Any lotto sum frequency diagram tends to look like a bell shaped
curve.But it is not a perfect normal curve, often called "the bell
curve". The difference is especially evident at the mean peak
value,which is truncated or flatened.. The 5/55 lotto part of
Powerball and the UK 6/49 game may serve as a test.
The frequency data for a complete set of the sums of all possible
combinations of any lotto, n/N, can be found on the internet, just ask
RP, or at university libraries.
In the case of PB the sums range from 15 to 265 with a symmetrical
distribution around the mean 140. Thus there are 250 distinct sums,
but it is sufficient to to plot the frequencies of every 5th sum.
The very peak value 140 has a frequency of 39361 and at 135 and 145
it is 39001 while at 130 and 150 it is 37917 and for 125 and 155 it
is 36176.Thus the peak from 135 to 145 is rounded off. But how can it
be determine that the the total curve is not a perfect normal curve?
This must be based on the fundamental characteristics of the Gaussian
curve function, which can be found in any good text book on
statistics. Certain criteria must be met, such as:
First, a plot of the percent cumulative frequencies on a standard
arithmetic probability graph paper must yield a perfect straight line.
Second, there should be a distinct inflexion point at one std.dev.
.where the curveature changes from convex to concave as determined by
the tangent to the curve.
Third, the peak value can be used to calculate the std.dev.
Fourth, three s.d. deviations from the mean must cover exactly99.73%
of all data and two s.d must cover must cover 95.44% and one s.d will
cover only 68.26%.
The sum frequency data for 5/55 meet none of these criteria exactly
but approach them , which is best illustrated with the std.dev.
The formula for the normal,Gaussian, curve shows that the peak value
must be equal to 40% of the Total divided by the std.dev. For 5/55 it
would be 0.4x3478761/39361=35.255. But the actual s.d. based on every
five sum from 20 to 260 is only 34.157. This value could be predicted
with the formula s.d.=square root of (N+1)(N-n)n/12, which yields
34.1565 for 5/55.
Thus 3 s.d would be 102.5 and the cumulative frequencies for the sums
less than 38 is only 1603, but should
be0.5x(100-99.73)x3.48x10^6=4696.
An identical analysis for the 6/49 game will be posted elsewhere.
All or some of the above discussion might have appeared here or some
other place but would be new to recent readers. It must be well known
and understood by those versed in the mathematics of lotto,with a
notable exception for johnph77, who insists that the std.dev. must be
14143, a clearly preposterous value, that any half smart highschool
student of Statistics 101 would be able to see as absurd ,because the
s.d. must be about 1/6 to 1/8th of the range 250(265-15).
Robert Perkis feel free to to use this poster with revisions as an
article at Lotto-Logix.
Stig Holmquist
