Linearization of Data - Example

When data is collected from an experiment, the exact mathematical relationship is not always readily apparent from a graph of that data. Data that yields a curve for its graph could appear to be represented by 1 1 many different functions. Both y = and y = have very similar graphs. As do y = x2 and y = x3 (when 푥 푥2 looking at them in the first quadrant only). Linearization of data is a method for determining which relationship is the correct one for the given data. The equation y = mx + b is the mathematical representation of a linear relationship. It is called linear because a graph of that function is a straight line. The variables x and y do not have a direct relationship in the equation y = mx2 + b, so this is not a linear function; its graph is a curve. But, y is directly proportional to x2. This is called a “direct square proportion.” The graph could be made to be linear if, instead of graphing x versus y, x2 versus y is graphed instead. The x factor changes, but the y remains the same. (See Data Table III below, and its accompanying graph on the following sheets.) This could be applied for any form of the 1 relationship between x and y: y = m√푥 + b, y = m + b, y = mx3 + b, etc. 푥 The following data charts and accompanying graphs are an example of the process of the linearization of data using “data” from a mock lab experiment. Data Table I shows an example of the direct results of data collection. The following three data tables are the values that will be used for the linearization of that data. Graphs will be constructed using those values and will be used to determine the exact relationship between the two variables. (Do some of the calculations of √푥, x2, and x3 on your own to show yourself how Data Tables I – IV were constructed.) The graphs on the following four pages are graphs of the different “data” from each of these data tables.

Data Table I Data Table II Data Table III Data Table IV x versus y √푥 versus y x2 versus y x3 versus y 2 3 x y √푥 y 푥 y 푥 y .5 .423 .707 .423 .250 .423 .125 .423 1 .860 1.00 .860 1.00 .860 1 .860 1.5 1.69 1.22 1.69 2.25 1.69 3.38 1.69 2 4.00 1.41 4.00 4.00 4.00 8.00 4.00 2.5 7.81 1.58 7.81 6.25 7.81 15.6 7.81 3 13.5 1.73 13.5 9.00 13.5 27 13.5 3.5 21.4 1.87 21.4 12.3 21.4 42.9 21.4 4 32.0 2.00 32.0 16.0 32.0 64.0 32.0 4.5 45.6 2.12 45.6 20.3 45.6 91.1 45.6 5 62.5 2.24 62.5 25.0 62.5 125 62.5

Graph Analyses

Data Table I Graph Analysis Notice that the axes are labelled x and y on this graph, because it is a graph drawn from the values directly obtained in this “lab.” Looking at the graph of x versus y, it is plain to see that it is a curve, and does not represent a linear relationship. This means that the data does not represent an equation of the form y = mx + b, or in other words, a direct relationship.

Data Table II Graph Analysis Notice here that the axes are labelled √푥 and y. Also notice that the “x” axis changed, but the y axis did not.

If we look at the graph of √푥 versus y, it is also pretty plain that y is not directly proportional to √푥 either, because the graph is again a curve and not a straight line. So, the data is not from an equation in the form of y = m√푥 + b.

Data Table III Graph Analysis Now the axes are labelled x2 and y to reflect the fact that those are the quantities being graphed. Here again the “x” axis changed, but the y axis did not. This graph looks a little more like a straight-line fit, but is still curved. This means that y is not directly proportional to x2 either, and the data is not from an equation of the form y = mx2 + b.

Data Table IV Graph Analysis Notice again what the axes are labelled in this graph, and that the y axis is again unchanged. This final graph, x3 versus y, is a straight line and shows a direct relationship between x3 and y. This means that the data is from an equation of the form y = mx3 + b. The slope of the graph is m and can be ∆푦 calculated with the equation 푚 = . In this case, the slope is m = 0.495. Seeing as this is a hand-drawn graph, ∆푥 we can safely assume that the slope is actually 0.50. The graph has the origin as its y-intercept, so b is simply 0. Substituting these equations into the generic equation y = mx3 + b, we get an equation of y = 0.50x3. If you try some of the data points from the original x versus y chart, you will see that this is indeed the equation the data was drawn from.