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Unformatted text preview: Example Repon for PHYS 450 Lab
by
Graham Whitcomb, T.A. Purdue University
January 16, 2007 Abstract:
For the purpose of demonstrating my preferred method of report writing, a random number generator was tested (a die) and the results were analyzed. Only 18
measurements were made. Which statistically random features are observable using this
limited number of measurements, and which are not, is investigated by this report.
(Throughout the report, hints will be given in italics.) It is uncovered, through analysis,
that 18 measurements is an insufficient amount of data for observing the equal chance of
obtaining any one of the six possible, random results. However, the average result of
random integers 1—6 is observed to be exactly 3.50, as predicted. Objective/Theory/HypOthesiS: (Call it what you like, but explain thoroughly the
reasoning behind your experiment and brieﬂy, what the actual experiment was.) A regular die, with six sides, supposedly generates a random result of l, 2, 3, 4, 5,
or 6 upon every cast. If the die is perfectly balanced and properly cast, it can be assumed
that all six results are equally likely, upon each cast (a perfect random generator). From
the laws of statistics], it is known that it can also be assumed that over a large number of casts, the six possible results should appear in approximately equal (large) numbers, with
the result count for each number becoming closer to one 6th of the total number of results
the more times the die is cast. If possible results are the integers 1 through 6, and they are appearing in equal numbers, then it is easy to see that the average of all results should be (1+2+3+4+5+6)/6 = 3.5 For this experiment, the same die was cast 18 times. This die was given to me by
a friend, and of unknown (to me at least) origins and manufacture, so there is certainly
reasonable suspicion that it should not be perfectly random. The die was cast by simply
cupping both hands around the die, leaving plenty of space, shaking the die vigorously
for about 15 seconds, and then casting it on the linoleum floor of my dorm room. The
result was recorded each time. ( I ’m purposely using passive voice. I really don’t care if
you choose to write in passive or active voice, but don’t switch between them! Formal
scientific articles are traditionally written using passive voice.) It should be noted that
the number 18 was chosen for the number of measurements, because it is a reasonably
small number (I don’t have all day to spend throwing dice), but it isjust large enough that
perhaps the law of averages will apply, and the 6 possible results may be seen in close to
equal numbers. ( I ’m pointing out what I expect to see happen.) The results will be
analyzed to examine the frequency of each possible result, the average, and what
conclusions can be drawn, from the statistically low value of 18 measurements about
what results a large number of measurements might yield. (This last paragraph is also about how long your description of the experiment should be.
I am interested in good analysis and logical, well reasoned conclusions, not in having the lab handout reproduced for me.) I Statistics of random numbers is a simple and widely known topic, so I willforego a real reference here. If
you are including any subjects unique to thisﬁeld, then you should deﬁnitely include a reference. Data: (Just display the data, and any relevant information pertaining to it.) Table 1: A Histogram of Results
6 7 v 7 — —..
a 2 a, 6 7
cc 4 7 7 i
.c: i
g LLI 3 7 7 7 :c} a 2 7 7 7 g 1 “1* * o , l
1 2 3 4 5 6
Value Generated Here the data appears in histogram form ( with columns indicating the frequency of
each individual result. This is not the norm for the class, but it suits this particular
experiment very well. You MUST report all ofyour datafrom the experiment, and I
prefer the use of graphs or charts over simple tables wherever possible. However, if
tables or lists will save you time and still show the data in a clear and concise manner,
you may use them.) A list of results as they occurred was recorded, however each die
was cast in almost exactly the same manner (a randomization process) and so the order
and timing of results did not seem relevant. This original list can be produced upon request. (If there is any sort of error or uncertainty associated with your data, it should be
carefully catalogued here as well. I specifically chose dice for this, because they achieve
indisputable integer results. How well you treat uncertainty is something you will be graded on.) Analysis: (Do the math, and discuss what you can derive from your data.) It can be seen, from Table 1 above, that the 18 result counts were very random
indeed. Results of 2 and 3 were obtained 5 times, whereas results of 1 or 5 were recorded
only once each! Had the number of results obtained in our experiment been large
enough, each result should have occurred with a frequency close to one sixth the total
number of results, or 3 times. In fact, not one result (or number) was obtained three
times! And only two results, 4 and 6, are even within 1 count of what was expected. SO
it can immediately be observed that 18 is far too small a number for us to expect to see each result occurring with equal frequency. However, when the average result is taken:
(1*1 + 5*2 + 5*3 + 2*4 +1*5 + 4*6)/18 = 63/18 = 21/6 2 3.50 (You don’t have to show every line ofarithmetic, nor evety algebraic step. However, you
must make it clear to me what steps you are taking to get your results. Sample
calculations can be extremely helpful toward this end. I’m not doing much math for this
example report, so it’s important to show it all, just to have something to show. I would
also like to point out that ifwe were averaging many values ofthe SAME thing, being
measured many times, the standard deviation becomes extremely important, so important
that I am pointing it out, even though it is irrelevant to this example.) 3.5 is exactly the average value that was expected! (It ’s okay to throw ideas around in
the Analysis section, just repeat what you decide on in the conclusion.) How such an
exact, predicted value has been obtained by this experiment has several possible
explanations: A. Dumb Luck ~ It may have just so happened, with my random die throwing, that these
18 numbers somehow added up to exactly 63. Stranger things have happened. A
repetition of the exact same experiment could quickly demonstrate whether or not 63 is a
“fluke” total for these values. (But YOU will not be at liberty to repeat your experiment;
therefore dumb luck alone is never enough ofan explanation! So I will oﬁer other
explanations, including the correct one.) B. The significance/size of 18 — It may be that while 18 measurements was not enough to
show the “smoothing” out of the number/value counts into equal one sixth portions. it
WAS large enough to give an appropriate average. And so, repetition of the experiment,
with 18 measurements, may not come to exactly 63, but it should be close. (This is
almost right, and if answered well enough, something like this could receive full credit.
But this is wrong, because in this situation I 8 is really not a very large number, and even
the insinuation that it is large enough to exhibit characteristics of good statistics is
worthy of points deduction.) C. We took the Average — It IS actually the case that it was what was being studied that
made the difference. (This is also the only answer that ﬁts with the passive voice used in
the rest of the paper, as it would be satisfactory by itself.) For the first part of the
Analysis, it was the expected results, specifically the number of times each expected
result occurred, that was being studied. However, for the second part of the Analysis, the
Average was calculated instead. Let’s consider what has to be done with a random result
or even a result where some uncertainty is involved: When the individual results, or how
often they occur, are being considered, an attempt is actually being made at predicting
each result! Saying that each of the six possible results will occur exactly 3 times is a tall
order to fill, if we consider that with every roll of the die a 2, for example, has just a one
in six chance of coming up. It being that there are 18 “attempts” at a result in which to
get exactly three 2’s makes it slightly more likely, but out of all possible results there are
very few that include exactly 3 of each ones, twos, threes, fours, fives, and sixes, and only about as many results that have close to those numbers. (I won’t calculate out all
these probabilities I’m mentioning here, and neither should you in our Class, unless it 's
assigned, but it's not hard lfyou’re curious enough to try it. Doing extra work like that is
worthy ofextra credit.) However, there exist a MUCH larger number of ways to
combine l8 integers, one through six, such that they add up to 63, and about as many that
would add up to be within a digit or two of 63! So by simply studying a little statistics
and probability, we learn why the average came out right, when the results seemed really
“wrong” at first... So, what has really been discovered here is the power of taking the Average. It
should be observed, even from this simple example, that achieving the expected, specific
results, using such a small number of data points, is virtually impossible. However,
taking the statistical average is such a powerful tool, that accordance with expected
results can be checked with good precision, even if relatively few data points have been
taken. This is one of the reasons statistics and other numerical methods are so often employed by scientists. Conclusion: (Review everything that was important, and wrap things up.) So our 18 measurements of random integers 1—6 were made and analyzed. At
first, they appeared to be irregularly random, possibly even favoring the numbers 2 and 3.
Perhaps the die was weighted toward those numbers? But upon further study, the
randomness of the data can be explained by 18 measurements only being a very small
amount of data to take, if indeed you wish to study such things. It can be concluded (in
my estimation) that if say, 100 such measurements were made, we would get counts of
each result that are much closely equal. The reasoning behind this is that while appearing
haphazardly random, the average result of the data being taken shines a lot of light on the
matter. The average is exactly what was predicted, 3.50! This confirms that while the
data appears at first glimpse to be “noise”, it is in fact purely random noise and should
average out over a large number of measurements. A second part to the experiment could
be performed using a number of measurements much greater than 18, in order to confirm
this hypothesis (a good suggestion). Both the unreliability of low statistics (working with only a small amount of data)
and the power of statistical operations, such as taking the average, were evidenced by this
experiment. Hopefully, it has been an informative read. ...
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