Absolute and Relative

Think of a large number. Did you? Good. Now is it really large? Large enough?

While reading the questions, you must have felt something amiss. A Large number of what? Large enough for what? There is no number that is large by itself - you can easily multiply it by a thousand, and presto, your original number becomes puny compared to it. There is no limit beyond which every number is a large number. You can talk about larger number or smaller number. These words are relative. The numbers themselves are absolute. The funny part is that a good relative comparison can be done for only absolute quantities. This is a very subtle point. It is like if you want to compare height, you want to look down and see if they are standing on the same ground. The word “large” lives in the gray area between absolute and relative. You have to ask large number of what? Ten million dollars can certainly be large amount of money, but the same number of raindrops might not even be registered. The word large challenges us by not only to make sure that the count is substantial, but it is significant. Sometimes this distinction is a little slippery.

Let’s look at the crime statistics in a hypothetical area. We read that in the last 10 years number of crimes went from 2000 a year to 2100 a year. You would say the situation has clearly deteriorated. The media will sound alarms, and theories of how exposure to television and free access to guns leads to increased violence would gain ground. But as any other numbers, they are relative. 2100 is larger than 2000 - any fifth grader can tell you that. The question is, do the numbers stand on their own, are they absolute? If you are smarter than a fifth grader, you would notice that in the same 10 years, the population of that area went from 2 million to 2.3 million. The crime rate has actually gone down, from 10 in 10000 people to about 9 in 10000. This is about 10% improvement. So the crime rate, is an absolute quantity that can compared from a society to society, from time to time. It is something like density. We usually say iron is heavier than water - what we really mean is iron is denser than water. If you took 10 pounds of iron and 10 pounds of water, they will weigh the same. Density is in some sense more absolute properties of the two materials - it is not relative to the amount.

The problem is that we usually talk about anything in numbers, and since numbers in themselves have the sense of absoluteness, it is easy to give that absoluteness to what they measure. But the most important thing to remember is this numbers are absolute only when they don’t have units. Once they acquire units, they lose that high priestly position, and become pedantic. One must look at the units to determine whether we can compare them without any problem.

When can we compare two numbers? The simple answer is that you need to do apple to apple comparison. All things being equal, 2100 crimes are more than 2000 crimes. However, whenever some of the things change, you need to take the change into account. In our example, the population changed. You can take this change out of the equation by dividing – like finding density. The reason for this is we want useful information by doing comparison. We want to know whether the crime situation in that particular area has improved over the given 10 year period. The assumption is that the number correctly represents the crime situation. This assumption is really a big assumption. In fact it incorporates multiple assumptions like
- The population did not change
- Reporting stayed exactly the same - (for every 5 crimes, 4 were actually reported)
- The crimes were similar in composition (there could be 1000 murders and 1000 petty theft 10 years ago, and now there are 300 murders and 1800 petty thefts)

Usually for a large enough population (there go the relative term again...) compositions remain similar over a short enough time. But if over the 10 years improved technology or improved awareness, and increased belief in justice system, let's say higher proportion of crimes got reported, then in fact the real crimes might be going down, but the reporting might increase. This is an irony of good things leading to bad perception.

I am not trying to say that nothing is comparable. You can do meaningful and useful comparisons. You just need to make sure we take the right relationships into account.