Graphing

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Graphical Analysis:  Introduction

In any scientific work much time is usually spent in determining relationships between two quantities such as distance and time or voltage and current. Let us assume that during the course of an experiment a number of length (L) measurements were made for a number of different temperatures (T). One might then immediately wonder if there was some pattern (some relationship) between the two quantities L and T. Looking at only the numbers from the measurements, one would have difficulty in discovering any patterns. Hence one forms a pictorial representation, a graph, to display the data. Figure 1

The graph is usually prepared with the independent variable (the quantity that is varied or set by the experimenter) along the x-axis and the dependant or measured variable along the y-axis. In lab work, this might not be followed but is commonly followed in published work. Then if the data of Temperature and Length follows a linear relationship, a straight-line equation y = mx + b will describe the experiment and can be easily determined from the slope and intercept of a a graph fit. Note that the parameters are determined from the graph and not from the data.

If the graph of L and T does not follow a straight line but rather some sort of non-linear curve, then the analysis becomes more complicated. From the graph alone one can infer the general trend of the data and obtain unknown values by interpolation. However, this still leaves one without an exact description of what is happening in the experiment. In our situation we can only determine the equation of a straight line. Therefore, once we have established that a relationship does exist (ie. the data forms some sort of curve on the graph), then the objective is to find some way of transforming the data into a linear relation. This might be done by guessing as to what the relationship might be or by using some theory to suggest an equation or by using a little of both - theory and guesswork. The following are some of the standard relationships and methods used in analyzing them.