Reversion to the Mean
- Thread starter secxces
- Start date
Paul:
I did (once upon a time) have a chart of the percentage of Heads in umpteen coin tosses, but can't find it now.
However, I've just made up a spreadsheet that'll toss 10,000 coins and show the percentage of heads, like so:
It's rather large, 2 Mbytes, 'cause it contains 10,000 percentages. Anyway, if'n y'all like to play, I've stuck it here:
http://www.gummy-stuff.org/Excel/coin-toss.xls
Each time you press F9 you get another 10,000 tosses.
It's fun!
I did (once upon a time) have a chart of the percentage of Heads in umpteen coin tosses, but can't find it now.
However, I've just made up a spreadsheet that'll toss 10,000 coins and show the percentage of heads, like so:
It's rather large, 2 Mbytes, 'cause it contains 10,000 percentages. Anyway, if'n y'all like to play, I've stuck it here:
http://www.gummy-stuff.org/Excel/coin-toss.xls
Each time you press F9 you get another 10,000 tosses.
It's fun!
gummy, this is awesome, never had so much fun ... ;-)
But what about the cumulative no of heads? Your chart shows (if I am reading it correctly) that after n tosses the percentage of heads fluctuates about or is near the men - as statistically expected.
Lets say after a while you have 10 heads in a row after having had a perfect series of alternate flips. So the line would be creeping along the mean at first and then move up to 10 - at which point the series continues with, say, a normal distribution.
The graph I remember plotted this over a zillion tosses and it became apparent that the deviation from the mean becomes larger and larger. The amplitude just increases dramatically.
Now I am not talking about the gamblers fallacy and do not argue against statistical principles, I was just impressed by these enormous fluctuations ...
paul
But what about the cumulative no of heads? Your chart shows (if I am reading it correctly) that after n tosses the percentage of heads fluctuates about or is near the men - as statistically expected.
Lets say after a while you have 10 heads in a row after having had a perfect series of alternate flips. So the line would be creeping along the mean at first and then move up to 10 - at which point the series continues with, say, a normal distribution.
The graph I remember plotted this over a zillion tosses and it became apparent that the deviation from the mean becomes larger and larger. The amplitude just increases dramatically.
Now I am not talking about the gamblers fallacy and do not argue against statistical principles, I was just impressed by these enormous fluctuations ...
paul
Paul:
Theoretically speaking, a series of all heads would have no long term effect on the percentage of heads.
Eventually, those heads would be buried by the later tosses
... assuming there are a jillion later tosses.
Anyway, I've modifed the spreadsheet so you can force a series of heads:
http://www.gummy-stuff.org/Excel/coin-toss.xls
Give it a try!
P.S. Although I could do a million tosses, the spreadsheet would need a Mack truck to transport.
You could, however, add the number of heads in each 10,000-toss scenario.
Theoretically speaking, a series of all heads would have no long term effect on the percentage of heads.
Eventually, those heads would be buried by the later tosses
... assuming there are a jillion later tosses.
Anyway, I've modifed the spreadsheet so you can force a series of heads:
http://www.gummy-stuff.org/Excel/coin-toss.xls
Give it a try!
P.S. Although I could do a million tosses, the spreadsheet would need a Mack truck to transport.
You could, however, add the number of heads in each 10,000-toss scenario.
thanks again, gummy, for your effort.
But what we really need is the cumulative value of heads AND tails. Lets say we have a sequence of 5 heads in a row and then tails heads tails, the number would be 5 -1 1 -1 and the result 4.
This is the graph we are looking for. Could we just add a column in your spread sheet to calculate the cumulative value of heads and tails and plot it?
paul
But what we really need is the cumulative value of heads AND tails. Lets say we have a sequence of 5 heads in a row and then tails heads tails, the number would be 5 -1 1 -1 and the result 4.
This is the graph we are looking for. Could we just add a column in your spread sheet to calculate the cumulative value of heads and tails and plot it?
paul
Okay, the latest spreadsheet counts Heads, Tails and the Difference: Heads - Tails.
If (for example) you force 500 heads, the cumulative Heads will be greater than the cumulative Tails by (roughly!) 500.
Perhaps that's the chart you remember seeing.
Eventually, however, those 500 heads will have diminishing effect on the percentage of Heads in a jillion tosses.
That is, in N tosses:
heads = h + 500
tails = N - h - 500
heads + tails = N
and
fraction of heads = (h + 500) / N = h/N + 500/N
and the 500/N part diminishes to 0, leaving h/N which'd approach 0.50 (or 50%) as N approaches infinity.
If (for example) you force 500 heads, the cumulative Heads will be greater than the cumulative Tails by (roughly!) 500.
Perhaps that's the chart you remember seeing.
Eventually, however, those 500 heads will have diminishing effect on the percentage of Heads in a jillion tosses.
That is, in N tosses:
heads = h + 500
tails = N - h - 500
heads + tails = N
and
fraction of heads = (h + 500) / N = h/N + 500/N
and the 500/N part diminishes to 0, leaving h/N which'd approach 0.50 (or 50%) as N approaches infinity.
Take a look at range contraction. AMD today was a good example. Yesterday it had a very narrow range. All the contraction means is that you are more likely to get a break in either direction...an expansion of the range. Same with Volatility contraction. Plot the 100 day Volatilty and then plot a shorter period like 10day or 5 day. When the shorter period Volatilty dips below the longer, you are likely to get an expansion in Volatilty, (a reversion back to the 100 day) you just don't know in which direction. A lot of option tading is based on this principle.