Empirical formula for linear function

Empirical formula for linear function

The simplicity of constructing a straight line and of thus expressing a law of proportionality between the increment of a function and that of the independent variable makes a straight line graph an extremely convenient means of arriving at an empirical law, i.e. one following directly from the experimental data, without special theoretical investigation. We represent graphically on a sheet of millimetre paper the table of experimental results and thus obtain a series of points. If we wish to obtain an approximate empirical formula so as to study a functional relationship in the form of a linear function, we now draw a straight line, which, if it does not pass through all the points given (which, of course, is almost always an impossibility), is made at the least to pass between these points in such a way that as nearly the same number of points as possible appear on one side of the line as on the other, all the points being reasonably close to it. Error theory, and observation theory, study more accurate ways of drawing the line mentioned, as also for judging the degree of error arising with such

an approximation. In less accurate investigations, however, such as in technology, the drawing of the empirical line is most simply carried out by the "taut string" method, the nature of the method being evident from the name. Having obtained the line, we use direct measurement to determine

its equation:

which gives the required empirical formula. When obtaining this formula we must not lose sight of the fact that different scales are very often in use for the magnitudes x and y, i.e. lines with the same slope on the axes X,Y may represent different numbers. In this case, the slope a will not be equal to the tangent of the angle between the line and the axis X, but will differ from this by a factor numerically equal to the ratio of the units of length used in representing magnitudes x and y.

The result is:

(Here, and subsequently, the sign ~ denotes 

approximately equal to.)

the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek