Use of graph in analysis

Analytical chemistry frequently needs to establish whether — and how — two or more measured properties of a system are mathematically related to each other, and the most direct way to uncover such a relationship is to plot the experimental data on a graph. A classic example used to illustrate the method is the relationship between the temperature and the volume of a fixed amount of gas: a set of experimentally measured (temperature, volume) pairs, when simply plotted as individual points on graph paper, will nearly always show some scatter due to unavoidable small errors in each measurement. If those scattered points are joined to each other directly, in the order they were taken, the result is a meaningless zig-zag line that reveals no real trend, because it treats every bit of experimental scatter as if it were a genuine feature of the relationship. The proper approach instead is to draw a single smooth curve that passes through the general run of the points — called the average curve — rather than forcing the curve to touch every point exactly; in the temperature–volume example this average curve turns out to be a straight line, from which the relationship $V \propto T$ can be inferred. Deciding whether a drawn curve is actually a good fit to the data is done in a specific, mathematically even-handed way: a perpendicular is dropped from every data point to the curve, and the length of that perpendicular is taken as that point's deviation from the curve. Perpendiculars for points lying above the curve (positive deviations) are considered separately from perpendiculars for …