A logic question regarding mathematics education

[Minoosh]

I'm teaching a high school-level statistics course. There is no class set of textbooks but I have a teachers edition I'm loosely using as a curriculum guide.

I was browsing through "measures of dispersal" and found this sentence:

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

I asked students if they could spot a logical flaw in the sentence. They couldn't immediately see it. That "because" bothered me. It seems both clauses are saying the same thing: Because the mean deviation is smaller, the mean deviation is smaller. It's not cause and effect; the book is just stating the same thing 2 different ways. "Begging the question" IMO.

Am I reading that sentence correctly, and is my analysis valid?

I'd like to challenges students with more such logic questions but it's really not the focus of the course. I have quite a bit of leeway though.

 

[psionl0]

Because the mean deviation is smaller, the mean deviation is individual deviations are smaller.

ftfy.

There is no logical flaw here except that the statement implies that all individual deviations are smaller.

 

[Startz]

I suspect the point of the question is that even though deviations are smaller on average for B, A might have some deviations that are larger than any in B.

 

[psionl0]

I suspect the point of the question is that even though deviations are smaller on average for B, A B might have some deviations that are larger than any in B A.

ftfy. I thought I made a similar point (though I didn't mention outliers).

 

[Startz]

(A) Thanks for the fix.
(B) Yep, you said it first. I hadn't read your post carefully enough.

 

[Minoosh]

It still seems to me that it's saying the individual data points collectively are closer to the mean. But I'm here to learn.

 

[psionl0]

It still seems to me that it's saying the individual data points collectively are closer to the mean. But I'm here to learn.

Yes, your text didn't spell out that a small mean deviation doesn't necessarily mean that ALL data points are close to the mean.

The text should have read:

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

 

[BowlOfRed]

Am I reading that sentence correctly

I would say that while yours is a valid interpretation, that outside a formal logic course I would prefer another one.

Instead of reading the statement "Because A, B" as describing why B exists, I would say it is describing the chain of how we can know or understand that B exists.

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.

 

[Minoosh]

Yes, your text didn't spell out that a small mean deviation doesn't necessarily mean that ALL data points are close to the mean.

The text should have read:

Yes, that makes sense. Most (but not necessarily all) will be closer the mean. Thanks for your response.

 

[jt512]

What kind of nonsensical textbook are you using? "Mean deviation" is meaningless. Mean deviation from what? The mean deviation from the mean? That is necessarily 0 for any distribution that actually has a mean. So it would be impossible for two distributions to differ in "mean distribution." The authors are either utterly misguided or you haven't given us enough context to understand what the authors are talking about.

 

[psionl0]

What kind of nonsensical textbook are you using? "Mean deviation" is meaningless. Mean deviation from what? The mean deviation from the mean? That is necessarily 0 for any distribution that actually has a mean. So it would be impossible for two distributions to differ in "mean distribution." The authors are either utterly misguided or you haven't given us enough context to understand what the authors are talking about.

Mean deviation is actually a thing. Just as variance is the mean of the squares of the deviations, mean deviation is the mean of the absolute values of the deviations.

https://www.quora.com/What-is-difference-between-standard-deviation-and-mean-deviation

 

[jt512]

Mean deviation is actually a thing. Just as variance is the mean of the squares of the deviations, mean deviation is the mean of the absolute values of the deviations.

https://www.quora.com/What-is-difference-between-standard-deviation-and-mean-deviation

Mean deviation is 0 (assuming we're talking about deviations about the mean). Mean absolute deviation is another thing.

 

[psionl0]

Mean deviation is 0. Mean absolute deviation is another thing.

Ah. It's a dictionary thing.

 

[jt512]

Ah. It's a dictionary thing.

I have no idea what a "dictionary thing" is. It is not a "thing" in any statistics textbook I have ever encountered, and I have encountered scores of them, at least.

 

[jt512]

Yes, your text didn't spell out that a small mean deviation doesn't necessarily mean that ALL data points are close to the mean.

The text should have read:

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

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.

 

[Minoosh]

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.

"Mean deviation" per this book uses absolute values. However we quickly moved on to standard deviation to avoid the pitfall that you are talking about.

From Wolfram MathWorld:

The mean deviation (also called the mean absolute deviation) is the mean of the absolute deviations of a set of data about the data's mean. For a sample size , the mean deviation is defined by. ... Mean deviation is an important descriptive statistic that is not frequently encountered in mathematical statistics.

From a website called mathisfun.com, I got a very good explanation of the pitfalls of this measure, including your point that defined one way "mean deviation" is useless. But that definition is not universal.

I like to see where these conversations go because it shows a passion for clarity that I share, but also because I see the disagreements and they make me think. However, I can't be puzzling this out in real time with large groups of students. With small groups I can, but that usually isn't the situation.

 

[Minoosh]

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.

 

[Minoosh]

deleted

 

[Minoosh]

Yes, that makes sense. Most (but not necessarily all) will be closer the mean. Thanks for your response.

I can also see why this statement *wouldn't* be true. There aren't any crazy outliers in this data set, so while not bulletproof, the statement does happen to apply to most problems these students will encounter.

I am using the statistics section of the only teachers-edition textbook I have, supplemented by Internet resources, and with my understanding informed by some of the discussions in this sub-forum. I passed what to me seemed like a very hard test (VHT) and as a result I am considered qualified to teach mathematics to teenagers. Others may disagree, but I am by definition qualified.

This is not college. I don't stand up doing examples on the white board while everyone scribbles in silence. Not even close. There is no textbook. The best single resource I can refer students to is probably khan academy.org.

 

[psionl0]

I have no idea what a "dictionary thing" is.

Sure you do. I have never heard it described other than "mean deviation". The word "absolute" goes without saying since mean "signed" deviation is by definition zero.

Of course, mean deviation is seldom used in statistics because variance or standard deviation have more useful mathematical properties.

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.