## Topic 6.4: Extrapolation (Error Analysis)

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The error associated with extrapolation is beyond the scope of this course, but if it is assumed that the data is normally distributed, it can be shown that the error is quite easy to calculate, and for examples, when the data is known to be linear, then the error of extrapolation only increases quadratically as you move away from the average of the x values and the corresponding coefficient is significantly smaller than that for using an interpolating polynomial with only two points. In fact, it can be shown that any extrapolation using an interpolating linear function has no statistical significance. It is like using a single point to estimate a mean: you cannot say anything about the error associated with your estimator.

To demonstrate this last point, consider finding the average height of all humans by taking just one human. While 5'11" may be an approximation to the mean, no information about how good this estimator is is available. Even if we randomly sample two humans and average their heights, we can find bounds for the actual average which are correct 19 times out of 20 (assuming we have a random sample).