# Comparing the means of two groups when a large number of data points have the same value

I have data on the profitability of wagers made according to two different strategies.

One strategy was to bet $1 on the underdogs in each of a sequence of sporting matches. They have data like this: Lost \$1
Lost \$1 Won \$4.50
Lost \$1 Won \$3
Lost \$1 ...and so on. Of the 1400 data points from wagers on underdogs, 1000 of them result in a loss of the \$1 stake while the profitable wins come in varying amounts but are all at least a profit of at least $1.10 or so. The other strategy was to bet on favorites. There the data looks like this: Won \$0.20
Won \$0.50 Won \$0.10
Lost \$1 Won \$0.80
Won \$0.20 ...and so on. Here only 400 of the bets result in a loss of \$1, and the vast majority of bets result in a profit of some small amount (always less than $1). The actual data is viewable here. Ultimately I'm interested in comparing strategies, and so I want to compare the means of the "bet on favorites" strategy with the "bet on underdogs" strategy. Theory tends to suggest both should lose money, but that betting on underdogs should lose more. What I'm wondering is what sort of test I should do to compare means, given the nature of the data. I wasn't sure what the implications would be of the fact that in one of my groups most of the data points are fixed to a particular value (-1). If nothing else it creates a massive skew in my dataset. I thought that maybe doing a Mann–Whitney U test might work well here, but I was unsure on this point, especially as my data is paired in the sense that each game generates one favorite to bet on and one underdog to bet on. Also, Wikipedia states that "the Mann–Whitney U test should not be used when the distributions of the two samples are very different". My distributions are really different since a huge number of the underdogs resulted in a loss of$1, whereas that is not true of the favorites.

• You could regard the stake as always lost and just focus on the actual winnings which would have clumping at zero not minus 1. Then do a search for zero-inflation (which is also a tag) on this site and see if that helps you. Most of the methods are for count data which might be a complication. – mdewey Oct 13 '18 at 12:49
• I did find a question that seemed to exactly match what you were saying. The accepted answer said to "use the Mann-Whitney U-test if the samples are not paired". My data is in a sense paired, since every match generates one favorite to bet on, and also one underdog to bet on. Should I therefore be using the Wilcoxon signed rank test? – user1205901 Oct 13 '18 at 22:16
• The idea of looking at this as paired data is intriguing, but doesn't sound right, since in this case there is a deterministic relation between the two members of each pair, not two correlated random events. There is only one random event! At least, one cannot use a usual paired test. Maybe modeling the win of either favorite or underdog as binomial, and the win/loss conditional on the binomial outcome? – kjetil b halvorsen Nov 16 '18 at 12:53