Donald Burrill wrote:
> Hi, Bob. Welcome back! Thanks. It is fun to see so many familiar names. > > > Don't have time for an extended discourse just now, but am thinking > about it. Did have one quick reaction, below: > > On Thu, 19 Feb 2004, Robert Frick wrote in part: > > > So I was looking for a formula that did not involve addition. Maybe > > the first moment (terminology?) is the more natural formula? The > > first moment, if I have my terminology correct, is the number such > > that summation of each number minus the first moment is zero. > > Sorry to disillusion you, but "summation" surely involves addition. I noticed my argument had a little flaw there <grin>. I think my claim is this. When you define the average the way I did, then summation is in the formula, but it isn't needed to calculate average, it's just needed to check the average. So if I give you the set <60, 62, 65, 69, 69 , 71> you guess 65, and if that's right, the summation of the differences adds to zero. You don't even need to divide by n. So my argument is that the summation isn't needed to calculate the average, it's just needed to check it. So it isn't a salient part of the formula. > > > > That formula does not make addition salient -- the only addition is in > > the error term, and everything is supposed to cancel out to zero. > > Only after having done (or at least thought about doing) a subtraction > for every datum. And subtraction is a subset of addition: if one > permits the objects being added to be signed numbers, "subtraction" as a > separate operation becomes entirely unecessary. Yes, I am tutoring a high school student whose foundation is so bad that he does not see that subtraction is just undoing addition. But for meaningfulness, subtraction usually beats addition. One example is on an interval scale, such as Farenheit, where subtraction is meaningful and addition is not. But I think that's a red herring, the real issue is that subtraction is often going to be meaningful when addition is not for many meaningful measures. For example, if one student has a 70% on a test and another student has an 80%, the difference is meaningful, and the sum is not. Similarly, if we are given 2 heights, the difference between the numbers is the difference in heights. Here, the addition of the two numbers is meaningful -- it represents the height when they are standing on top of each other. But when students calculate an average, and they first add 10 heights, are they really supposed to be visualizing 10 people standing on one another? And does that really help them understand the concept of average? No. When we ask students to add numbers to calculate an average, the sum is not meant to be meaningful. If we ask students to subtract a number from their guess of the center (or the real center), the are calculating a meaningful number -- the amount that number differs from the supposed center (or real center). Now I recall how addictive this list can be. Thanks again for everyone's attention and comments. Bob . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
