Poisson Distribution

A total of 2,608 time units were monitored (for a total time of 326 minutes, or about five and a half hours), and over that period, 10,097 emissions were recorded (in adding up Rutherford's count, remember not to count the "57" in his top row, which is the number of units in which no emissions were recorded). That gives an average rate of 3.87155 emissions per unit of time. The nearest r values supplied on the Poisson Table are r = 3.5 and r = 4.0, neither of which is markedly close, so we will need to calculate a new column for r = 3.87155, the rate per 7.5-second unit of sampling. Nothing could be easier, always assuming that your calculator has an e*x key. The resulting column is given below. We then multiply each value by the total number of units involved (2,608), to get the Expected (E) values for that group of samples. Finally, we put beside them Rutherford's Actual (A) or observed values. The result looks like this:

	r = 3.87155	E	A
p(0)	0.0208	54.2464 ~ 54	57
p(1)	0.0806	210.2048 ~ 210	203
p(2)	0.1561	407.1088 ~ 407	383
p(3)	0.2014	525.2512 ~ 525	525
p(4)	0.1950	508.56 ~ 509	532
p(5)	0.1510	393.808 ~ 394	408
p(6)	0.0974	254.0192 ~ 254	273
p(7)	0.0539	140.5712 ~ 141	139
p(8)	0.0261	68.0688 ~ 68	45
p(9)	0.0112	29.2096 ~ 29	27
p(10)	0.0043	11.2144 ~ 11	10
p(11)	0.0015	3.9120 ~ 4	4
p(12)	0.0005	1.3040 ~ 1	0
p(13)	0.0002	0.5216 ~ 1	1
p(14)	0.0000	0.0000 ~ 0	1
TOTAL		2608	2608

The fit is rough enough to look real, but overall, it is obviously close. Nothing strongly suggests a departure from a Poisson process, that is, a random variation process. Hence the answer given above.

Rutherford is one of the great figures in modern science, and nobody is going to be very inclined to doubt his data, or his interpretation. Very good. But strictly speaking, the data as supplied are summary in nature, and therefore conceal a great deal of possibly relevant information. Our inference as given above is in principle vulnerable to some of that concealed information. For example, if a rolling average of 100 units of time showed a tendency toward a decreasing or increasing rate, over the five and a half hours, that would suggest caution. Also, one wonders what would have happened if the team had returned to that same film of polonium after a year, and observed it for another five and a half hours with the same equipment. And then in a different part of the Earth's magnetic field, still with the same equipment. Only with such further information can the present inference be accepted as strongly indicated.

It is a statistical bad habit to skip these confirmatory observations, the legislative checks and balances of the scientific life. We will return to this methodological issue in Problem 20.

Update 2007. Various objections were in fact raised about the original experimental data, among them the angle subtended by the equipment, the ability of observers to count by eye, with a low-power microscope (these counts would later be made by optical detectors), information sometimes coming in at the rate of two scintillations per second; and lag time on the mechanical counter used by Rutherford's team (the counts would later be registered electronically). These points have been addressed up by subsequent experiments, and even closer matches of data with Poisson predictions have been obtained. In the end, Rutherford's inference of a random process seems to be confirmed.

Part of genius is knowing when you're right, even if the data set is incomplete or the recording mechanism is open to technical objections. This part of genius Rutherford possessed.