Physics 452

Data Analysis and Error Propagation

 

Assume that a sample has been received from the NCSU reactor that contains a single radioactive isotope that decays with a half-life of a few hours.  Using a suitable detector capable of counting the gamma radiation from the sample, the following measurements were made:

An initial measurement is made for ten minutes, followed by 30 additional 10-minute measurements at 2-hour intervals.  The sample is then removed from the room and a background count is taken with the same detector under the same conditions, this time for 1 hour to improve the statistical accuracy of the measurement. 

A sample data set for you to analyze inside an Excel document is provided in the users folder on the lab computers. It is called template.xls. Copy it to a different name.  To avoid having the numbers change whenever Excel recalculates, select the data values (C17:C47), copy them and paste special onto the same location as ‘values only.’ 

 

Assignment:

 

1)  Determine the uncertainty in each individual measurement and in the background measurement.

 

2)  Use the appropriate method of error propagation to assign an uncertainty to each data point after subtraction of the background.

 

3)  Use a weighted least squares analysis of the corrected data to determine:

            a)  The mean life of the radioactivity and its uncertainty.

            b)  The initial activity and its associated uncertainty.

 

4)  Based on the least squares analysis:

            a)  Calculate a reduced c2 for the least squares fit, comment.

            b)  Find the probability that the reduced c2 for another dataset having the same distribution would exceed your reduced c2.  Use the appropriate table in Bevington[1] and comment on the value.

 

5)  Suppose a NIST reference reported a lifetime for the same isotope you analyzed as:   t = 19.25 ± 0.15

hrs. Use the following questions based on the 3 points above to help guide your explanation for the

analysis of your set of radioactive decay data.

a)  Which measurement is more precise (yours or the value from NIST)?  Why?

b) Does your measurement agree with this expected lifetime?  What level of confidence do you have in this comparison?

c) If greater precision is required for your measurement, what suggestions do you make to reduce the error in the lifetime measurement?

 

The template.xls worksheet also includes aids to help you complete the task in a timely manner.  For comparison purposes only (since it is actually the wrong treatment with unequal errors), a fit using unweighted least squares also appears on the spreadsheet.  You should comment qualitatively on how the weighting properly accounts for the errors by examining the respective plots.

 

Write a brief, yet thorough, report that includes all equations used, the original and calculated data, uncertainties, the results from the least squares analysis (including uncertainties in the fit parameters), the quantities listed above, and a plot that shows the fit to the data (include error bars).  The report should address all the numbered points above, but should not have number, answer … format.  Rather, it should be written as one coherent textual report with an introduction, methods, results (here is where the figure and caption go), discussion, conclusions, and references sections.  Equations, tables and formulae should be included in the text as appropriate.  Write the report on the analysis rather than the experiment (i.e. the methods are the error propagation, etc. procedures, and the discussion is of the fit values, reduced c2, etc. and your interpretations of them).  Each student should complete the assignment individually and may submit it to the instructor for revisions as many times as desired before the final due date listed on the course syllabus.

 

Remember to keep the "big picture" in mind when you write your conclusion for this assignment.  The reasons for spending all the effort to determine the error in a measurement are to:

    1)  Properly convey the precision of the measured value.

    2)  Facilitate comparison with theoretical or other experimental values.

    3)  Determine the primary source of error so that future measurements can be made more precisely.

 



[1] Bevington, P.R. and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, New York (1992).