EPTSC2Gears is a wonderful tool for a number-crazed mathematician like myself. Not only does it show cool graphs of winrate over time (a nice bouncy profile) or APM development (rising slowly but steadily) or present winrates per matchup (damn you, TvP!) and per map (<3 Shattered Temple), it also comes with some statistics that are not directly involved with ones process to improve.
Recently I took a peek at the breakdown of my replays by day. And there I noticed something odd... I suck on wednesdays. (Technically I'm not terribly good on any day of the week, but you get the point) The numbers do not lie:
monday - 52.8%
tuesday - 57.1%
wednesday - 39.4%
thursday - 49.3%
friday - 52.0%
saturday - 51.4%
sunday - 54.5%
This data has been gathered from all my 1v1 games since october 2010, removing any games where the opponent insta-quit upon the game starting. A total of 1208 games on the EU ladder, with an overall win rate of 50.7%.
Now just stating a win percentage and claiming that my wednesdays are particularly bad is a bit sloppy. After all, there's a certain randomness to the whole thing. While an SC2 match by itself has very little randomness, the matchmaker can match you against opponents that play a style you know how to beat or against a favourable race. So is that the case on my wednesdays? Bad luck with the opponents I get?
Math intermezzo! Standard deviations
To determine this, a second variable is needed. The standard deviation. For those of you not familiar with statistics: The standard deviation is a common measure for the spread in a data set. In this case, we are computing the standard deviation we would expect assuming a 50% win rate. If we assume a 50% chance to win each game, then for any set of games, the expected standard deviation of the number of wins is the square root of the number of games divided by 2. Divide this value again by the total number of games in the data-set to find out the relative standard deviation.
An often used term that means standard deviation is 'sigma', so I'll be using that. The higher the sigma-value of a data-set, the more the win rate deviates from the expected value and the less likely it is that this is purely due to chance. If the win rate follows a normal distribution (which looks like a Gauss- or bell-curve), the chance of finding a win rate at a distance of at most 1 sigma from the expected value is 68.2%. In other words: a deviation of more than 1 sigma from the expected value happens with a 31.8% chance. Pretty common.
A deviation of more than 2 sigma happens only 4.2% of the times in a normal distribution. So if you measure a deviation of this magnitude, you can already start wondering if it is truely due to chance or if there's another reason. Note that the often-used 95% confidence level almost coincides with the 2 sigma deviation. If a measurement is at least 2 sigma away from the expected value you can say with a 95% CL that the expected value is wrong.
Less talk, more numbers!
Now that the background is in place, lets run the numbers. I computed the standard deviation for my weekday-winrates and got the following:
mon => 52.8% (4.5%) (0.62)
tue => 57.1% (7.7%) (0.92)
wed => 39.4% (4.3%) (2.5)
thu => 49.3% (4.3%) (0.16)
fri => 52.0% (4.1%) (0.49)
sat => 51.4% (2.9%) (0.48)
sun => 54.5% (2.8%) (1.6)
The first column is the win rate, the second column the standard deviation and the third column the deviation from the expected value (50%) in terms of sigma. You may wonder why the standard deviation is not the same for every day. The answer is simple: the number of games played is different. The more games you play, the more likely it is that your winrate will be close to the expected 50%. Play 2 games and your results can vary wildly: 0%, 50% and 100% are all very likely winrates. Play 1000 games and (unless you're top GM or bottom bronze), you will have a winrate very close to 50%. So the more games in a dataset, the lower the relative standard deviation (or: the narrower the bell curve is around the 50% mark).
The main observation from the data is that most days behave quite well. 5 out of 7 days have a deviation of less than 1 sigma from the expected value. That's quite good. Sunday is on the high side, with 1.6 sigma, but this is still not a very large deviation and can happen due to random chance. Wednesday, on the other hand, is a different story. 2.5 sigma deviation means that such a result has approximately 1% chance of occurring naturally if the win-chance is 50%. Even though you can't rule out pure luck, this value seems to be unlikely.
Is that it?
So could it be that I actually play significantly worse on wednesdays? The story continues with a dramatic twist as we discover that for most people, midnight doesn't necessarily mean the end of the day.
Stay tuned!
