Probability & Statistics/Rounding off ages

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QUESTION: This is just a question and has no basis in reality. This question may sound weird, but I hope you won't laugh at it. This is NOT a homework question. I just want to know that can "rounding off" numbers produce 100% correct answer.  

For example, in a family a person dies. There are groups which are controversial about the age when he died. I mean to say that there is a difference of opinion that what was his age when he died.

Old son, young son, nephew, and cousin which say that their great grandfather died at the age of 70. (They say that they are authentic).

Another group which consists of Old brother, young brother, and uncle which say that he died at the age of 73. (They say that they are also authentic, but this group is the correct one.)

One more group in the family which consists of aunt, mother, wives, and daughters says that he died at the age of 75. (They also claim that their sayings are authentic).

It is very important to note that all these people claim to be authentic.

NOW the sister comes up and says that she has rounded off all of these ages and came to conclusion that their great grandfather died at the age of 73.

My main question is that since all of them claim to be authentic, is there a contradiction in these reports according to mathematics or not? Is the sister 100% right when she rounded off all these reports, or does the contradiction still exists?

Thanks.

ANSWER: Briefly: No. The sister has no reason to believe that by averaging the three possible ages that this should result in the correct outcome.

What if one group said 70 and one group said 74? Regardless of which is correct, the average is 72 which is not one of the choices.


You are confusing statistical sampling (which is a valid scientific & social scientific methodology) with simply taking averages of possible outcomes.

If I have a set of two dice that I roll, I can roll them 1000 times and based on what I know about dice, I will roll a total of 7 about 167 times. I will roll a total of 2 only about 28 times. I might be wrong, maybe you will roll a total of 2 more times (29 or 30 or more), but the most likely number of times you will roll 1+1 on the dice is 28 out of 1000. This is how probability and statistics can estimate real life quantities. See also : http://www.edcollins.com/backgammon/diceprob.htm



However, do not have a set of statistical data. You have a single unknown quantity, the age of a man. That quantity is not a statistical or random variable of any sort. It is simply a number that is in dispute. Asking people what they think the disputed number is will not give you a set of data you can analyze statistically to find the correct value.

It would be like rolling the set of dice only once and hiding the outcome. We can argue which outcome is more likely or disagree about what outcome we think occurred, but there is only one possible outcome. No amount of combining our guesses will settle the argument because there are no statistics or data here -- just one number that people cannot agree upon.

---------- FOLLOW-UP ----------

QUESTION: Respected Clyde Oliver,

Thanks a lot for your answer. Your answer was comprehensive and easy to understand. One of my friend told that the sister is 100% accurate when she rounded off those ages because they just got lucky that their "recollections" averaged to 72.66, which, if rounded, would be 73. So the sister in this question was 100% accurate when she rounded off those ages of deaths. Do you agree with that? Does the contradiction still exists, or did the sister just guessed it correctly?

Personally, I didn't understand my friend's logic. If 9 is the closest number to 10, that doesn't make 5+5=9.


I am just a humble person.

Thanks.

Answer
It doesn't matter if the sister took the average age from the three groups and was lucky enough to average those three groups' numbers to get the exact age down to the second. It is irrelevant. There is no reason to believe the sister is correct in any way -- the scenario assumes she is correct, and that the "middle" of the three groups is correct, but we could easily tell the same story and say that the correct age was 70. Or maybe the correct age is 85 and they are all very wrong. There is no way to know. Mathematics and statistics do not work in the way your friend says, and this sister is -- at best -- making an unsubstantiated guess that is not necessarily any more likely to be correct than any other guess. It might be a good way to settle a family dispute, but it is not a valid application of statistical or mathematical methods.

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Clyde Oliver

Expertise

I can answer all questions up to, and including, graduate level mathematics. I do not have expertise in statistics (I can answer questions about the mathematical foundations of statistics). I am very much proficient in probability. I am not inclined to answer questions that appear to be homework, nor questions that are not meaningful or advanced in any way.

Experience

I am a PhD educated mathematician working in research at a major university.

Organizations
AMS

Publications
Various research journals of mathematics. Various talks & presentations (some short, some long), about either interesting classical material or about research work.

Education/Credentials
BA mathematics & physics, PhD mathematics from a top 20 US school.

Awards and Honors
Various honors related to grades, various fellowships & scholarships, awards for contributions to mathematics and education at my schools, etc.

Past/Present Clients
In the past, and as my career progresses, I have worked and continue to work as an educator and mentor to students of varying age levels, skill levels, and educational levels.

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