Formalizing the problem. Identifying the unknowns One issue to consider is what are the unknowns in this situation? The natural answer would be to say that the unknowns are x and y. This is not actually the case, however, as we have some values given to us and we're trying to find out the line that best fits those points on a graph. We don't really care about any particular value of x, what we care about is the coefficients a and b that will tell us what kind of line this is going to be. So in fact we are trying to solve for a and b that will give us the nicest possible line for these points. So the question is how to find the best-fitting line and its parameters, a and b. And of course, we have to decide what we mean by "best." We want to minimize some function of a and b that measures the total error we are making when we choose this line compared to experimental data. There are various ways to do it: some valid ones give us different answers. You must decide what it is you prefer, such as measuring the distance to a line by projecting perpendicularly or measuring the difference between experimental and predicted values of y for a given value of x. In the conventionally used way of measuring distance, you minimize the distance between two points. But if you have one bad data point, such as a measurement in which someone fell asleep while conducting the experiment, that point may dominate your results and give you misleading information. In statistics, it is often important to deal with outliers. A point that is very distant from others may not be a valid data point and so it is more appropriate to measure the average distance of all points or give more weight to distant outliers. There are several possible answers, but one of them gives us a particularly nice formula for a and b, so that's why it's the universally used one. The least-squares method measures the sum of the squares of the errors. Because of aesthetic considerations, we choose to plot data in a linear fashion, even when it is not linear. Linear plots are also easier to solve than nonlinear ones and therefore more appealing in that regard as well. If you have a method that minimizes the total square deviation between observed and predicted values, then that's probably the best one. The sum of the squares of the deviations from a model's prediction to observed data is known as the sum of squared residuals. Minimizing this value can be important when trying to find an accurate model to predict observable data. The sum of the squares of the deviations from a model's prediction to observed data is known as the sum of squared residuals. Minimizing this value can be important when trying to find an accurate model to predict observable data.