IU Physics P451/P551, Spring 2004
Homework
Homework Set 1,
Due 1/22.
You are measuring a voltage under the same experimental conditions with a high-precision DMM (digital multimeter). You take 10 measurements of this voltage and record the numbers into this text file. There are random errors in the measurement (from noise, etc) and the readings from the high-precision DMM is not as precise as the number of significant figures would indicate. Using the voltage data that you have recorded, determine:
  1. The mean value of the voltage,
  2. The estimated error on this mean value δ V
You do this again under the same conditions and record the numbers into this text file. Using this data, determine:
  1. The mean value of the voltage,
  2. The estimated error on this mean value δ V
The theoretical value for this voltage is 6.10 V (with negligible error).
  1. What is the probability that each of these 2 measurements is consistent with this value?
Solution:

  1. The mean value of the voltage = 6.593 (see Preston & Dietz, p9, formula 1)
  2. The estimated error on this mean value δ V = 0.36 , the standard deviation of the mean (P&D, p9, formula 3). After considering this error and following the rules of significant figures, you would report the measurement of the voltage as: 6.6+-0.4.
    (If you were going to use this voltage and propagate this error for a subsequent calculation, you should keep another decimal point (i.e. 6.59+-0.36).
  3. The mean value of the voltage = 6.116 (see Preston & Dietz, p9, formula 1)
  4. The estimated error on this mean value δ V = 0.102 , report as: 6.12+-0.10.
  5. For the first case: 6.6+-0.4 is 1.25 sigma from the "theoretical" value of 6.1. The probability of a deviation in the measured value of 1.25 sigma or larger is obtained by integrating the gaussian distribution from 1.25 sigma to infinity (and from -1.25 sigma to -infinity). This yields 0.21. So the probability of this measurement with a deviation of 1.25 sigma (or larger) is 21%.
    For the second case: 6.12+-0.10 is 0.2 sigma from the "theoretical" value of 6.1. The probability of a measurement with a deviation this large (or larger) = 84%
Homework Set 2,
Plotting and Fitting, Due 1/29.
  1. Using a plotting/fitting program of your choice...
    1. Make a histogram of the voltage data (100 pts) from Problem 1, found in this text file.
    2. Fit this data to a Gaussian distribution.
    3. Report the mean, sigma, and errors on these quantities.
    4. How are these related to the mean and standard deviation calculated in Problem 1?
  2. You are measuring (with a Hall-probe gaussmeter) the magnetic field of our 12-inch electromagnet as a function of current supplied to the windings. You record the current, measured magnetic field, and estimated error on from the gaussmeter in this text file. The error on the current (the independent variable) is assumed negligible. Again, using a plotting/fitting program...
    1. Plot this data with error bars.
    2. Fit this data to a straight line (y=mx+b), show the fit, report the (two) fit parameters with errors. Make sure that the fit is a considers the errors properly -- not many programs do, by default. (See the links page for help if you use Sigmaplot.
    3. Report the "goodness-of-fit" parameter ("chisquare"). Does the data fit well to a straight line? Does this "chisquare" value tell you anything about these errors?
Make sure your plots are well-labled. Explain your work and what program(s) you used. Even though the computer does the work, make sure you understand how these fits work ("Method of Least Squares").
Solution:

  1. Voltage Histogram:
    1. Using your favorite plotting/fitting program... Your histogram should look something like this: Voltage Histogram
    2. The gaussian fit is shown.
    3. The mean/sigma and errors are also shown on the plot:
      mean = 6.17+-0.13, sigma = 1.03+-0.11
    4. These are quite similar to the values found from the simple calculation in the previous problem as the should be. In the limit of large N, they will converge to the same values.






  2. Magnetic Field vs Current Measurement:
    1. The plot should look something like this (I used PAW): Mag Field vs Current
    2. The fit is shown on the plot. The resulting fit parameters (and errors) are: slope = 0.170 +- 0.006, intercept = 0.19 +- 0.09.
    3. The "chisquare" is shown on the plot: chisqare = 2.16 for 8 degrees of freedom (DOF= degrees of freedom = number of points - number of fit parameters). Sometimes the normalized chisquare is reported (normalized chisquare = 0.27). The data fits well to the straight line, perhaps too well. The chisquare, on average, should be about equal to the number of DOF. The fact that this chisquare is much smaller than this is a clue that perhaps the errors are overestimated?
Homework Set 3,
Plotting and Fitting II, Due 2/5.
  1. If the results of HW#2 were not satisfactory, resubmit. Help with Sigmaplot is available on our links page.
  2. Once again, you are measuring (with a Hall-probe gaussmeter) the magnetic field of our 12-inch electromagnet as a function of current supplied to the windings. You record the current, measured magnetic field, and estimated error on from the gaussmeter in this text file. (Note: this is a different file than in the previous problem) The error on the current (the independent variable) is assumed negligible. Again, using a plotting/fitting program...
    1. Plot this data with error bars.
    2. Fit this data to a straight line (y=mx+b), show the fit, report the (two) fit parameters with errors.
    3. Report the "goodness-of-fit" parameter ("chisquare"). Does the data fit well to a straight line? Does this "chisquare" value tell you anything about these errors?
Make sure to do this fit taking the error bars into account properly. Either figure out how to get Sigmaplot (or another program) to do it. Or use PAW. Here is a macro to get started. ivb.kumac You'll have to take the values from the text file above, remove the text (PAW cant read the text), and put into a file called ivsb2.vec. Then "exec ivb.kumac" to run it.
Solution:

  1. Voltage Histogram: See HW Set 2, problem 1.
  2. Magnetic Field vs Current Measurement:
    1. The plot should look something like this (I used PAW): Mag Field vs Current
    2. The fit is shown on the plot. The resulting fit parameters (and errors) are: slope = 0.153 +- 0.004, intercept = 0.58 +- 0.08.
    3. The "chisquare" is shown on the plot: chisqare = 7.05 for 8 degrees of freedom (DOF= degrees of freedom = number of points - number of fit parameters). Sometimes the normalized chisquare is reported (normalized chisquare = 0.88). This is a reasonable chisquare value as can be seen on the plot. The spread of the points around the fit line looks reasonable considering the error bars. Compare this to the analogous plot in HW set 2.
Rex Tayloe
Last modified: Thu Feb 5 11:29:19 EST 2004