We merely infer it from the given rate. All such inferences are hazardous.
Consider. If all we really know is that r = 0.2000 per hour, we have no information about how the 40 dropped dishes were distributed among the 200 hours of the probation period. We may be inclined to assume a von Bortkiewicz type of data set for the 200 hours of Asa's probation period. Well, what would the data set have looked like if it were of the von Bortkiewicz type? We can answer that question. Given a rate of r = 0.200, it follows that over a period of 200 hours (say 25 eight-hour shifts, or about one month), we would by the Poisson Table expect to find, on average and rounded to whole numbers:
164 hours in which 0 dishes were dropped
33 hours in which 1 dish was dropped
3 hours in which 2 dishes were dropped
and probably none in which 3 or more dishes were dropped, an event for which, over 200 hours, we have only p(3) = 0.2200 (that is, a probability of 0.0011 times 200), which rounds down to zero. This is the supposition on which our answer to Problem 4 was based. If that supposition is true, then it follows that Asa never dropped a fully loaded tray. But the supposition is just a supposition.
We can substitute another supposition. Suppose instead that all 40 dropped dishes were on one overloaded and spilled tray. Asa's frequency profile would then not match the above Poisson situation. It would instead have the following form:
199 hours when 0 dishes were dropped
0 hours when 1 dish was dropped
0 hours when 2 dishes were dropped
[and so on, until we reach]
1 hour when 40 dishes were dropped
The average rate r in this case still works out to 0.2 dishes per hour, but we are no longer in Poisson territory, and the rate r accordingly cannot be used to calculate specific dish probabilities.
If Asa's dish-dropping during his probational 200 hours was in fact 1 tray and not 40 separate dishes, then it is more accurate to report his rate as 1 incident of dropping in 200 hours, or 0.005 incidents per average hour. It is still not Poisson, and we can't use our Poisson tables to make predictions about it, but knowing that fact (1 accident in 200 hours) gives much more assurance about the Chancellor's dinner than does the other rate (1 accident every 5 hours), and knowing that the facts had this texture would affect our subjective judgement in that case. Maybe we don't have to send Asa out for lemons after all?
Why do restaurants keep tabs on dishes rather than on incidents? Maybe because they think in terms of cost, and it is the dishes, not Asa's wounded dignity, that cost money.
The Poisson Distribution only applies if the occurrences in question are independent of each other. But with waiters dropping dishes, there may easily be linkages: one event (one unseen slippery spot, on the way back to the kitchen) may produce more than one broken dish. We may state a general rule: You cannot tell, merely from the average rate r alone, without other information, anything about the distribution. We will presently find, in the problem at the end of this Lesson, that you also cannot tell, merely from the distribution, anything about the clustering. You have to know the territory, you have to see the ground, you have to look at the map. You have to know where the numbers are coming from.
Some people, having gotten hold of the numbers, will proceed to work solely from the numbers. But statistics is not about numbers; it's mathematics that is about numbers. Statistics is about reality.
Statistics is Copyright © 2001- by E Bruce Brooks
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