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Background: I have multiple sets of team's little league soccer scores made up only of whole numbers [0-12 inclusive], eg:

A: 1, 2, 4, 7
B: 0, 0, 0, 1, 1, 1, 4, 7
C: 0, 1, 2, 2, 2, 3, 4

Due to a positive skew, I would like to improve upon using the mean for prediction of next score.

  • At about 4 or more pieces of data for a set, I can make a fairly accurate prediction using the mean average of the set, [diminishing returns for reducing error starts very rapidly]. But is definitely open to improvement at a fundamental level.
  • Most sets will be made up of 4-60 numbers and have a very nice bell shaped distribution for 90% of the data from 0-5 inclusive but have a positive skew for the remaining 10% from 6+. The data points from 6+ are not outliers necessarily but they most certainly do interfere with the mean for future predictions [consistently over predicting on average by about 5-7%].

Using the mean of previous scores as a rudimentary prediction. "Next" is the next score achieved by the team in the set.

    Mean - Next - Error
A:  3.50    4     0.50
B:  1.75    2     0.25
C:  2.00    0     2.00

Issue: How does one test the error of a probability based prediction without a specific number to compare against like a mean / median / mode?

Probabilities are rounded [may not total to 100].

     0   1   2   3   4   5   6   7 -  Next - Error - Intuition / aim?
A:   0  25  25   0  25   0   0  25      4      ?       3.0
B:  38  38   0   0  13   0   0  13      2      ?      ~0.8 ??
C:  14  14  43  14  14   0   0   0      0      ?       2.0

Obviously predicting probabilities like the above even with slightly larger sets that form smoother distributions is meant for a very different kettle of fish predictions eg Set C: ~71% chance of scoring less than 2.5 goals.

  • Ideally a method to output a simple as possible [tacky is great] not necessarily best practise, it could be an estimated prediction of what a team will score higher than 50% of the time and lower than 50% of the time so that an error can be produced similar to the mean prediction and hence compared, or...
  • Some other way without producing an exact prediction for the probabilities for each set but would allow direct comparison of error with the mean prediction.
  • Most of the sets have 22-42 scores but the solution needs to work effectively even with a very limited number of data points [eg 4-9] if using the probabilities to infer some sort of approximate prediction.

Already attempted / notes:

  • Median / mode are not effective alone. Using mean but: Trimming / omitting semi-outliers [6+, 7+, 8+] has not reduced error in a useful manner either.
  • Combining the opposition's defense into the fold is not an issue and does not change the problem at hand, it just means instead of measuring the error of the teams' attack it will be goal difference instead.
  • Example of a tacky [and convoluted] solution and more difficult to implement than desirable: taking the two [or N] highest frequencies and averaging them based on their frequency. Unfortunately this falls apart almost immediately as can be seen by the above sets eg Set A, which two scores are the highest? Set B, (0*38+1*38)/(38+38)=0.5 is about as bad as the mean prediction of 1.75. Set C, what happens when the set has an odd number of unique data points?
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    $\begingroup$ You lost me early on: what are you predicting? How are these data being generated? What do the numbers measure or represent? You need to explain these essential contextual elements of your question if it is to become anything other than a mathematical abstraction (which then permits only purely mathematical solutions). Also, in its present abstract form, this question is difficult (or maybe impossible) to understand because some technical terms do not seem to be used in conventional ways (such as "mean average" or "conditional probability"). $\endgroup$ – whuber Dec 28 '15 at 14:25
  • $\begingroup$ Little League soccer scores. Defense has to be taken into account of course not that really exist with the little guys. By "mean average", I should just say mean. Conditional probability is usually used for classification I take it but it could still be used on any arbitrary set of numbers with a non-"normal distribution?" Thanks. $\endgroup$ – Evad Dec 28 '15 at 14:47
  • $\begingroup$ Thanks, but I'm still as much in the dark as before. Soccer is a game of two teams, whence a score must be a pair of counts--but you are discussing only single values. $\endgroup$ – whuber Dec 28 '15 at 14:49
  • $\begingroup$ Presumably, defense of Team-B can be averaged with the attack of Team-A. Using a very convenient set, Team-A scores 2 goals 30% of the time & 3 goals 70% of the time; Team-B concedes 2 goals 70% of the time & 3 goals 30%. Combining, Team-A could reasonably be expected to score 2 goals (70+30)/2 = 50% of the time and 3 goals 50%, at least at a basic level especially if both Team-A & Team-B had played 20+ games. It is my understanding that attack & defense are usually equally weighted even by big sports statisticians eg pace is assumed to be simply the mean, possession in soccer too. $\endgroup$ – Evad Dec 28 '15 at 15:16
  • $\begingroup$ I just clicked about your issue with "conditional probability". I am going to pair up Team-A's prediction with Team-B's prediction [ignore defense for the moment], repeating the above Team A scores 2: 30% & 3: 70% while Team B scores 2: 40% & 3: 60%. So ignoring too momentarily that eventually a "clean sheet" will occur. With that data only: A vs B:: 2-2: 12%; 3-2: 28%, 2-3: 18% & 3-3: 42%. Obviously the goal difference has to be used / is the end goal, so determining the error would be no different than that of the single set of attack / score predictions of Team-A alone. $\endgroup$ – Evad Dec 28 '15 at 15:34

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