In my experiences with excel, I find the skew comes from the limited number of sample draws versus the total number of possibilities. The theoretical models tell you what would happen after the dust settles, where the numbers are expected to be frequency-wise... observations don't agree because there are only usually a few hundred or a few thousand results and millions of possibilities.
The theory helps set ranges, but observation helps narrow them down to useful size.
here's an example from the MUSL powerball...
5/53
Theoretical ranges...
(Sorted Order)
First ball- 1 to 49 - 48 numbers to pick from
Second Ball- 2 to 50 - 48 numbers to pick from
THird Ball- 3 to 51 - 48 numbers to pick from
Fourth Ball- 4 to 52 - 48 numbers to pick from
Fifth Ball- 5 to 53 - 48 numbers to pick from
Observed Range to date...
First ball- 1 to 32 - leaves 31 numbers to pick from
Second Ball- 2 to 45 - leaves 43 numbers to pick from
THird Ball- 6 to 50 - leaves 44 numbers to pick from
Fourth Ball- 8 to 52 - leaves 44 numbers to pick from
Fifth Ball- 10 to 53 - leaves 43 numbers to pick from
Observed Range with spikes trimmed (3 or more occurrences in boundary numbers)...
First ball- 1 to 31 - leaves 30 numbers to pick from
Second Ball- 5 to 34 - leaves 30 numbers to pick from
THird Ball- 14 to 46 - leaves 32 numbers to pick from
Fourth Ball- 17 to 50 - leaves 33 numbers to pick from
Fifth Ball- 32 to 53 - leaves 21 numbers to pick from
We know all high or all low spikes can and do happen, but to be within 80% of the actual observed ranges of all 205 powerball draw results since the new matrix went into play (10-9-2002), the numbers above better serve your immediate needs.
There are also gaps within those ranges...
Observed Range with spikes trimmed (with missing #s and #s drawn less than 3 times)...
First ball- 1 to 31 (21-22-23-24-25-26-27-28-29-30) leaves 20 numbers to pick from
Second Ball- 5 to 34 (27) leaves 29 numbers to pick from
THird Ball- 14 to 46 (15-41-42-43-44) leaves 27 numbers to pick from
Fourth Ball- 17 to 50 (18-19-23-24-36) leaves 28 numbers to pick from
Fifth Ball- 32 to 53 (37) leaves 20 numbers to pick from
powerball itself is a tough nut to crack...
range is 1 to 42, in theory... and 1 to 41 observed (using 3 or more hits to count)...
if you remove the missing and low frequency numbers, you get
Powerball - 1 to 41 (12-14-24-26-27)
picking numbers from the above lists will not guarantee a win, you just put yourself in the observed history "Hit Zone"... Unfortunately the numbers don't know what ranges they are supposed to fall into, so luck is still a piece of the puzzle... 1% system, 98% luck, 1% actually buying a ticket to be in the game...
good luck and keep the math flowing... cool forum!