how do we think about the situation and try to strategize to make the right choice?
Since we know a deck of cards has 4 of each rank of cards (Ace to King), and a total of 52 cards, we can find the probability of the pair Queens and Jacks occurring, where we lose, probability of them having another 10, where we tie, and the remaining is we win. We count the number of “remaining” cards that player 2 could have and divide it by the number of cards left (not including the 2 in their hands since those also “remain”) to determine the probability of a result. That means 3 Queens + 3 Jacks gives a total of 6 losing-cards, 2-10s gives a total of 2 tie-cards, and then the rest of the 37 cards out of 45 we win. Recall the expected value formula and apply:
Now you would interpret this as the expected value of betting is a gain of $68.89. If you solely used this method of deciding to bet, then you would bet.
Conclusion
Overall, to answer the question of what do you expect the mean to be equal to, the answer is the expected value! One important piece of information I did not mention above is the fact that this formula only applies in the discrete case, which means we look at outcomes that are finite, like winning/losing, heads/tails, number of heads or tails, etc. The other case would be the continuous, but in essence it’s the same thought process of outcome times the probability of the outcome. I hope you’ve got a good fundamental look at the expected value and what it equates to. Maybe you could potentially use this knew found knowledge to try to beat others in games of odds. Maybe this even got you questioning if the actual relationship between the mean and expected value are the same.
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