Investigation 6.1: Interpolation

Your instructor will give you an assorted set of at least ten cylinders. Using a string and a ruler, measure the diameter and circumference of each cylinder and enter the data in the table below.

CYLINDER MEASUREMENTS Circumference (cm)        Diameter (cm) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

These pairs of values can be presented as ordered pairs on a graph. Using the following instructions for graphing to guide you, plot the circumference as a function of the diameter of the cylinder. Another way to say this is "plot the circumference versus the diameter." Still another way to put it is "plot the circumference on the vertical axis (y-axis) and plot the diameter on the horizontal axis (x-axis)." The independent variable is always plotted on the horizontal x-axis while the dependent variable goes on the vertical y-axis.

Graphing Techniques

1. Place a title on the graph paper somewhere near the top of the page. If your graph is going to be wider than it is tall, then the title should still be at the top of the page.

2. Select a scale for each axis so that the graph will cover more than half the page in each direction. Your graph should be centered on the page.

3. On each axis indicate the scale divisions, the name of the variable being plotted (circumference and diameter), and the units of measurement.

4. The origin is at the lower left-hand corner of the graph and usually has a value of zero. The numbers increase from left to right along the horizontal (x) axis and from bottom to top on the vertical (y) axis.

   NOTE: Not all graphs follow this rule. Since the larger the positive number for the magnitude of a star the dimmer it is, magnitude numbers plotted on the vertical (y) axis start with larger, positive numbers at the bottom and end with smaller and negative numbers at the top! [Remember, the brighter the magnitude of a star, the smaller the number.]

5. Circle data points to represent graphically uncertainty of data and to ease the drawing in of the "best fit curve" (a straight line is considered a "curve" in this context). We will refer to these circled data points as "error circles."

6. After all data are entered on the graph, draw a thin line that best represents, as you infer it, the total accumulation of data. Follow the trend of the data points with a smooth curve. Your line should either go through, or as near as possible to, as many error circles as possible. Start your line at your first data point and end it with the last point. If you continue your line either to the origin or beyond the last point, then make it a dotted line. There may be a measurement that doesn't seem to fit the trend you see. If so, should you remeasure that data set and try to include it in the trend of the data set, or should you simply ignore it? This is science and there is always error in measurements. Most graphs will NOT be drawn dot-to-dot. When you draw dot-to-dot, you are giving more importance to individual measurements than to the collection of all measurements. In variable star astronomy, it is the accumulation of all the data that is significant, as it is in the measurements of the cylinders.

Now you are ready to answer the following questions by analyzing the results on your graph.

1. What shape is your "best fit" curve? What does this tell you about the relationship between the two variables you plotted, circumference and diameter?
2. Choose a diameter value that you did not measure, that lies along the line you drew but is not a data point on the graph. Reading from the vertical axis, what would be the circumference for this diameter? You have just used interpolation to determine an answer. Anytime you can get a number you did not actually measure between two points you did measure, you are interpolating. Determine what the circumference would be for a cylinder with a diameter that is 5 cm larger than your largest measured diameter. For this you need to go along the dotted line outside of your last data point. This is called extrapolation.
3. Draw straight lines to both the x-axis and the y-axis from two different points along the line you drew through the data points, choosing two points where no data are plotted. For each axis, subtract the smaller value from the larger one, then divide the value for the y-axis by the value for the x-axis. This will give you the slope of the line, or the "rise" over the "run." Does this number look familiar?