Finding Uncertainty in a Graph S Slope

Finding Uncertainty in a Graph’s Slope:

Finding uncertainty in a graph’s slope is a simple as finding the range of possible slope values. The uncertainty would be equivalent to (±(half the range)). Finding the range of possible slope values requires that you determine what the maximum possible slope (max slope) and the minimum possible slope (min slope) would be for your data.

Step 1: graph data as usual. If the data is linear, go on to step 2. If it’s not (i.e. it’s originally a parabola), then there’s a mid-step to linearize your graph—we’ll go there later.

Step 2: In LoggerPro, you’ve already entered your data to create the original graph. Now, you need to create a New Data Set:

And re-name the data set—call it “Max Slope Values”, then repeat Step 2 for a data set you’ll call “Min Slope Values”:

Step 3: Calculating data point values

The logic behind calculating the values for new data points is that, ultimately, we want to create a line that would have a “maximum” slope, and another line that will have a “minimum” slope (is this process always perfect? No, but it works for the general purpose of finding the uncertainty in your actual slope!)

For both the max slope and the min slope, you need to start by determining which of your data points are both closest to your linear fit AND at the extremes of your data. Typically this means that your bottom-left data point (closest to the origin) will be used, as well as the data point the farthest up and to the right. Sometimes you will choose the second closest to the extremes, but not too often.

Finding Max Slope Data Points:

For lower-left data point:

Manipulated (X) variable + uncertainty = new X coordinate

Responding (Y) variable – uncertainty = new Y coordinate

For upper-right data point:

Manipulated (X) variable – uncertainty = new X coordinate

Responding (Y) variable + uncertainty = new Y coordinate

Finding Min Slope Data Points:

For lower-left data point:

Manipulated (X) variable – uncertainty = new X coordinate

Responding (Y) variable + uncertainty = new Y coordinate

For upper-right data point:

Manipulated (X) variable + uncertainty = new X coordinate

Responding (Y) variable – uncertainty = new Y coordinate

Step 4: Getting Data Points visible on graph:

Double-click in the graph to pull up the “options” window. Choose the tab labeled “Axis Options”

Expand the data sets in the “Y-axis” section, and put a checkmark in EACH of the MASS boxes (for all 3 data sets):

Step 5: Getting lines on your graph:

Click on the “Linear Fit” button at the top of LoggerPro’s menu bar. Check the two boxes for Min and Max slope:

Final Product:

Sometimes, if your error bars are larger, the lines will be more obviously a maximum and a minimum. Report your graph with all 3 lines present.

Final Answer:

Assuming the question was: “What is the density of the unknown blue liquid?”, the answer will be the slope because density is the ratio of mass to volume. The uncertainty in your slope is equal to half the difference between the max and min slope values.

Slope = 0.783 g·mL-1

Uncertainty = (0.793-0.781)2g·mL-1 = 0.006 g·mL-1

Final Answer: The density of our unknown blue liquid is 0.783 ± 0.006 g·mL-1