STATISTICAL ANALYSIS OF EXPERIMENTAL DATA
Gary L. Bertrand
University of Missouri-Rolla

   This is an exercise in which you will produce "experimental data" which will be subject to random errors and may be subject to systematic errors. The "data" will be produced by trimming a large rectangle to match the dimensions of a smaller target rectangle.

   The arrows above the large rectangle move a vertical saw left or right, and the arrows to the left move it up or down. The minimum size of a movement is 5 pixels.  When you are satisfied with the position of the saw, click on chop and the cut will be made. You can trim more later but a cut cannot be undone. [If you find you have cut off too much, click on new sample to reject this set of data.]  When you are satisfied with the cuts you have made, click on record data. The width and height of your rectangle will appear to the left of the screen with either the area or the perimeter calculated from these dimensions.

   Perimeter or Area determines whether to display the perimeter or the area calculated from the recorded width and height.  Both will be included in the printout.  These calculations are included to allow comparison of the uncertainties in the calculated values to the uncertainties in the observed values.

   Click on new sample to get another large rectangle to work with. The target rectangle does not change. The target rectangle can only be changed by changing between Practice and Assignment modes or by clicking on New Target.

   The program may be operated in Practice or Assignment modes:

   In Practice mode, the program displays statistical calculations for your measurements. All of this data may be printed, along with details of the calculations.  The target values may be seen after five observations have been performed.  This allows comparison of the target value to the observed mean and its standard deviation.

   In Assignment mode, the statistical information is not displayed. The printout is formatted to help you perform these calculations.
 

Note:  Measurements may be made either "freehand" or by using some sort of measuring device (a notecard or a slip of paper might be used to reproduce dimensions).  Actually, it is a good idea to do one set of five or more measurements each way.  Even with a measuring device, there will be some variation in results due to the 5-pixel limitation, and there may be a systematic error due to parallax or to screen distortion.

The statistical method (see background) provides rules for estimating uncertainties in values calculated from experimental measurements.  This is especially important when the experimental measurements are performed in different experiments.  In this experiment, the height and width of the target rectangle are being measured at the same time, allowing the perimeter and/or the area of the rectangle to be calculated for each measurement.  However, this would not be possible if one person were measuring only the width and another was measuring only the height of the rectangle, with the results to be combined to calculate the perimeter and/or the area.  In this experiment, the perimeters/areas from the individual observations may be averaged and uncertainties may be calculated directly AND the heights and widths from the individual observations may be averaged and their uncertainties calculated.  This provides a test of the rules for estimating uncertainties in calculated quantities.

Suggested Procedure:  Have a Calculator handy.
 

1. The program opens in PRACTICE MODE, showing the AREA of the cropped rectangle.  Move one of the saws into position and click on chop.  Move the other saw into position and click on chop.  Click on Record Data.

2. Click on New Sample and repeat step 1 until at least five sets of data have been recorded.

3. Note that the area calculated by multiplying the mean width by the mean height is the same as the mean area.

        Calculate the standard deviation in the area from the standard deviations of the width and height:
           smean,area = area x sqrt [ (smean,width/width)2+ (smean,height/height)2 ].

        Compare this value to the value that the program has generated by calculating the observed areas.

4. At this point you may click on target values to compare to your observations.  It is not unusual for the observed means to differ from the target values by more than one standard deviation.

5. Click on t-Factors and note the t-factor for the 95% confidence level for the number of degrees of freedom (df) of your data.

6. Click on prepare for printout, and print these results.  Note how the program has calculated standard deviations, and used the t-factor to calculate the uncertainties at the 95% level of confidence.

7. Click on menu.  Remain in PRACTICE MODE and click on Perimeter to put the program into Perimeter Mode.  The rectangles are now blue instead of red, but the dimensions of the target rectangle has not changed.  Perform the measurements as before to generate at least five more sets of data.  If you did the first part of this exercise without any measuring devices, use a notecard or scrap of paper to try to improve the reproducibility of your measurements.

8. Note that the perimeter calculated as twice the sum of the average width and average height is the same as the mean perimeter.

        Calculate the standard deviation in the perimeter from the standard deviations of the width and height:

          smean,perimeter = 2 x sqrt [ (smean,width)2+ (smean,height)2 ]

        Compare this value to the value that the program has generated by calculating the observed perimeters.

8. Click on prepare for printout, and print these results.



9. Click on menu.  Click on ASSIGNMENT MODE .  Note that the dimensions of the target may have changed.  Generate at least five sets of data in either Perimeter or Area mode.

10. Click on t-Factors and note the t-factor for the 95% confidence level for the number of degrees of freedom (df) of your data.

11. Click on prepare for printout, and print these results.

12. Perform the indicated calculations.  The printouts from the PRACTICE MODE may be used as a reference.

close