C. Theoretical vs Empirical

The independent and dependent variable relationship can be either theoretical or empirical.

1. Theoretical

A theoretical relationship is based strictly on a mathematical equation; there are no observed or measured data. There is an exact relationship between the two so there is no error.

A basic example in surveing is a proposed grade line in an alignment design. A +3.0% grade begins at station 10+00 with elevation 800.0.and ends at station 13+00. For every 100 ft horizontally, elevation changes 3.0 ft vertically. Expressed as an equation:

We select a station and compute its elevation; elevation is the dependent variable.

Set up a table for elevations at full stations, Table I-1

Although the table shows only four data pairs, because theirs is an exact relationship, we can add as many as we like: 801.5 ft at 10+50, 804.5 at 11+50, etc.

The table is one way to look at the data, plotting it is another way to visualize it. Using the same scale for both axes, Figure I-2, results in a very flat plot that is difficult to interpret. That's because elevation changes only 9.0 ft while stationing changes 300 ft, a 1:33.33 ratio.

Figure I-2 Grade; Same X and Y scales

Increasing the elevation scale exaggerates the plot vertically and makes the data grade line more apparent. Figure I-3 uses a vertical exaggeration of 10:

Figure I-3 Grade; Vertical exaggeration 10

Because a theoretical plot is based on a mathematical equation, there are infinite data points. When the graph is drawn, only the line or curve is shown, not individual data points.

2. Empirical

Empirical data is based on measurement or observation. The dependent variable is measured at a specific independent values. Being measured, the dependent values are is subject to errors; the independent ones are considered error-free.

Let's use another surveying example. A profile level is run along a section of center line on existing terrain. The results are in Table I-2.

Plotting the data, using a vertical exaggeration of 4, Figure I-4:

Figure I-4 Profile Elevations

It looks like a straight line will fit ... but will it? Since two points define a straight line, we can construct a whole bunch of grade lines, Figure I-5.

Figure I-5 Multiple Straight Lines

Any single line fits two points perfectly, but misses the other two, sometimes by quite a lot.

Draw a best-fit line which may or may not go through individual data points but meets some specific criteria, Figure I-6

Figure I-6 Best-fit Straight Line

Because this is a linear relationship, a best-fit line represents acceptable compromises but must still be straight.