1. Uncertainty and Error

2. Error An error is a “mistake” in a measurement. Types of errors:
Random
Systematic
Avoidable
Most errors are not really mistakes, but they are factors that affect measurements. Avoidable errors are mistakes, and should not occur.

3. Random Error
As the name implies, a random error is something that occurs because of a random event. In this measurement it is possible that someone walking by could cause vibrations or air currents that would randomly change the number. Performing multiple trials is a good way to minimize the effect of random errors.

4. Systematic Error
A systematic error is one in which the measurement is consistently off due to poor calibration or other similar event. If this balance consistently reads 0.5% too low, for example, that would be a systematic error.

5. Avoidable Error
An avoidable error is a mistake and is something that should be avoided. In this example the balance is not zeroed, which can easily be done to “avoid” the error.

6. Uncertainty
All measurements have a degree of uncertainty. This is a limitation of the equipment used to make the measurement. If you are only making one measurement you can estimate the uncertainty from the scale used. In this case the uncertainty could be ±5.0ºC (one full division) but a better uncertainty would be ±2.5ºC (half of one division). The temperature would be recorded as 27 ±2.5ºC.

7. Calculating Uncertainty From Multiple Trials
If multiple trials are performed that should reasonably result in the same measurement, these formulas can be used to calculate uncertainty. They cannot be used if the trials are measuring results that are expected to vary even if there are no errors (if a variable is changed between trials, for example). The absolute and relative uncertainty are a measure of the precision of the measurements, but they do not indicate anything about the accuracy.

8. Accuracy and Precision
Low Systematic Error
High Systematic Error
Low
Random
Error
High
Accuracy is indicated by a low uncertainty (either absolute or relative). Precision can be determined by a percent error or percent difference.

9. Sample Trial # Length (m) 1 31.5 2 32.5 3 33.5 4 30.5
Assume these are the results of four students measuring the length of the same hallway. It is reasonable to assume there is only one length of the hallway and all the students should get the same result. The variation in the results is the result of errors.

10. Average
The average of a sample is usually an accurate representation of the “true” value of the measurement
Unless the actual value is known independent of the measurement (a fundamental constant, for example) there is no way to determine the exact value (“true” value) due to uncertainty. The average is the best value to use in this case.

11. Absolute Uncertainty Is this good?
The absolute uncertainty tells us that it is probable that the “true” length of the hall is between 31.0 m and 33.0 m.
Is this good?

12. Relative Uncertainty
The relative uncertainty is a useful value because it is a percentage which makes it easy to compare to other measurements. The absolute uncertainty can be misleading – an absolute uncertainty of ±1.0 cm probably sounds good, but if the measurement is 5.0±1.0 cm then the relative uncertainty is 20.0%. If the measurement is ±1.0cm then the relative uncertainty would be 0.02% a much higher degree of precision.

13. Rules for Working With Uncertainties
When adding/subtracting values work with Absolute Uncertainties When multiplying/dividing values work with Relative Uncertainties
These are not the best methods for working with uncertainties, but they are good enough for us to use in this class.