A logic question regarding mathematics education

[jt512]

Another term that baffles me is "experimental probability." That will come up soon. I don't know why what actually happens is defined as probability at all. Why isn't it just "what really happened"?

I welcome your thoughts.

With terms like "mean deviation" and "experimental probability," I fear that you are using a terrible textbook. What textbook is this? And can you provide the context in which they use the term "experimental probability"?

 

[psionl0]

Another term that baffles me is "experimental probability." That will come up soon. I don't know why what actually happens is defined as probability at all. Why isn't it just "what really happened"?

I welcome your thoughts.

You will probably find a better definition than I can provide but for now just consider "experimental probability" as the probability deduced from running experiments.

It is what you have to do if you can't calculate a probability. You can run an experiment repeatedly and the proportion of "favourable" outcomes becomes the probability of that outcome.

A typical example is polling. You may find that the "experimental probability" that a person respects Donald Trump is 0.0000000000001

 

[Saggy]

'Because tire B has a smaller mean deviation than tire A, the individual values for tire B deviate less from the mean.'

Rewriting .... without the confounding 'deviation' ....

Because tier (?) B has smaller average height than tier A, the individuals in tier B are shorter than those in tier B.

Is it true? Clearly not. As it reads it says that all individuals in tier B are shorter than every individual in tier A.


You flip a coin 50 times, it comes up tails 20 times. The experimentally determined probability that a coin flip with the coin will produce tails is 20/50.

 

[jt512]

Even assuming that by "mean deviation" the authors meant mean absolute deviation (MAD), the above is not true. Given a set of observations of random variables X_i and Y_i, i=1,...n, MAD(X) < MAD(Y) does not imply that most of the X_i's are closer to their sample means than most of the Y_i's are to theirs. Either the authors are wrong, or we need to see the context in which they made the statement to understand what they meant.

You are welcome to provide a counter example.

X <- rep(c(-1, 1), each=5)
print(X)
#  [1] -1 -1 -1 -1 -1  1  1  1  1  1

Y <- c(-10, rep(0, 8), 10)
print(Y)
#  [1] -10   0   0   0   0   0   0   0   0  10

# MAD(X)
mean(abs(X - mean(X)))
# [1] 1

# MAD(Y)
mean(abs(Y - mean(Y)))
# [1] 2

# |X_i - mean(X)| = 1 for all X_i :
abs(X - mean(X))
#  [1] 1 1 1 1 1 1 1 1 1 1

# Proportion of Y_i's with |Y_i - mean(Y)| ≤ 1
mean(abs(Y - mean(Y)) <= 1)
# [1] 0.8

 

[Minoosh]

With terms like "mean deviation" and "experimental probability," I fear that you are using a terrible textbook. What textbook is this? And can you provide the context in which they use the term "experimental probability"?

That isn't even from the same textbook. khan academy, that wolfram site, many other Web references recognize both "experimental probability" and "mean deviation." I don't slavishly follow one source and I didn't invent these terms. Neither did the book I'm working out of.

 

[jt512]

That isn't even from the same textbook. khan academy, that wolfram site, many other Web references recognize both "experimental probability" and "mean deviation." I don't slavishly follow one source and I didn't invent these terms. Neither did the book I'm working out of.

I'm trying to help. You're the one who said you didn't understand the term "experimental probability." Neither do I, because there is no such thing in statistics. Perhaps there is in engineering.

 

[psionl0]

print(X)
#  [1] -1 -1 -1 -1 -1  1  1  1  1  1

print(Y)
#  [1] -10   0   0   0   0   0   0   0   0  10

I thought that you were going to come up with a data set with massive outliers.

It might make you logically correct but that doesn't make the inference unreasonable less contrived cases. Your example simply highlights one of the problems of the mean - that it is badly affected by extreme outliers.

In the course of experimentation, outliers like that are often discarded as being unrepresentative of the set and distorting any reasonable conclusion might otherwise be made. In this example, set Y would be seen to have a lower deviation than a raw calculation might indicate.

I'm trying to help.

 

[jt512]

I thought that you were going to come up with a data set with massive outliers.

I don't know why you would think that.

It might make you logically correct...

That's what it means to be correct in mathematics. The statement, "Because MAD(X) < MAD(Y), most of the observations in X are closer to their means than most of the observations in Y" is false.

but that doesn't make the inference unreasonable less contrived cases.

And that doesn't make the statement true.

Your example simply highlights one of the problems of the mean - that it is badly affected by extreme outliers.

My example shows that the statement is false.

In the course of experimentation, outliers like that are often discarded as being unrepresentative of the set and distorting any reasonable conclusion might otherwise be made. In this example, set Y would be seen to have a lower deviation than a raw calculation might indicate.

 

[psionl0]

I don't know why you would think that.

Because there is no way that you could find a real world example that defies the general maxim.

Keep on "helping".

 

[Modified]

Because there is no way that you could find a real world example that defies the general maxim.

Keep on "helping".

"Real world" examples like this happen all the time. Having a small number of extreme outliers is a common thing.

 

[jt512]

Because there is no way that you could find a real world example that defies the general maxim.

It doesn't matter, she's teaching statistics, not engineering. Regardless, samples with most of the observations in the tails are routinely seen in real-world applications of Bayesian statistics.

 

[Minoosh]

I'm trying to help. You're the one who said you didn't understand the term "experimental probability." Neither do I, because there is no such thing in statistics. Perhaps there is in engineering.

My first recollection of it was when I was working out of a study guide intended to help students pass the state's then-standard exit exam. That was in 2014. It puzzled me, but I Googled it and it seemed to be a thing. Now I have a teachers edition of a 2001 Algebra 2 textbook from Holt, Rinehart and Winston. A couple of hours ago I saw it referenced on Khan Academy. I see it on IXL Math, another site that helps students practice. There are many references online.

From page 629 of the BAB (big-ass book):

In mathematics, the probability of an event can be assigned in two ways: experimentally (inductively) or theoretically (deductively) ... Experimental probability is approximated by performing trials and recording the ratio of the number of occurrences of the event to the number of trials.

It doesn't make sense to me either but there it is.

There no longer is an exit exam. There's testing, but it won't keep a student from graduating. It might hurt my data but not theirs.

 

[Minoosh]

Dollar-store dice are not really fair, at least not in the number of trials I've been able to do in class (usually with small groups). We kept at it and the number 4 was definitely overrepresented. It seems like casinos would be picky about that kind of thing.

 

[jt512]

My first recollection of it was when I was working out of a study guide intended to help students pass the state's then-standard exit exam. That was in 2014. It puzzled me, but I Googled it and it seemed to be a thing. Now I have a teachers edition of a 2001 Algebra 2 textbook from Holt, Rinehart and Winston. A couple of hours ago I saw it referenced on Khan Academy. I see it on IXL Math, another site that helps students practice. There are many references online.

From page 629 of the BAB (big-ass book):

In mathematics, the probability of an event can be assigned in two ways: experimentally (inductively) or theoretically (deductively) ... Experimental probability is approximated by performing trials and recording the ratio of the number of occurrences of the event to the number of trials.

It doesn't make sense to me either but there it is.

It makes no theoretical sense, and so it shouldn't be there, or anywhere else. The problem is the word "assigned." It treats probability as something we give to the coin, rather than a property of the coin that we are attempting to infer.

In classical statistics the probability of an event, say that event that a tossed coin lands heads, is the long-term relative frequency with which the event occurs. Roughly speaking, if we flipped the coin literally infinitely many times, the probability would be the proportion of times that the coin landed heads. However, we can ever only flip the coin a finite number of times, so the true probability of it landing heads is unobservable. This leaves us with two alternatives: (1) we can attempt to deduce the probability from the physical properties of the coin, or (2) we can estimate the probability by carrying out an experiment.* But in neither of these cases are we "assigning" to the coin its probability. That probability is an inherent property of the coin itself. Saying that we assign probabilities to the coin obfuscates the fact that we are attempting to ascertain a property of the coin.

*A third method, Bayesian statistics, would combine (1) and (2).

 

[jt512]

Dollar-store dice are not really fair, at least not in the number of trials I've been able to do in class (usually with small groups). We kept at it and the number 4 was definitely overrepresented. It seems like casinos would be picky about that kind of thing.

Although I don't know why you made that post in this thread, it is an interesting factoid. I'm surprised, though, that you were able to detect a bias using the number of trials that you could feasibly do in class. I'm also surprised that four was favored. I would guess it would be either six or one, since this pair of opposing faces are the most dissimilar of the three opposing pairs.

 

[Modified]

Although I don't know why you made that post in this thread, it is an interesting factoid. I'm surprised, though, that you were able to detect a bias using the number of trials that you could feasibly do in class. I'm also surprised that four was favored. I would guess it would be either six or one, since this pair of opposing faces are the most dissimilar of the three opposing pairs.

Usually one is favored, by quite a lot. If four was favored, then the die minus the pips must have been significantly asymmetrical, either in material density or shape. Casino dice, of course, are extremely close to fair when new.

 

[psionl0]

In mathematics, the probability of an event can be assigned in two ways: experimentally (inductively) or theoretically (deductively) ... Experimental probability is approximated by performing trials and recording the ratio of the number of occurrences of the event to the number of trials.

It doesn't make sense to me either but there it is.

It is straightforward enough if you don't get bogged down in semantics. You toss a coin a number of times and count how many times it lands up heads. By dividing the number of heads by the number of tosses, you get an estimate of the probability that the coin will land up heads.

To be accurate, you would say that this probability is approximate (p(heads) = 0.5 doesn't mean that every second toss will be heads). A more detailed discussion will involve "confidence intervals" but there is no need to overload the students at this stage.

 

[Minoosh]

In a (non-formal) proof, I might say: "Because the triangles are similar, the obtuse angle of each is equal". More precise would be to add in "...we can deduce.." or "...we can assume..." or "...we therefore know..." in there, but in most cases this is understood.

This example is very helpful. I don't need to be picking apart every assertion in the book ... they are close enough, and though nitpicking would be fun with a bright student, it's not necessary for my purposes.

 

[Minoosh]

Usually one is favored, by quite a lot. If four was favored, then the die minus the pips must have been significantly asymmetrical, either in material density or shape. Casino dice, of course, are extremely close to fair when new.

Dollar store. They could have been lopsided to begin with, plus the heat where I live can warp plastic if I leave supplies in the car.

Although I don't know why you made that post in this thread, it is an interesting factoid.

It started as a longer post about the relationship between engineering and statistics. These exercises are supposed to demonstrate the law of large numbers. We probably didn't do enough. But I was also pretty sure that properly engineered dice would eventually demonstrate true randomness.

It is straightforward enough if you don't get bogged down in semantics. You toss a coin a number of times and count how many times it lands up heads. By dividing the number of heads by the number of tosses, you get an estimate of the probability that the coin will land up heads.

I do get bogged down in semantics, way too much. In my searches I found a paper called "Why are normal distribution normal?" that I enjoyed although I could not understand all of it.

 

[jt512]

It is straightforward enough if you don't get bogged down in semantics.

I do get bogged down in semantics, way too much.

There is no such thing as getting bogged down in semantics in mathematics. A mathematical field is basically a set of definitions and a set of theorems derived from the definitions. If a definition or a theorem is not precisely stated, the theorems on which it depends will be ill-defined.

Minoosh, your mathematical intuitions are good. You caught the fact that the statement about the mean (absolute) deviation implying something about the distribution of the individual deviations was false. You caught the fact that there was something weird about the concept of an "experimental probability." The fact is that the sources you are getting this stuff from are mathematically sloppy, and you're intuitively grasping the sloppiness and objecting to it.

I wish I remembered what statistics textbook I had in high school (it's probably out of print anyway). It was about as mathematically rigorous as a non-calculus-based statistics text can be. You need to find a book like that to educate yourself on the subject. Then you will not be fooled by "experimental probabilities" and false assertions about mean (absolute) deviations.