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119) Errors in unison ?

Ludwik Kowalski (12/20/2003)
Department of Mathematical Sciences
Montclair State University, Upper Montclair, NJ, 07043




After reading final comments made by Dr. Kirk Shanahan (see Unit #118) I composed another essay, similar to that of Unit #116. The previous essay focused on systematic errors, the one shown below focuses on random errors. The main point is to illustrate Kirk’s view that these two kinds of errors can not always be separated. As before, the content is inspired by numerous e-mail messages received from Kirk in the last two weeks; the form is inspired by Galileo Galilei.

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1) Student:
My friend, an chemical engineer, discovered an unusual radioactive rock. The radioactivity of that rock increases when the temperature goes up. Is this possible?

2) Teacher:
We do not know everything about nature; new discoveries are likely to be made anywhere and at any time. But I would be highly surprised if his claim turned out to be validated.

3) Student:
Why would you be surprised?

4)Teacher:

Because a large number of great scientists, including Marie Curie, looked for such phenomena, when radioactivity was discovered, and their results were always negative. That is one reason. The second reason is that such discovery would be in conflict with what we already know. A change in temperature, even as large as several thousand degrees, has no effect on what is going on in atomic nuclei. And radioactivity, as you know, is a nuclear phenomenon.

5) Student:
But didn’t you tell us that unexpected discoveries are always possible? My friend is willing to make a demonstration. Would you be interested in seeing it?

6) Teacher:
Yes, I would. And I will try to open-minded. Keep in mind, however, that the burden of proof is on him. Most likely we will find an error in his presentation. But who knows, perhaps the claim of your friend is valid. If it is then everything we know about nuclear phenomena would have to be reexamined. Let us hear him.

7) Engineer (one week later):
Thanks for allowing me to use the laboratory. Everything is ready. Should we go and start collecting data?

8) Teacher:
I would prefer you to first tell us exactly what you want to do. Where is your “magic rock?”

9) Engineer (reaching in his pocket):
Here it is. Looks like a common stone but I will show you that it is radioactive.

10) Teacher:
What kind of counter are you going to use?

11) Engineer:
A Geiger counter, it is not very different from the one I saw in your lab. I will place this rock below the counter, set the timer, and start counting, for example, for several minutes or several hours.

12) Teacher:
Good. Why would you need to count for several hours instead of several minutes?

13) Engineer:
I count longer when I want better precision. Radioactivity is random, if if the number of counts is N then the standard deviation is the square root of N. The percentage error decreases when N becomes larger. Longer counting times result in larger N. The counting rate, equal to N divided by time, is proportional to the source activity.

14) Teacher:
That is right. Let me elaborate on this. Suppose we are trying to determine the activity C of a source with a counter when the background is not negligible. The source activity, if any, is expected to be A - B, where A is the number of counts recorded when the source is present and B is the background (when the sample is removed). If A is not very different from B, then even small random fluctuations in A and B will have strong effect on C=A-B. The standard deviation of the mean C (s3), the standard deviation of the mean A (s1) and the standard deviation of the mean B (s2) are related as:

s1^2 = s2^2 + s3^2

Suppose a single set of two measurements yields: A=100 and B=25. In that case s1=10, s2=5 and s3=sqr(s1^2+s2^2)=11.2. We would only be able to say that C is 75 plus or minus 11 (15% error). To increase the level of precision we could increase the counting time, for example, by the factor of one hundred. With A=10000 and B=2500 we would be able to say that C is 7500 plus or minus 112. (1.5% error). The longer we count the more reliable the result. Ernest Rutherford, who discovered atomic nuclei, is often quoted to say: “If experiment requires statistical analysis, then one should do a better experiment. ” He was probably referring to Geiger counter experiments in which N were too small.

15) Student:
Does it mean that any level of precision is possible?

16) Teacher:
Yes, but only “in principle.” This is an important point; I suspect that we will have to return to it later. Meanwhile let me say that your use of the term “precision” is appropriate. This term should not be confused with the term “accuracy.” Today we are talking about random errors and we identify precision with the standard deviation of the mean, s. Mathematicians do tell us the s approaches zero when N approaches infinity.

17) Engineer:
But real counters will break, sooner or later. Is that was the reason for adding the “in principal?”

18) Teacher:
Yes, this was one of the reasons. I think we are ready to hear about the actual data you collected so far. How large was your “signal”, N1, when the rock was below the counter?

19) Engineer:
It was 18043 in two hours.

20) Teacher:
How large was your background, N2, when the rock was taken away?

21) Engineer:
It was 2180 in one hour.

22) Teacher (addressing students):
In terms of counts per minute we can say that A was 150 (plus or minus 0.7%) and B was 36 (plus or minus 2%). There is no doubt that this rock is radioactive; the difference between the signal and the noise (background) is much larger than random errors. I think it is time to see the real experiment.

23) Engineer (turning toward the apparatus):
To make sure that the geometry is the same as in my earlier experiment I will place the rock on this stand. I am setting the timer for six hours. Now I am pressing the start button. Do you hear the clicks? We are not going to sit here for six hours; the counting will stop automatically and we will continue tomorrow.

24) Teacher:
May I suggest that we stop the counting in an hour. This may already reveal something significant. Meanwhile let me explain what I would like to verify. The best way to do this is to use our own counter. Let me turn it on and place our radioactive cobalt source below it. As you can see, it counts rapidly. What I want to do is to see how the counting rate depends on the high voltage applied to the Geiger tube. Please write down these results; each refers to one minute of counting.

voltage -->  700    800   1000   1100   1200   1300   1400
counts  -->    0     37    372    400    458    540    974
Next plot the number of counts against the voltage. What you see is called the “plateau curve” for our counter. It shows that the dependence of the counting rate on voltage is not very strong between about 1000 and 1300 volts. That is why we normally use the counter at 1200 volts. Ideally the curve should be flat , in that way changes in voltage would not affect the counting rate. But no counter is ideal, some counters are better than others in that respect. They also differ in terms of stability of the voltage. Ideally a voltage should remain constant but in reality it changes unpredictably.

25) Engineer:
My counter stopped counting. It displays 11707. In terms of counts per minute it is 195 (plus or minus 0.92%).

26) Teacher:
This is this not consistent with 150 (plus or minus 0.7%) that you reported before? The difference between 195 and 150 is highly significant. Would you say that bringing the rock into this room changed its radioactivity?

27) Engineer:
I think that the background in this room is higher than in my basement.

25) Teacher(after the background was measured):
As you can see the background in this room is nearly one half of it was in your basement. What else might be responsible for the discrepancy?

26) Engineer:
I do not know. What do you think?

27) Teacher:
One possibility is that the plateau curve of your counter is not much better than the curve of our counter and that the voltage applied is not stable. Suppose the voltage drifted from 1200 to 1400 volts. Have you tested stability of your power supply?

28) Engineer:
No I haven’t. I simply assumed that the voltage does not drift too much.

29) Teacher:
That is a good assumption when things seem to be normal. But you are claiming something totally new and unexpected. You should check and double-check everything. One thing I would recommend, besides measuring voltage continuously, is to determine the plateau curve before and after each serious measurement.

30) Engineer:
Thanks for constructive criticism; I agree with you. I will start another sequence of experiments.

31) Teacher:
That is good; let us know what happens. Let me use this occasion to comment on the term “in principle,” used above. Mathematically it is correct to say that random errors can be reduced by as much as we want. It is only a matter of making a larger and larger number of measurements and calculating averages. This,however, is true only when systematic and procedural errors are absent. But hidden variables, such as the unknown voltage instability, should always be suspected. In your case trying to improve precision by increasing the numbers of counts would be counterproductive. It does not make sense to be preoccupied with tiny cracks in one window when another window, in the same room, is widely open. Likewise, using highly accurate ammeters or voltmeters is likely to be counterproductive in an environment dominated by random errors and by effects of hidden variables. In other words, as often emphasized by Kirk Shanahan, the issues of accuracy (systematic and procedural errors) and the issues of precision (random errors) can not always be separated.

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