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BLOSSOMS MODULE
FLAW OF AVERAGES
Rhonda Jordan
Dan Livengood
R: Hello and welcome to our Blossoms module on the flaws of averages. My name is Rhonda Jordan and Im a graduate student here at the Massachusetts Institute of Technology in Cambridge, Massachusetts, USA.
We begin our module with two illustrations of some limitations of averages, which helped motivate us to make this module. Our first example is as follows:
On Thursday, I performed a modern dance, on campus, in this white dress. Then, Friday night, I went to MITs elegant grad gala in my favorite black dress. For Saturday night, I was going dancing with some friends and I decided to wear something that was the average color of the two dresses. So what did I end up wearing? This. Which has neither the elegance of the black dress, nor the simplicity of the white dress. My friend Dan, who is also a graduate student at MIT, has a quick story about a day he had recently that will help illustrate what we mean by the flaws of averages.
D: Thanks Rhonda! Hi! My name is Dan Livengood. And like Rhonda Im also a graduate student here at MIT.
For our second example, Id like to tell you a quick story about a day that I had recently. On that day, I spent four hours in the morning here at MIT. Then, in the afternoon, I spent four hours across the river in downtown Boston. So, if I were to stand in my average location over those eight hours, where would I be? Yep, Id be standing on the Charles River. At least there was a sailboat for me stand on! Hello again! Im Dan.
R: And Im Rhonda. As we said earlier, we are both graduate students at the Massachusetts Institute of Technology in Cambridge, MA, USA. Today were here to talk with you about the flaws of averages.
D: Now before we start we do want to be clear that were not trying to say that averages are bad. In many situations, averages provide a very good, single descriptive number of a situation. For example, whats the average height of all the students in your class?
R: Exactly! Now, Im sure you can come up with a number of other situations where the average number is a good description.
D: The main point of this Blossoms module is simply to point out a few pitfalls that could arise if youre not attentive to detail when youre using and interpreting averages.
R: So lets get started and consider a case where the average may not always be a good description of the real situation!
D: Sounds good.
R: So Dan, lets imagine that youre at the edge of a river that you want to cross. But, theres a sign. The sign says, Average river depth one meter. Now, given this sign, would you cross the river?
D: Hummm. Well, if I saw the sign I would think, Im 1.8 meters tall. Im taller than the average river depth. So I should be able to walk across and keep my head above the water all the way on average, right?
R: Thats true, if this is your situation. If this is the water and this is the riverbed, it goes straight across at one meter, then youll definitely be able to keep your head above the water.
D: OK. Well I can think of another example where Id be able to keep my head above the water. So again, if this is your water, and this is your new riverbed shape, if for the first half of the river its 0.5 meters deep, and then for the second half of the river its 1.5 meters deep, on average its still 1 meter deep. With only 1.5 meters as the maximum depth, Id keep my head above the water.
R: All right Dan. But heres the kicker. Lets say this is your riverbed. OK? And you start at zero meters here and the riverbed goes all the way down to two meters. The average depth is still one meter, but in this two meter area, if you cant swim, I think youd have a hard time keeping your head above the water, Dan.
D: You know, youre right, Rhonda. So class, what other riverbed shapes can you come up with? And for each one that you come up with, am I going to be able to keep my head above the water if I just decide to walk across the river, especially if I cant swim? Hummm.
R: Well let you guys discuss this with your teacher and your classmates. Well see you later!
R: Welcome back! I hope you had a nice discussion with your teacher about how the average is not always a good description of the actual situation. That is our flaw of averages number one.
D: And moving right along, our flaw of averages number two is that the function of the average is not necessarily the same as the average of the function.
R: Dan, what do you mean by that?
D: Well Rhonda, it just so happens that Ive got a nice little example to help explain this flaw of averages. And I think youre going to like it.
R: Sounds good.
D: So I have two plates of cookies.
R: Im SO hungry! Can I have some?
D: Sure. Ill let you have one plate and then Ill have the other.
R: OK. But like I said, Im really hungry so can I have the plate that will fill me up the most?
D: Sure! What Ill tell you is that underneath A is a plate with two circular cookies that have an average diameter of 7 cm. And underneath B theres a plate with two circular cookies that have an average diameter of 8 cm. So which plate would you like to have?
R: Oh Dan, thats easy. I want plate B. The average diameter of those cookies is larger.
D: OK. Lets look at the two cookies that youve just chosen. Now would you like to see the two cookies on plate A?
R: Sure, why not? Lets see.
D: Well, here you go. On plate A youll see that we have one VERY large cookie and one very small cookie.
R: Well, wait. The area on this plate is larger than the area of cookies on this plate. Thats not fair, Dan.
D: Well, thats the exactly the point of this flaw of averages. So class, do you understand why Rhonda ended up with a plate of smaller area of cookies even though she picked the plate with the larger diameter on average? Talk this over with your teacher and your class and well see you when you get back.
R: Hey Dan, whats that over there?
D: Whats what?
R: That, right there
D: I dont know.
D: Welcome back! So far weve discussed two flaws of averages. The first one was that the average is not always a good description of an actual situation. The second flaw of averages is that the function of the average is not always the same as the average of the function.
R: Now the third flaw that wed like to introduce is that the average depends on your perspective.
D: All right. So what do you mean the average depends on your perspective?
R: OK. Take me for instance Dan. So Im a dance teacher and I teach two dance classes, one beginners class and one advanced class. My beginners class has 45 dancers, but my advanced class has five dancers. So if we took the average of these two numbers, we need to add 45 + 5 and then divide by 2. We get 25. So Dan do you see how thats 25?
D: I do see how thats 25.
R: OK. But what if we actually interview all of the students and ask each one of them how many dancers are in your class including yourself? If we take the average of all of their responses, do you think wed get the same answer?
D: Well, yeah, I dont know. I dont see why it would be any different than the 25 that we just calculated.
R: Dan, I actually did ask all of my students this question. And these are their 50 responses: forty-five 45s, and five 5s. So I took the average of all of these numbers and I got 41.
D: Wait a minute! 41? But why is that so different than the 25 we calculated earlier?
R: Dan, thats exactly and precisely the key lesson of this segment. So class, why is it that when we calculate the average from my perspective you get 25 dancers. But if you calculate the average from the dancers perspective you get 41? Talk this over with your teacher and your classmates and well see you when we come back.
R: Hello again! We hope you all have enjoyed our module on the flaws of averages. Lets review what weve done today. Our first flaw of averages is that the average is not always a good description of the actual situation.
D: Our second flaw of averages is that the function of the average is not always the same as the average of the function.
R: Now our last flaw is that the average depends on your perspective.
D: Now, these three flaws that weve discussed today are not necessarily the only flaws of averages, and we encourage you to look and find other examples where averages can be misleading.
R: But we wanted you all to have a sense of a few of the pitfalls that might arise if you are not attentive to details when calculating and interpreting averages.
D: As we leave you today, we have one last example of that first flaw of averagesthat the average is not always a good description of the actual situation.
R: So in our free time Dan and I love to perform. Im a tap dancer,
D: And I sing.
R: And so were going to put the two together for you.
(Song and dance.)
R: Now if we took the average number of taps that I just tapped,
D: And take the average note that I just sang,
R: Youll be listening to the following:
(Another example.)
D: Now hopefully you agree with us that the first version was much more interesting than the average!
R: So now were going to say our good-byes and leave you with a longer version of our song and dance. Happy studying!
D: Goodbye!
(Song and dance.)
D: Hello there! Welcome to our flaws of averages Blossoms Modules teachers segment. Rhonda and I had a lot of fun putting this together, and were glad that youre interested in discussing this with your students. So, we wanted to go over a few of the things that we had in mind with the module. To begin with, we mention this in the video and we just want to stress that were not trying to say that averages are bad! Averages do have their place. At one point we discussed the average height of the students in your classroom. Thats just one example of how the averages could be very helpful. But we wanted to go over a few of the pitfalls that you could run into if youre not careful with using averages. So well go over each of the flaws of averages that we discussed. And dont forget you may have other flaws of averages that youve come up with, or that you think of, that you could go over with your students. So feel free to adjust and adapt this in any way that you find is helpful to explain this to your students. But again, we want to go over some of the things that we had in mind for you to go over in between the different segments. So well start off with our flaw of averages number one, that the average is not always a good description of the actual situation. Well be right back.
D: And were back to discuss our first flaw of averages, that the average value may not always be a good description of the actual situation. So what did we have in mind with this one?
You may have noticed that there are a few examples along the way to illustrate this point. The two of our opening segments, myself on the river, Rhonda in her dresses, both were designed to help illustrate this point. And we wanted to go over some of the things we felt that you could talk over with your students after this first segment. One of the concepts that you could talk about is that the average value may not be an actual outcome of the situation. So what do I mean by that?
Lets take this coin here. If I flip a head, Im going to give you a dollar. If I flip a tail, Im going to give you zero dollars. So if I were to flip this coin many times, on average Id be giving you 50 cents. But wait a minute, 50 cents is not one of the outcomes. The outcomes are either a dollar or zero dollars. So thats conceptually what we were trying to get at with this one.
Another concept that you could discuss with your students is that depending on the situation, the average may be exactly the same, but the distribution may be different. So if you have a more advanced class, one of the things we had in mind was you could discuss the normal distribution, exponential distribution, whatever kinds of distributions you want to look at. For example think of the crossing of the river. We had the flat line, we had the sloped line, we had the sort of double table top line. Both of those were essentially three different distributions that all have the same average value. So those are the kinds of things that we were thinking you could discuss with your students after this first flaw of averages. You may have some others and feel free to check out the website which will have some more examples of what we had in mind for these. But for now, thats it for this part. And now well go on to the second flaw of averages, that the function of the average is not necessarily the same as the average of the function.
D: And now we can talk about our second flaw of averages, that the function of the average is not always the same as the average of the function. So if youve looked at our other segments for the module you probably remember the example with the cookies. So we had plate A with two cookies and plate B with two cookies, where the average diameter for plate A of the two cookies was 7 cm and the average diameter of the two cookies on plate B was 8 cm. And as Rhonda helped us beautifully illustrate, intuitively you would think that the plate B set of cookies would be bigger in terms of area since the diameter is bigger. But as you saw, the cookies on plate A were actually bigger when we uncovered them. So that was just a nice way to illustrate this point that the function of the average is not always the same as the average of the function. Now weve left the math details to a file on the website. So if youre interested in discussing this in further detail with your students, feel free to go look at that page. We've also got another example about wind turbines, and Im sure you can think of many other examples of functions where again the function of the average is different than the average of the function. So we'll now move on to our last flaw of averages, that the average value depends on your perspective.
D: Our last flaw of averages to discuss with you here in the teachers segment is that the average depends on your perspective. So from the main video our example was Rhondas dance classes. And weve got a number of other examples on the website that you can look at. I wanted to bring to your attention one of the examples, though, about Lake Woebegone. Now Lake Woebegone is a fictional town from a radio program here in the US, and one of the key phrases about Lake Woebegone is that all of the children are above average. Now this may sound like an impossible statement, but again it all depends on your point of view. If the average that youre discussing is a national average, it could be possible that Lake Woebegones children could all be above average. We have more details on the website so feel free to go look at those. We hope youve enjoyed working with this module. Rhonda and had a wonderful time putting this together. We also hope you enjoyed our little song and dance at the end. Thats for your entertainment and for your enjoyment with your students. Also, if you have any curious students who are wondering, Did they actually figure out what the average note was that Dan sang or the average number of taps that Rhonda tapped? We actually did! And weve got a file on the website once again that has the full details, so feel free to discuss that over with your students on how we did it. And again, if you have other flaws of averages that come to mind, please feel free to share them or other examples that work out in between the two segments. Please dont hesitate to use those as well! We hope you had a wonderful time with our module, and once again Rhonda and I had a great time putting it together. Take care and have a good day!
END OF TRANSCRIPT
Blossoms Module PAGE 8
Flaw of Averages
6/24/09
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