Guidelines for Data and Graphing in Lab Reports

Transcription

1 Position (m) Position (m) Guidelines for Data and Graphing in Lab Reports Graphs are a useful mode for representing information. However, unless the graph contains certain pieces of information, the representation may not be as useful. All graphs should contain: Title: This is usually written as the vertical axis variable vs. the horizontal axis variable. For example: Position vs. Time. One should then add to the title additional information to clarify. For example: Position vs. Time for a Truck Moving up a Hill. Note: The most general direction is to place the dependent variable on the vertical axis and the independent variable on the horizontal axis. However, this general guideline is not followed under certain circumstances such as: When time is to be graphed, it is usually placed on the horizontal axis. When one wishes to compare the graphed results (equation of the line/curve) to some equation. In this case one may wish to graph so that the equations come out in the same form. If the given equation is R = L/A then one may select to graph R as the vertical axis even though it might have been the independent variable in the lab. And as always, if the instructor asks for a particular graph, that is the one you do whether it follows the general guideline or not. Axes Labels: Each axis should be labeled with the name of the variable it represents. Following the variable name should be the units in parentheses. For example, Position (m) is the correct label for an axis representing Position measured in meters. Appropriate Scaling: Some scaling divisions are better than others. Choose a scale division that will allow you to plot points accurately The scale division values should be marked next to the axis with small hatch marks (or darkened lines) indicating the exact location on the axis of the value. Notice that the labels correspond to only major gridlines to reduce crowded labeling. The unlabeled, or minor gridlines should correspond to an exact value. The ones above would correspond to 2, 4, 6, etc. Inappropriate scaling would give minor gridline values such as 1/3, 1/6, 1/9, etc. Point Protectors: These are small boxes or other shapes that surround the data point. Notice how the line turns invisible as it passes through the point protector so that we are still able to distinguish the point. Thus, the point protector "shields" or "protects" the point from being obscured. Use a different type of point protector for each set of data when more than one set is graphed on the same graph grid. Position vs. Time for a Truck Moving up Hill Time (s) Time (s) Phys NSCC: Davene Eyres

2 Data and Graphing Guidelines for Lab Page 2 Best Fit Line/Curve: When it is appropriate to draw a line/curve through data points on a graph, the line/curve should be smooth, representing the data trend as opposed to a dotto-dot discontinuous line. You will need to consider whether a straight or curved line is best from all the information you have about the situation. With a multiple line graph, the lines/curves need to be distinguished from each other. Differently shaped point protectors may be sufficient, but to better distinguish the lines, use different colors or line weights. Key or Legend: When there are more than one set of data points graphed on the same grid, there should be a key or legend to distinguish between them. Sometimes it is adequate to just label the line. Special Guidelines when using Excel for graphing: Use x-y scatter: When selecting the type of graph to use, you must use x-y scatter plotting. Choose the option that does not include lines between the points. The option for a line graph will not work for the type of data we generally have in science. As an exercise, you should try it and see if you can see why it is a bad choice. Best-fit line/curve: For introductory course, you should hand draw all best-fit lines/curves. Draw a line or curve so that it best illustrates a smooth trend. For more advanced courses, you should use tools appropriate to the course such as regression analysis or the trend line feature in Excel. Some instructors will also require an r 2 value as well. Point Protectors: This is generally not an option with Excel. The principle still holds, however, that the point plotted must be visible and readable. Printing the graph: If you are using Word for your report, you can copy and paste your graph into your document directly. If not, you can print separately. In either case, make sure that your graph is large enough to allow someone (me) to read values off your graph with ease. Axes labels and grid lines: You may have to select major and possibly minor grid lines for each axis. Make sure that there are enough lines that someone could read values off your graph to a reasonable degree of accuracy. Equation of the Line: For introductory courses, do not use the option in Excel that makes the equation for you. You will have to handwrite it after calculating the slope and reading the intercept off your graph. Use the following guideline for writing the equation of a linear line. The variable you graphed. Do not include units. The slope with units. y = mx + b The variable you graphed. Do not include units. = + Read the vertical axis value from your graph. Include units.

3 Data and Graphing Guidelines for Lab Page 3 For more advance courses, you might be instructed to use the equation feature in Excel. If so, be sure to check the options tab to make sure that the equation you are getting is the one you expect! You should then use the textbox feature to retype your equation with the correct variables and with the appropriate units. When writing equations of curved lines, you will still follow the linear equation guidelines regarding including variables in your equation and appropriate units for values such as slopes, constants, and intercepts. Data Tables: Include Measured Data: If you measured something, it needs to be in your data section. Check to see that everything is included somewhere in the data section and is easily located by someone reading your report. Don t forget the one time data. That is, something that you measured only once, or something that is part of the equipment set-up. Include Uncertainty: If you measured something, it has uncertainty. You must determine the amount of uncertainty for each type of measured data and report it. Use the value +/- range format. Include Initial Conditions: These are special pieces of data. They are things like where is x=0? When did you start the clock, t=0? How did you arrange for v=0 at t=0? (This one might need to be explained in the procedure.) Include Derived Data: If you used the measured data to calculate other values, these need to be included in a table as well. You may add columns to the measured data table or make a separate table. Make sure that example calculations are included in your report. Include Qualitative Data/Observations: It doesn t have to be a number to be data. Other observations are equally important to record. Make sure that all observations are recorded in an easy to read format.

4 Data and Graphing Guidelines for Lab Page 4 ANALYZING GRAPHS Overview A graph is a type of pictorial description that shows the relationship between one variable and another. The specific dependence of one variable on another is called the relationship between two variables. The dependent variable is dependent on the independent variable it changes only when the independent variable changes. For example, the distance a ball rolls down a track is the dependent variable because how far it goes depends on the amount of time that passes. Time, however, is unaffected by is independent of the ball s position. When graphing, plot the dependent variable on the vertical axis, and the independent variable on the horizontal axis. Unless you have something like time which usually goes on the horizontal axis. The shape of the line can tell us whether the position is increasing or decreasing as the time instant increases. Consider the three Position vs. Time graphs in Figure 1. All three graphs show that the position increases as time increases. However, it is clear that the relationship between them is not the same in each case. What is needed, therefore, is a better way of expressing this relationship. For this, we need to recall some algebra. Each of the lines in the graphs above represents a general line shape that can be described with an equation. Let s take a look at the straight-line graph first. Graphs that are Linear: Recall the equation of a straight line as y=mx+b where: x x x t t t Figure 1 - Three Different Graph Shapes y refers to the variable represented by the vertical (dependent) axis x refers to the variable represented by the horizontal (independent) axis m is the slope of the line b is the vertical axis intercept

5 Data and Graphing Guidelines for Lab Page 5 Example Use the Position vs. Time graph at the right to answer the following questions. Assume this is for a toy car that is moving across a table top. a.) Find the slope of the line. Be sure to include your units. b.) Explain why it was not correct to use the point (0,0) in calculating the slope. c.) Find the vertical axis intercept. 20 Position (cm) 10 Position vs. Time d.) What is the vertical axis variable? Assign a 1-letter equation variable to it time (sec) e.) What is the horizontal axis variable? Assign a 1-letter equation variable to it. f.) Write the equation of the line. Be sure you do not use the equation variables x and y in your equation as you did in algebra. This graph is not a y vs. x graph. What quantitiy did you graph? Your slope with units. y = mx + b What quantitiy did you graph? = + Read the vertical axis value from your graph. Include units.

6 Data and Graphing Guidelines for Lab Page 6 Graphs that are Curved: The Method of Multiple Linear Parts: Choose linear sections of your line (2 or 3 sections). Write an equation that represents the relationship shown by each straight-line section of your graph. When you state the equation be sure to give the range within which each equation is valid. The Method of Linearization: This technique involves guessing the form of the function and re-graphing a function of one or both variables to form a linear representation. Then the equation can be written following the format for a linear equation. See Excel file for example and more specific instructions. Regression Analysis: Regression analysis can be titled a trend analysis in spreadsheet programs. When using this option, one must make sure that the form of the equation is reasonable. For example, don t accept a linear equation for a set of data that is clearly curved. The form of the trend line is not x (m) the only thing wrong with these graphs. They have formatting y = 6.681x R² = x (m) Linear (x (m)) problems including: The legend is not necessary for a one line graph. The title is not correctly formatted. There are no vertical gridlines. There are no axes labels. The equation is not formatted properly If R 2 is shown, it should be discussed in the analysis section. Don t accept a function that is a complex polynomial to get a trend line to hit all your points. All the curves (black line) to get the dots all on the line are not the trend. Instead find a smooth trend (red line)

7 Data and Graphing Guidelines for Lab Page 7 Don t accept a function that is clearly too dependent on an outlier data point. The black line does NOT show the trend of the data. Excel used a clear outlier which made the trendline that Excel gave to be too shallow. Instead, use what you know to draw the proper line. In this case, the red one much more closely matches the trend of the data points. Also, notice all the graphing guidelines that are missing from this graph. Can you find them all? Most are listed on the last page example. Did you read carefully? Go back and check Reading Graphs: There are only Three types of data that can be read from any graph. The information from the coordinate, the slope, and the area. The analyzing lab data, be sure to check each of these for meaning. Be careful to note, in the lab directions this term, when you are directed to consider one or more of these in the analysis. Later (after the first few labs in General Physics I) you will be expected to look for these on your own. Coordinates give information about the variables that are graphed. Special point to note are intercepts, maximum, and minimum points. When reading a coordinate, interpolate between the minor axes to read the value to the best accuracy possible. The Slope of the line may have meaning. The units can be a clue as to the physical meaning of the slope. The slope should always include the units. Students should be able to find the slope of a linear best-fit line by selecting 2 points on the line. To find the slope of the best-fit line when it is curved, draw the tangent line at the point of interest and find its slope. Remember: Do not use the point 0,0 unless it is ON the best-fit line. And You always need 2 points to find a slope. The Area under the best-fit line (or over if it is below the axis) may have meaning. The area should always include the units. The units can be a clue as to the physical meaning of the area. For areas under curved lines, approximate it with a number of rectangles or triangles.

8 Data and Graphing Guidelines for Lab Page 8 Solutions to Graph Equation Sample Questions: a) Find the slope: I used the two points (0 s,4 cm) and (2.2 s,20 cm). Notice that the units are put with the values when calculating the slope. b) Explain why it was not correct to use the point (0,0) in calculating the slope. The point 0,0 is not ON the line for which we are writing the equation. c) Find the vertical axis intercept. Look at the graph and see where the line crosses the vertical axis intercept. It crosses at (0 s, 4 cm) d) What is the vertical axis variable? Assign a 1-letter equation variable to it. The vertical axis variable is the horizontal position. It is not vertical since the object is not moving up or down. It is a horizontal position since it moves sideways along a table top. The vertical axis variable is x. Notice that the vertical AXIS is about a horizontal VARIABLE since the motion is HORIZONTAL. e) What is the horizontal axis variable? Assign a 1-letter equation variable to it. The horizontal GRAPH AXIS represents time with a symbol, t. f) Write the equation of the line. Notice: It has the two variables x and t. The slope and intercept have units. Units always go on the numbers in the equation. The variables x and t do NOT have units. ( )