Mean and Standard Deviation in Gaming
In the previous post, we talked about the long term expectation of games derived from the probabilities of outcomes in casino games.
How about measuring results in the short-term such as when the results of shoes or player activity appear inconsistent with the expected values? Let’s face it, they usually do!
For this, it is useful to apply the principles of the central limit theorem, in particular, the concepts of mean and standard deviation.
But first…
Central Limit Theorem
The central limit theorem explains that the outcome of repeated experiments will follow a bell-shaped pattern. In terms of casino games, we can translate this as meaning that the outcomes of play will follow a certain pattern of winnings and losses.
(Source: http://schools-wikipedia.org/)
This is a bell-curve, so named due to its shape.
The bell-curve’s centre is known as the mean or average (µ).
Notice that the bell-curve is divided into 6 parts from -3σ at the extreme left to 3σ at the extreme right. The σ symbol means standard deviation. Thus, the bell-curve is divided into units of 6 standard deviations.
The central limit theorem explains that for an ideally NORMAL situation, approximately 68.2% (34.1% on either side) of all outcomes will occur within the -1σ to 1σ region, with approximately 27.2% of all outcomes occurring between the -1σ and -2σ regions (13.6%) and the 1σ and 2σ regions (13.6% as well). The remaining 5% thereabouts is divided into two and occur between the -2σ and -3σ and 2σ and 3σ regions.
From the bell-curve, assuming you believe it, we can assume that the majority of outcomes will occur within the -1σ to the 1σ region.
We can translate this into casino gaming, where we can infer that this bell-curve would also apply to the results of games.
So, how would we know?
Mean
The most important part to this question lies in determining the mean or average of the data. Let’s take this player report as a sample:
| Player | Win/(Loss) |
| Player 1 | $100 |
| Player 2 | ($500) |
| Player 3 | ($300) |
| Player 4 | $200 |
| Player 5 | ($100) |
The mean of the win/loss for all 5 players is = $100 + (-$500) + (-$300) + $200 + (-$100) / 5 = -$600 / 5 = -$120
This means that on average, these players lost $120 each. So, -$120 becomes our middle point in our bell-curve.
Standard Deviation
Now, assuming that an average $120 loss is also true for all players, we can now find the standard deviation. The calculation is as follows:
| Player | Win/(Loss) | Win/ (Loss) – Mean | (Win/ (Loss) – Mean)2 | Variance | Standard Deviation = √Variance |
| Player 1 | $100 | $100 – ($120) = $220 | $220 x $220 = $48,400 | $48,400 + $144,400 + $32,400 + $102,400 + $400 / (5-1) =$328,000 / 4 = $82,000 | Square root ($82,000) = $286.3564 |
| Player 2 | ($500) | ($500) – ($120) = -$380 | -$380 x -$380 = $144,400 | ||
| Player 3 | ($300) | ($300) – ($120) = – $180 | -$180 x -$180 = $32,400 | ||
| Player 4 | $200 | $200 – ($120) = $320 | -$320 x -$320 = $102,400 | ||
| Player 5 | ($100) | ($100) – ($120) = $20 | $20 x $20 = $400 | ||
| Mean | ($120) |
With a standard deviation of $286.3564 or $286.36, we can apply the probabilities in the bell-curve to the results of games in the short-term.
| Probability | 2.3% | 13.6% | 34.1% | 50% | 34.1% | 13.6% | 2.3% |
| Standard Deviations | -3σ | -2σ | -1σ | Mean | 1σ | 2σ | 3σ |
| Result | -3 x $286.36 + (-$120) =-$979.08 | -2 x $286.36 + (-$120) =-$692.72 | -1 x $286.36 + (-$120)=-$406.36 | -$120 | 1 x $286.36 + (-$120) = $166.36 | 2 x $286.36 + (-$120) = $452.72 | 3 x $286.36 + (-$120) = $739.08 |
What this now means, is that based on our data, the chance of a player losing $120 is 50% and most players will win between $166.36 or lose $406.36. Wins or losses outside of this range are less likely and thus, more suspect.
This being the case, we actually now have Player 4 winning $200, which is more than the expected win of $166.36 – we should probably have a look at his play.
How we will do that will be discussed in a later post.
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