Extrapolation
Extrapolate : To estimate the value of a variable outside its tabulated or observed range.
To infer an unknown from something that is known; conjecture

What is extrapolation?

Whether you realize it or not, you use extrapolation all the time. If you regularly observe over a period of years that someone you know lies everytime he speaks, you will reach a point where you will conclude that any story he tells is a lie. You do this by "extrapolating" or forming a "conjecture". You cannot be certain that he will lie, but it appears to be a safe assumption. You might also notice that everytime you eat spicy chili, you get heartburn. After 10 episodes of heartburn after eating spicy chili, you might "extrapolate" that you will get heartburn the 11th time you eat the chili.
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Good usage of extrapolation

Extrapolation is also used in Chemistry. Consider the Water Phase Diagram below. The left part [axis] of the diagram is atmospheric pressure. The bottom part [axis] is temperature. Notice that the lower left-hand line does not touch the temperature nor pressure axes. This is because we have no measurement here of what phase [solid-liquid-gas] water is in at ZERO pressure with various temperatures. But we can "extrapolate" that water would be solid at zero pressure and any temperature below -30 degrees C. We would also "extrapolate" that water would be a gas at zero pressure and any temperature above -30 degrees C. Even though we did not actually observe the phase of water at zero pressure, we can make a reasonable conjecture based upon a large amount of data as presented below.

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Poor usage of extrapolation

If, however, you make LARGE extrapolations based upon relatively SMALL data, then you can easily make a mistake. Take for instance our lying friend. Suppose he lied twice. Should we extrapolate that he will lie every time he speaks for the rest of his life? Or should we decide that all chili will forever give me heartburn after one episode of discomfort?

Charles Darwin also used extrapolation to come to his conclusion of the evolutionary origin of species. He studied finches on the Galapagos Islands. He noticed that the finches were different on each island. On islands where the birds could find food by sticking their beaks into crevices in trees, the finches had long beaks. On islands where woods were scarce, the finches had short, fat beaks that allowed them to burrow for food. He imagined that at one time the finches were the same, but over time developed differently shaped beaks to adapt to their environment. On this point he is absolutely right. This is what is known as "micro-evolution", where a certain species can only survive with a certain trait. Therefore all without the trait tend to die out. This means that the environmental stress "selects" for animals with a certain trait. But it should be noted that an animal cannot express a trait for what it does not already possess in its DNA. So micro-evolution means that if a long beak is needed for survival, then those with naturally long beaks will live and pass on this trait to their offspring.

Charles Darwin and Extrapolation

Charles Darwin took his observations about natural selection and extrapolated that a species could not only express a longer beak, but a species could change into a higher species, and over time could change into a reptile, primate, or even a man. This is a case where Darwin used a relatively small amount of data [change in size of beaks] to extrapolate a very very large amount of data, even coming to a conclusion upon the origin of mankind from the lowest of life forms.

Darwin was not careful with his scientific extrapolation, and therefore came to erroneous conclusions.
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