Adventures in Measurement

It's often important to quantify the properties of objects around us. For example, if we were selling apples, we would need to have a way to quantify them in order to set consistent prices. We might decide to quantify the apples by number, weight, or volume. Maybe we'll sell 3 for $1, or 50¢ per pound, or $5.00 per peck.

The study of the science of measuring is called metrology. Metrologists concern themselves with several important questions: How can we quantify the properties of objects? What units of quantity are meaningful, and how do we measure them? How do we apply these methods of measuring in the real world? How can we create measurement systems and apply them in a manner that helps regulate trade, taxation, safety, etc.?

How do we assure that measures are accurate? For example, if I'm pricing apples by weight, how can my customers be assured that they're getting a fair deal?

The scale I use to measure the weight of the apples is calibrated to a certain accuracy, which assures that the "pound" of apples I sell you isn't lighter than the "pound" of apples somebody else sells you, at least within a certain margin of error. The pound is officially defined as 0.45359237 kilograms, and the kilogram is officially defined in relation to an artifact, the International Prototype Kilogram (IPK). "The IPK is the kilogram... [it] is made of a platinum-iridium alloy and is stored in a vault at the BIPM in Sèvres, France." The United States owns three replicas of the IPK, housed at NIST (National Institute of Standards and Technology).

How accurate is the scale I use to measure apples? There are NIST standards which the scale would have to meet before I could use it to sell apples. At a minimum, it would have to be accurate enough that the uncertainty of the measurement (e.g., x lbs ± y lbs) would not impact the cost of the product. So, for an uncertainty of y when buying x pounds of apples, the cost of x+y lbs should be the same as the cost of x-y lbs.

How could that be, you may ask, since x+y ≠ x-y (unless y=0)? Well, when we deal with money, we don't pay fractions of a penny; merchants round to the nearest penny. In other words, if I charge 50¢/lb, then if I weigh out an amount of apples a that has a "true" cost c such that 49.5 ≤ c < 50.5. If c is defined by the previous inequality, and c = 50 a, then what are the upper and lower limits on a?

Well, if c = 49.5, then 50 a = 49.5, and therefore a = 0.99. At the other extreme, if c = 50.5, then 50 a = 50.5, and therefore a= 1.01. Thus my scale would need to be calibrated such that it measures one pound to within an accuracy of 0.01 pounds. If we convert this to relative error, its percent error must not exceed 1%.

For measuring apples, we would need a much higher accuracy than we would need for measuring the weight of trucks at a highway weigh station. We don't need to know the weight of the truck to within a hundredth of a pound, but we might still need the same relative error (i.e., 1%).

Also, sometimes we don't need to know the exact size of something, we just need a ballpark figure. For example, if we're trying to fit a couch into the back of a pickup truck, we just need to make sure that the couch is shorter than the length of the truck bed.

Other times, we want the most accurate measure we can get, but are limited by uncertainty or human error. For example, if we want to measure the length of an ant, but all we have to do it with is a yardstick, we will be limited by the size of the calibrations on the yardstick, and by the ability of our eyes to see something so small. If we had better equipment (i.e. a smaller, more finely calibrated ruler, and a magnifying glass) we would get a more accurate measurement.

One of the things that makes me roll my eyes every time I watch American football is the questionable way in which the status of the downs is sometimes determined. The referees have a chain of length ten yards, anchored by bright orange poles at each end, which is handled by the chain crew. The chain crew holds one end of the chain on the sideline at a point that is parallel to the location where the ball starts upon first down. The length of the chain is stretched along the sideline in the direction the team is driving, to determine whether the team has made a first down. Sometimes, when it is a close call, the chain crew moves the chain onto the field, and the referees measure with it, to see if the first down has been achieved. Sometimes, by a matter of inches, the team doesn't make the down.

There are some problems with the manner in which the status of the down is determined. We need to see the errors inherent in the system, which compound to make this measurement wrong.

First, the starting point at which the chain is placed is lined up visually with the location of the ball more than twenty-five yards away (In the NFL, the field is 160 feet wide). If the angle between the ball, the end of the chain, and the sideline is off by one tenth of a degree (i.e. it's 89.9° or 91.1°), then the difference between the starting point of the chain and the true starting point of the ball is off by more than 1.5 inches.

Second, when the down attempt is over, the referee generally places the ball in the place he believes is the point of farthest advance. How accurate is his placement? This is something I don't know, but it can't be more accurate than within a few inches. Let's say within three inches for the sake of argument.

Let's assume that the length of the chain is perfectly accurate. (It is probably off by some small factor; also, we are neglecting shrinkage or expansion due to temperature, but that's okay.) Even so, in the worst possible case, the starting point of the chain was 1.5 inches ahead of the starting point of the ball, and the referee put the ball down three inches behind the line of farthest advance, meaning that the difference between the true advancement and the measured advancement exceeded 4.5 inches. Given that a football is about 11 inches in length, that means the measurement could be off by more than 2/5ths of a football length! I've seen downs decided by less than that!

So what happens in football is that they think they have an accurate measurement of the lengths involved, but in reality, the fate of a team's advancement down the field is determined by little more than luck.

This is a real-life example of an organization needing a consultation with metrologists. If I were the football commissioner, that would be one of the first things I did.

Metrology is a fascinating subject. It's about more than weights and lengths; metrologists also determine how to measure volume, time, energy, work, and many more qualities. If I weren't a computational scientist, I think I'd want to be a metrologist. I hope you enjoyed this foray into metrology as much as I did!