What is the mean of this data: 6, 6, 6, 12, 18, 18, 42, 80, 80, 80? We can agree the answer is their sum (348) divided by their count (10), or [(3)(6) + (1)(12) + (2)(18) + (1)(42) + (3)(80)]/10, which comes to 34.8. So, what does this number mean (no pun intended) and why is it important? Certainly, it is the single number that can replace every term and still sum to the same 348. Moreover, ten rectangles of width 1, having these terms as height, would have a total area the same as a single rectangle of height 34.8 and width 10. Thus, the idea lends itself to notions ranging from average test scores to the Mean Value Theorem in the integral calculus.
A wonderful way to impress students with the significance of the mean is to demonstrate that it represents the balancing point (i.e., the centroid or center of mass) of a system of masses on a “negligible” lever. According to the physics, let us suppose we have such a negligible lever and that we have designated some position of it as its origin. Now, imagine that we place a mass of 3 kg at the 6 cm mark, 1 kg at the 12 cm mark, 2 kg at 18 cm, 1 kg at 42 cm, and 3 kg at 80 cm. Were we to place a fulcrum under this lever at the 34.8 cm position, then the “system” would balance because the first moment on the left would exactly equal the first moment on the right:
3(34.8 – 6) + 1(34.8 – 12) + 2(34.8 – 18) = 1(42 – 34.8) + 3(80 – 34.8)
(both sides are 142.8). In this we see a truly phenomenological reason that the center of mass C of such a system is determined by the famous formula C = the sum of the products of mass times position, divided by the total mass of the system. [In physics, the sum of the products of the mass times position is called the “first moment of the system.”] And this is precisely C = [(3)(6) + (1)(12) + (2)(18) + (1)(42) + (3)(80)]/10.
However, there is no such thing as a “negligible” lever. (That is, any lever we would use is itself a distribution of mass, so the true centroid can only be found using the calculus because we cannot discard the lever itself.) So how would students know that any of this is true?
But there is a way to do this, and it is very simple. All we need to do is to “take the lever itself out of consideration”; and here is how we do that: We need two meter sticks; call them L and R. One of them will be designated our “negligible lever,” L; the other, just our “ruler,” R. We begin by finding and marking the center of mass of stick L. Again, a ChapStick provides an excellent fulcrum. We must do this because the typical meter stick might not be so uniform that its center of mass is at exactly its center, the 50 cm mark. Often the actual centroid is half or more of a centimeter off of the 50 cm mark.
We now need 10 identical masses. Quarters (i.e., US 25 cent coins) work outstanding well; they are amazingly uniform. (Do not use, for example, ordinary washers. They might appear to be identical, but they are typically far from uniform in mass!) Now, lay down the “ruler” meter stick R. Place 3 quarters tangent to this stick at 6 cm mark; place 1 quarter tangent to the 12 cm mark; place 2 quarters at 18 cm; place 1 quarter at 42 cm; and place 3 quarters at 80 cm. Now, lay down the lever meter stick L, on the other side of these quarters in such a way that its center of mass, which we marked earlier, is perfectly aligned with our “mean” position, 34.8 cm, of the ruler stick R. (You should understand why, doing this, effectively removes the “negligible” ruler L from consideration as part of the “system” of masses.) Now, carefully transfer the 10 quarters onto our negligible lever meter stick, L. Once you do that, slide the ruler meter stick R up against L (being sure the centroid of L is right at the center of mass position, 34.8 cm, of R) to be sure that those quarters are still arranged tangential to R at the appropriate positions of R.
The students can then carefully raise the now “negligible” lever L, holding the 10 quarters, and place our ChapStick fulcrum under its centroid. The lever will balance when the students let go of it, and the crowd will go wild! [For the photo below, I did not have two meter sticks at home; I had one meter stick and one yard stick. So, I used the yard stick as the “negligible lever.” Note that its center of mass (black arrow) is off of its midpoint (the 18-inch mark) by about a quarter of an inch. The five stacks of quarters seen in the photo actually hold the ten aforementioned quarters. (The pink strips are covering some advertisements.)]
Again, we see that we have:
Q. A real world question. We place small stacks of quarters along various positions of a meter stick, and ask the students to “Determine the center of mass of this system.”
R. This problem is answered via the formula C = the first moment of the system (w.r.t. an origin) divided by the total mass of the system.
V. The solution is verified by seeing that the system really does balance, once the lever itself is removed from consideration as described above.