The Number Game: Uses of Quantitative Reasoning

Quantitative Reasoning  can be define practically as a psychometric test that is complex and extensively advanced. And as such, it readily measures a person’s ability to use mathematical skills in order to solve equations.

When you can measure what you are speaking about and express it in numbers, you know something about it, and when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind. (Lord Kelvin)

Quantification sharpens the image of things seen by the mind’s eye, both external phenomena and internal conceptions, but it is useless or worse until the right
things are in the field of viewpoint. (R. W. Gerard).


THE WORD “quantitative” from quantitative reasoning, means measurable in numbers, as opposed to “qualitative,” which refers to verbal description. Although not every aspect of science is quantitative, the sciences are certainly more quantitative than other intellectual pursuits like literature or philosophy and even Information technology in some cases.

Scientific discourse is also more quantitative than typical everyday conversations. Why should this be so? What is gained by the process of reducing qualities to numbers, and what is lost? One major advantage of quantification is exactitude. Instead of saying that an elephant is heavy or that an atom is small, we can provide a number for the mass of the elephant or the size of the atom. (Of course the matter is a bit more complicated because no measured number is really exact, but instead is only as good as the measurement that we made to get it.

Our number may not be exact, but we have clearly gained in precision by switching from a word like “heavy” to a number like 1850 kg for the mass of our elephant.

We have actually gained something even more important than precision because the word “heavy” has no meaning by itself. An elephant is heavy compared to a hummingbird but is not heavy compared to Mt. Everest.

Is an elephant heavy? Having a number for the mass of the elephant, we can now compare this number, quantitatively, to the masses of other things. We can say which is heavier, and we can say by how much. Our comparisons are unambiguous now, so we have gained in clarity as well as precision.These advantages are found in everyday life as well as in science.

Having a number for your bank account balance is probably better than feeling
wealthy or poor. In scientific work, however, the matter becomes crucially important because a defining characteristic of science is that we compare our understanding of nature with the observation of nature. A qualitative agreement between the two might be possible for many different understandings, not all of them correct.

Quantitative agreement between a predicted number and a measured number is much less likely to be the result of incorrect thinking. Physicists of the nineteenth century, for example, predicted that a hot hollow body with a small hole would emit infrared radiation out of the hole. Their theories also predicted, quantitatively, how much radiation should come out at each wavelength. Here is a collection of Quantitative reasoning questions and answers. Try it and see the result.

At short wavelengths, the predictions disagreed with the experimental measurements of the radiation. This seemingly minor quantitative discrepancy was the beginning of the revolution  we now know as quantum theory, the technological fruits of which include lasers and microelectronic computer chips.

Example: The Area Needed for Solar Cells
More subtle and less obvious advantages also result from quantification and numerical work. It’s possible to simplify complicated chains of reasoning, which would otherwise be difficult to carry out, by reducing them to a set of numerical computations.


Other Gains and Losses

If the phenomena we are trying to understand have been reduced to numbers,
then the behavior of the phenomena can be represented by mathematical
relationships among the numbers. Not only do the mathematical forms bring order and simplicity to complex phenomena, but manipulation of the mathematical relations can reveal new and previously unsuspected behaviors.

We have now crossed the bridge from quantification to the role of mathematical thought itself in the sciences, so we’ll not proceed further here. Instead, we will return to the question of whether anything is lost in the process of reducing the world to numbers. We certainly pay a price in the sense that we have lost both the raw sensory experience of
what we study and any aesthetic dimensions it may possess.

I prefer listening to the music generated from the quantitative information stored
on a compact disc, rather than counting the etch pits on the disc. In this sense, many things outside the realm of science cannot be quantified still retain their true meaning. A more interesting question is whether anything of genuine scientific interest is not amenable to quantification.

This question may be more controversial, but it seems clear to me that the answer is yes. Taxonomy, for example, is the science of classifying plants and animals into an organized system; it’s inherently descriptive and nonquantitative, yet central to science. But these larger issues are not our main concern here. Let’s return to the more practical questions of
how to utilize numbers effectively when they are appropriate to the problems under consideration.


How Good Are Measured Numbers?
When you see a number, do you ever ask yourself: “How well do I know this number?” Most people probably don’t (after all, a number is a number, right?). Let’s consider an example we are all familiar with, such as the nutritional information on a box of cereal. Suppose our Wonder Flakes box tells us that one 8 oz serving supplies 15% of our daily
requirement of carbohydrates. This sounds unambiguous enough, but think about it for a minute. Can that 15% possibly be the same number for a 250-lb person who does manual labor all day and a 140-lb person who sits at a desk all day? Presumably, 15% is some kind of average.

Even granting that we have an average number, is our number necessarily exactly 15% or might it be 14.5%? How about 16.2%? Could it be as far off as 12%? Do you think that even 15.01% is too far off? Or do you think it surely must be exactly 15% or else they wouldn’t have written 15%?

Actually, very few numbers are exact. The only examples I can think of are: purely mathematical numbers, quantities defined by convention (i.e., mutual agreement), and integers (I have exactly 10 fingers). Any other numbers that are the result of measurement are not exact. The proper interpretation of such a number is that it represents a range of
values around the actual written value. How wide is this range? In the case of our Wonder Flakes box, we unfortunately have no idea.

In fact, a number usually represents a range that we don’t know. Keeping this point
in mind is extremely important when evaluating the information that crosses your path each day. An unadorned number, with no information given about its precision, doesn’t tell you as much as it pretends to. Knowing this, you can interpret numbers more realistically and thereby (paradoxically) get more genuine information from them.

We have so far been discussing poorly known numbers, but some numbers are known very precisely. The charge of an electron, for example, is known to within a fraction of a part per million. Analytical chemists routinely measure masses of chemicals with extremely high precision. But even these numbers are still not exact. We need some way to express numerically how well or poorly a number is known; we need the concept of
uncertainty. The uncertainty of a quantity is the range within which we believe this quantity to lie. The uncertainty of a number is also sometimes called the error of the
number, the experimental error, or the error bar. However, statistical interpretations of uncertainty are only really meaningful when they are based on extremely large amounts of data, which is almost never.

So, in practice you are safer to interpret a quoted uncertainty as only a
rough estimate unless you have evidence that it’s statistically valid.

An Afterthought
Obviously, getting better (more reliable and/or more precise) information is always desirable whenever this is possible. But even a rough estimate can be informative and valuable, and we often learn what we need to know just from the order of magnitude of an estimated number. One of the difficulties of estimation techniques is that there are no simple rules to follow.

Not knowing whether we’ve gotten the right answer and not having necessary information to work with are both somewhat disturbing. The benefits of making estimates, however, are well worth the effort involved both in science and in practical affairs.