1. 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 crosses

2. 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 .

3. u v v - dv - s

4. v↑ u→
Plotting 1/v against 1/u will give a straight line.

5. <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

6. 1/u
1/v

7. 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.

8. 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
0.043
7.583
0.036
0.048
0.049
3.782
0.03
0.053
0.055
1.283
0.026
0.057
0.059
3.983
0.023
0.061
0.062
0.078
0.021
0.064
3.477
0.019
0.065
0.066
0.421
20.61
Values before ‘solving’

9. 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
4E-06
0.035
0.047
1E-04
0.03
0.052
0.011
0.026
0.056
0.023
0.060
0.059
0.614
0.021
0.061
0.062
0.533
0.019
0.064
0.063
0.1
1.313
After ‘solving’

10. 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?

11. 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.

12. These offsets could be due to systematic errors
The fact that the slope is 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.

13. Hall Effect
Hall Voltage proportional
to current and B field.
Sign of voltage depends
on type of charge carrier. B I + + + - - e-
Hall V

14. Hall Effect
y = x
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.

15. 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.

16. 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