MI 3 Light Intensity Lab
Name __________________________
Light Intensity vs. Distance
The following activity is designed to help you learn a bit about data collection and
display, curve fitting, and error analysis, and how Excel can facilitate this work.
You will be gathering light intensity data and plotting the intensity, I, vs. distance, d.
(See next page for complete instructions). Ultimately, you will need to determine a
model for the data, and be able to use your error analysis to convince the reader that your
model is indeed the best fit.
But first, a brief overview of data analysis:
Whenever you fit a curve to data points, most of the points on the scatter plot will not fall
exactly on the fitted curve. You might recall this from the belly button project in MI-2.
So the question becomes, "How close is our fitted curve to the data?" We begin by
looking at the difference between the y coordinate of your data point, yi , and the y value
that is predicted by your model, y i . We call these differences residuals, or ri  yi  yi .
Examining the residuals and making a residual plot (a scatterplot of  xi , ri  ) gives us a
better sense of how our model fits the data and may also alert us to characteristics of the
data that we did not previously notice. Often, we also compute the squares of the
residuals and examine their sum. Why would this sum be helpful? Why do we sum the
squares of the residuals, and not the residuals themselves?
There are two features seen in a residual plot that are typically used to help determine
whether the chosen model is appropriate. First, we look at the size of the residuals. What
would be indicative of a good fit? What would indicate a possible outlier? Next, we
examine a scatterplot of the residuals to determine whether the residuals follow any
particular trend or pattern. What message would the residual plot give you if a pattern
was apparent?
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MI 3 Light Intensity Lab
Name __________________________
1. Collect light intensity data every 0.5 meter from 0.5 to 3.0 meters, using a cbl and
a single light bulb. Record your data in Excel, labeling the independent variable
as d, and the dependent variable, as I.
2. Ideally, this experiment should be conducted in complete darkness except for the
light of the bulb. Determine how to adjust your data to compensate for ambient
light levels that are non-zero. Use only the cbl, light bulb, and meter sticks to
develop a compensation method, rather than trying to block the ambient light
sources.
3. Use Excel to graph two transformations of your data. You should transform the
data and evaluate both ln  I  vs. d and ln  I  vs. ln  d  and fit each with the best
linear fit. Your task is now to determine the relationship that best fits the data,
supporting your claim with error analysis.
4. Use Excel to create the appropriate error analysis. For each linear equation in step 3,
compute the residuals, the squares of the residuals, and the sum of the squares of the
residuals. Finally, create a scatterplot of the residuals, with the residuals from both
models shown on the same set of axes. Refer to the file named “mathworks
residual_shorter” on the student server > public > math > MI 3 (or on the Moodle
site) for more help on residual analysis. From this work, you should be able to select
the better model for your data, and justify your choice.
5. Transform the better-fitting relationship from step 3 above back to a nonlogarithmic form. Write a paragraph* to explain and show the calculations (Use
equation editor to enter the equations.)
6. On one graph in Excel, graph the non-logarithmic relationship from step 5 above
(using “curves” only) and the original data (using “symbols” only).
7. Turn in: Both in hard copy (12 point font, double spaced) and to turnitin.com:
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A paragraph (or two) that gives a brief description of the task as described.
Your work should stand alone from this assignment sheet. Include how
you will approach the problem.
Your *paragraph/work from step 5.
A concluding paragraph (or two) that reveals your chosen model for the
data and give justification for your choice, based on your error analysis.
8. Post your Excel file, named Yournames_light int.xlsx on the class Moodle site.
Your plots should all be labeled as described above, and your columns of data
should also have descriptive names. Your file should contain your work with the
data and graphs showing the linear fit for each, as well as the final graph created
in step 6.
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