I'm trying to decide which type of mean is best to use when playing Deal or No Deal (in terms of comparing it with the Banker's offer). I realise the standard answer would be the arithmetic mean, but the problem arises when, say, a contestant has lots of low-value boxes (£0.01, £0.50 etc) and one massive-value box (such as £1,000,000).
Needless to say, this horrendously skews the arithmetic mean, and the "size" of the expected value becomes almost solely dependent on that one giant number (i.e. if you accidentally pick it, the expected value whiplashes back down to a more representative value of the spread - that is, a vastly smaller number).
Out of curiosity I've drawn up a little Excel spreadsheet which reflects the running value of the arithmetic mean, the median, and the geometric mean as you play the online simulator. The arithmetic mean is always much larger than what is offered, making it impossible to say deal and "beat the mean", but the median is almost always so low that a contestant comparing what is offered to this value would end up saying "deal" on the first round every time.
My suspicion is that the geometric mean might be the best way of coping with skewed scenarios as described above, but it is also too low to be worth bothering with. Is there a way of shrinking the Banker's offer so that it can be properly compared with the geometric mean? Should I divide the geometric mean by the arithmetic mean and taken some sort of measure of how much they deviate? Or take some sort of root of the amount offered? Or is there an even more subtle way of finding a representative value of the remaining boxes which is immune to skew?