Measuring Things — Question 1

Question 1

What exactly is being measured?

Three students are asked to report “the size” of the same smooth, convex, rigid pebble. One reports its longest span, one its greatest width perpendicular to that span, and one its thickness perpendicular to both. Their correctly measured results are shown below.

1. Do the three different numbers establish that at least one student measured incorrectly? Explain.

2. Rewrite the instruction so that another laboratory can reproduce the intended longest-span measurement. Define the endpoint choice, orientation search, contact condition, and reporting unit.

3. A fourth student averages the three spans and calls the result “the diameter.” Calculate this average. Is it a legitimate reported quantity? Is it the requested longest span?

Do not assume that the pebble is a sphere or that its volume is known.

Question 1 - Solution

What exactly is being measured?

(a) Identify the measurands.

$$d_1=24.0\ \mathrm{mm},\quad d_2=18.0\ \mathrm{mm},\quad d_3=12.0\ \mathrm{mm};\qquad d_1\ne d_2\ne d_3.$$

A measurand is the quantity intended to be measured. Longest span, perpendicular width, and thickness are different measurands. Different numerical results therefore do not, by themselves, demonstrate measurement error.

(b) An operational definition.

$$d_{\max}=\max_{p,q\in S}\|p-q\|,\qquad S=\text{pebble surface}.$$

One valid instruction is: “Find the greatest straight-line distance between two points on the pebble’s surface. Rotate the rigid pebble and search orientations until no larger span is found; use opposing parallel contacts with only gentle contact, avoiding deformation. Report the maximum separation in millimeters, with the instrument resolution and a stated reading bound.” For this smooth convex pebble, the largest parallel-contact separation gives its longest span. The operational definition makes clear that a perimeter distance along the surface is not intended. Repeat the search to check that a larger orientation was not missed.

(c) A defined average is not the same quantity.

$$\overline d=\frac{24.0+18.0+12.0}{3}=\boxed{18.0\ \mathrm{mm}}.$$ This is a legitimate mean of these three specified orthogonal spans if labeled that way. “Diameter” alone is ambiguous for a nonspherical object. It is not the longest span, which is $\boxed{24.0\ \mathrm{mm}}$.

Check: Defining a quantity clearly is necessary even when every instrument reading is correct.