B position q with respect to time for a mass of 3kg

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Unformatted text preview: e can tell from Figure 4(b) that the object is at at . A grain of salt: our prediction graphs will not always need units. For example, if we are asked to draw a graph predicting the relationship of, say, the acceleration due to gravity of an object with respect to its mass, the label “Mass” will do just fine for our horizontal axis. This is because we are not expected to give the precise functional dependence in this situation, only the overall behavior. We don’t know exactly what the acceleration will be at a mass of , and we don’t care. We just need to show whether the variation is increasing, decreasing, constant, linear, quadratic, etc. In this specific case, it might be to our advantage to include 253 APPENDIX: REVIEW OF GRAPHS units on the vertical axis, though; we can probably predict a specific value of the acceleration, and that value will be meaningless without them. (b) Position q with respect to time τ for a mass of 3kg. The acceleration is constant. (a) Figure 4: Poorly- versus well-labeled and -captioned graphs. The labels and caption make the second graph much easier to interpret. Every graph we make should also have some sort of title or caption. This helps the reader quickly to interpret the meaning of the graph without having to wonder what it’s trying to say. It particularly helps in documents with lots of graphs. Typically, captions are more useful than just titles. If we have some commentary about a graph, then it is appropriate to put this in a caption, but not a title. Moreover, the first sentence in every caption should serve the same role as a title: to tell the reader what information the graph is trying to show. In fact, if we have an idea for the title of a graph, we can usually just put a period after it and let that be the first “sentence” in a caption. For this reason, it is typically redundant to include both a title and a caption. After the opening statement, the caption should add any information important to the interpretation of a graph that the graph itself does not communicate; this might be an approximation involved, an indication of the value of some quantity not depicted in the graph, the functional form of a fit line, a statement about the errors, etc. Lastly, it is also good explicitly to state any important conclusion that the graph is supposed to support but does not obviously demonstrate. For example, let’s look at Figure 4 again. If we are trying to demonstrate that the acceleration is constant, then we would not need to point this out for a graph of the object’s acceleration with respect to time. Since we did not do that, but apparently had some reason to plot position with respect to time instead, we wrote, “The acceleration is constant.” Lastly, we should choose the ranges of our axes so that our meaning is clear. Our axes do not 254 APPENDIX: REVIEW OF GRAPHS always need to include the origin; this may just make the graph more difficult to interpret. Our data should typically occupy most of the graph to make it easier to interpret; see Figure 5. However, if we are trying to demonstrate a functional form, some extra space beyond any statistical error helps to prove our point; in Figure 5(c), the variation of the dependent with respect to the independent variable is obscured by the random variation of the data. We must be careful not to abuse the power that comes from freedom in (a) (b) Figure 5: Graphs with too much (a), just enough (b), and too little space (c) to be easy to interpret. (c) plotting our data, however. Graphs can be and frequently are drawn in ways intended to manipulate the perceptions of the audience, and this is a violation of scientific ethics. For example, consider Figure 6. It appears that Candidate B has double the approval of Candidate A, but a quick look at the vertical axis shows that the lead is actually less than one part in seventy. The moral of the story is that our graphs should always be designed to communicate our point, but not to create our point. 255 APPENDIX: REVIEW OF GRAPHS Figure 6: Approval ratings for two candidates in a mayoral race. This graph is designed to mislead the reader into believing that Candidate B has a much higher approval rating than Candidate A. Using Linear Relationships to Make Graphs Clear The easiest kind of graph to interpret is often a line. Our minds are very good at interpreting lines. Unfortunately, data often follow nonlinear relationships, and our minds are not nearly as good at interpreting those. It is sometimes to our advantage to force data to be linear on our graph. There are two ways that we might want to do this in this class; one is with calculus, and the other is by cleverly choosing what quantities to graph. The “calculus” method is the simpler of the two. Don't let its name fool you: it doesn't actually require any calculus. Let’s say that we want to compare the constant accelerations of two objects, and we have data about their positions and velocities with respect to ti...
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This document was uploaded on 02/23/2014 for the course MANAGMENT 2201 at University of Michigan.

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