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Deductive reasoning and visual representations


August 24, 2016 by Akane Kuma

Recently, I’ve been working on my math IA titled: GDP per capita and life expectancy. The aim of this IA is to investigate the correlation between GDP per capita and life expectancy. While working on the regression line (GDP per capita on the x-axis and life expectancy on the y-axis), I manipulated the data (such as taking logarithmic) and even deleted some outliers.

In our daily life, we see mathematics represented visually in graphs, models, number lines, etc. However, from my own experience, these data are most of the times manipulated. Those who decide to show the representation would decide what scale and what information to show on it before we see it.

So, how useful and accurate can a graph be?

The most common representation I see are bar graphs. Those are used every day in weather forecasts. Numbers of sunny days, rainy days, cloudy days, etc are represented by them. When I see them on TV, I usually interpret them as percentages. “It rained last few days, so I might need to bring an umbrella.” “This week would not rain, because it has been sunny” This kind of deductive reasoning is most likely to be unreliable. However, the reporter uses the bar graph as conclusion of the next few day’s forecast—this is most of the times reliable, but fails sometimes. The nature of a bar graph is to record numerical information, such as numbers of car accidents, numbers of participants, etc. There is no if-then relation at all. From GDP per capita and life expectancy point of view, a bar graph cannot be used to “predict” something for it is only a simple numerical record of single event. The purpose of showing bar graphs in a weather forecast is obvious…….they want to “create” relation for independent events.

As manipulations are usually done with those data we see, most of the media are deciding what information to show as well. A single event can be presented in different ways by different media. For example, one island has several names…“Senkaku Islands”, “Diaoyutai Islands”, and “Diaoyu Islands”. Both China and Japan have their own historical evidence which they use as defense. If to focus on one of them only, the opportunity of hearing other voices, seeing from different perspective is lost. Belief would blind people—the standard of being “right” varies depending on the perspective you see things from.

Even a picture can be deceiving as well. Any realistic art teacher would tell you to focus on one scene and narrow down or to fix the scope. This naturally cuts of rest of the environment. A very serene pot of flower can be drawn in a disorderly room. Even an X-ray can be “inaccurate” in this sense—it focuses on specific part of a bone and leaves out all the rest. If a patient claims an injury, taking an X-ray might not help to diagnose. When someone presents a picture, I have a tendency of using deductive reasoning to imagine what the environment is like—when I see a picture of several flowers, I imagine a flower garden in most of the times.

So when I worked on GDP per capita and life expectancy, what I really was doing was working on deductive reasoning. Making a curvy regression line straight would give out an impression of strong correlation and deleting several outliers reinforces the straightness. Graphs and pictures would require interpreter’s deductive reasoning (of course, if the aim of looking at pictures and graphs is to test a theory, it would be inductive reasoning) and therefore the accuracy of graphs and pictures shall be bound with that of deductive reasoning.



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