How do you assign an error to a measurement? How are errors combined? How does one fit a curve to experimental data? Given a sample of 127 events which have an equal probability of having a value A or a value B what is the average number Of events of type A and the standard deviation on this? Noughts and crosses exercise 53 noughts 74 crosses
Looked at why we do an error analysis. How to assign errors to data. How to add errors. Need to think about how to do an experiment to minimise errors. Some statistics. Mean, Standard deviation of sample and population. Error on Mean. Weighted mean. Least squares fitting. Some examples of how to work with data .
u v v - dv - s
v↑ u→ Plotting 1/v against 1/u will give a straight line.
<v> dv 1/u 1/v d(1/v) 20.8 0.27 0.048 0.0006 17.9 0.32 0.040 21.0 20.8 0.27 0.048 0.0006 25.0 17.9 0.32 0.040 0.056 0.0010 30.0 15.8 0.42 0.033 0.063 0.0017 35.0 14.6 0.26 0.029 0.069 0.0012 40.0 13.3 0.37 0.025 0.075 0.0021 45.0 13.2 0.34 0.022 0.076 0.0020 50.0 12.4 0.36 0.020 0.081 0.0023
1/u 1/v
Account for systematic errors by assigning offset values uo, vo to the object and image distances u, v. Assign values of u0, v0, m and c in cells in the spread sheet. Make estimates for u0, v0 and c set m=-1 Tabulate and in Excel Evaluate Use SOLVER in EXCEL to minimise S by adjusting uo, v0 , m and c.
c= 0.085 m= -1.000 uo= -3.00 vo= -3.000 x y dy mx+c 1/(u-uo) 1/(v-vo) d(1(/v-vo)) S 0.042 -0.0005 0.043 7.583 0.036 0.048 -0.0007 0.049 3.782 0.03 0.053 -0.0012 0.055 1.283 0.026 0.057 -0.0009 0.059 3.983 0.023 0.061 -0.0014 0.062 0.078 0.021 -0.0013 0.064 3.477 0.019 0.065 -0.0015 0.066 0.421 20.61 Values before ‘solving’
c= 0.082 m= -0.989 uo= -3.46 vo= -3.305 Values after ‘solving’ x y dy mx+c 1/(u-uo) 1/(v-vo) d(1(/v-vo)) S 0.041 -0.0005 4E-06 0.035 0.047 -0.0007 1E-04 0.03 0.052 -0.0012 0.011 0.026 0.056 -0.0008 0.023 0.060 -0.0014 0.059 0.614 0.021 0.061 -0.0013 0.062 0.533 0.019 0.064 -0.0015 0.063 0.1 1.313 After ‘solving’
Now use modified data to plot graph and fit straight line to graph. Slope should be -1 and intercepts on the x and y axes should be the same. What would cause the above not to be true? Why should uo, vo be non-zero?
What do we learn from this graph? To few data near intercepts. Intercepts determine the focal length Plot data as you do the experiment to help you determine what next to measure.
These offsets could be due to systematic errors The fact that the slope is -0.989 and not exactly -1 is due to experimental errors. Note how much closer the slope is to -1 after allowing for offsets on the object and image distances. These offsets could be due to systematic errors or we may have to modify the theory. A more advanced lens formula incorporates principle planes which would explain the offsets.
Hall Effect Hall Voltage proportional to current and B field. Sign of voltage depends on type of charge carrier. B I + + + - - e- Hall V
Hall Effect y = 13.575x + 26.444 Residuals- difference between data and straight line fit. What do these tell us? The scatter will indicate the size of the random errors. A smooth curve suggests a higher order polynomial is needed to fit the data.
Additional data shows that a cubic term is also needed. Cubic term comes from the effect of heating which generated more electron hole pairs increasing the current without adding to the Hall Voltage.
The End Errors analysis is important It can take longer to process the errors than to get a final result. A result without a proper error analysis is not very meaningful. Do not quote more significant figures than are meaningful. Process your data as you do the experiment to determine where you need to concentrate your effort. The End