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+~~NOTOC~~
+===== 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.