## Analyzing Experimental Data

Last week we looked into how uncertainty arises when taking measurements. This week we will investigate how uncertainty arises within a set of data and how to analyze your results using tools such as plots and lines of best fit.

Tools such as Excel or Google Sheets are a great asset to scientists to help make calculations using an entire data set and these tools also provide easy access to plotting. In this course, using a spreadsheet can help to decrease the number of calculations that your group needs to do for each data point. You can use them to calculate the mean across multiple columns or to use the cell as a variable in any equation you define. Plotting is another tool that can save your group from multiple calculations, and it uses all of the data points in your set to create the result. With enough data points, using a plot to find your result can yield more precise conclusions than any one measurement.

In this workshop, you are tasked with obtaining the spring constant of the spring you are given using various masses as your data points. You will need to use your plot as a tool to obtain this. This will require you to discuss which equations will be helpful when you plot. You and your group will discuss the value of considering many data points within a measurement and why we look at a set of data to draw conclusions. This workshop emphasizes uncertainty analysis, again, as understanding uncertainty is a major learning goal of this course. However, this time your group will be considering a data set rather than a single measurement. It also adds to that understanding by further exploring the concept of uncertainty as it relates to a known model.

### Research Concepts

In order to be productive in class, it would be helpful to research before class:

- Force diagrams
- Lines of best fit for linear data sets and the generic equation for these lines
- Ways of representing this kind of data in graphs
- How to construct a scatter plot in Excel or Google Sheets (http://sheets.google.com)
- How to add error bars to plots in Excel or Google Sheets
- The difference between precision and accuracy

;#;
**Potentially Useful Equations**

$F_{spring}=kx$ and $PE_{spring}=\frac{1}{2}kx^2$

where k is the spring constant and x is the displacement from the spring's resting position
;#;

### Data Analysis

**Part 1 – Discussion of Model**

Your goal will be to find the spring constant of the spring that your group is given using a ruler and set of masses. But first, you should discuss how you will accomplish this by using the equipment and the physics knowledge that you have. To help guide your discussions, consider:

- What is the force diagram for this system?
- Using those equations, what variables do you already know?
- What variables can you measure with the given equipment?
- What variables do you need to find with equations?

These questions are important to consider before deciding your procedure. These can help your group create a plan for your experiment, since the overall goal of the experiment (finding the spring constant in this case) may not be possible by direct measurement.

**Part 2 – Take Your Measurements**

Now that your group has discussed what measurements to make, you should conduct your experiment. Record your data in a spreadsheet. Don't worry about any calculations you need to make just yet. Focus on recording the data that you can measure directly first. Some questions to consider before taking your measurements:

- What is the precision of your measuring tools?
- How did you choose to measure the displacement?
- Is there another way in which you could measure the displacement that would make your results have less uncertainty?

**Part 3 – Calculations and Plotting**

Plot your data as a scatter plot. Talk with your group about which variable should be on the x-axis and which variable should be on the y-axis. If your results are linear, add a line of best fit to your results. What is the equation for your line of best fit?

Recall the equation you found using your force diagram. Can you rearrange this equation to look like your line of best fit? If so, consider the questions below:

- What does your slope represent?
- What does your y-intercept represent?
- Do these results make sense?
- Can you find the spring constant from your plot?

**Part 4 – Representing Uncertainty in Plots**

We have discussed uncertainty in Workshop 1, but how does the uncertainty in our measurements translate to a plot? Error bars are the graphical representation of the uncertainty value that we assign to measurements.

Discuss with your group the uncertainty on your x-axis and y-axis variables. Then, add error bars to your plot.

**Part 5 – Taking More Data**

As we've discussed, the precision of your measurement often depends on the measurement device and how the measurement was taken. In some cases, the measurement you take will be the same each time it is taken, while others will show some variation between data points. If you find that the variations between points are larger than the precision of your instrument, you can increase the precision of your data by taking multiple measurements.

Repeat the measurement done in *Part 2*, two more times**.** Now you
will have a total of three datasets. Plot a line for each new dataset on
the same graph as before (so that you will have three lines on one
plot). Is there a trend between your lines?

Calculate the mean for each data point. On a new plot, create a line that uses the mean values from all three of your measurements (remember to use your tools in Excel/Sheets for this!). Use this to find your final conclusion for the spring constant. Your group should discuss what the equation for the line of best fit means and if it makes sense. Be sure to also consider if the size of your error bars should change now that you are using the mean value instead of a single measurement.

At the end of the day, you will be turning in your notebooks. Tutors will put particular emphasis this week on your figures, your discussion of the plan for your measurements, and the discussion of the equation/model to find your results.