1. Experimental Errors and Uncertainties

2. Errors and Uncertainties
Errors can be divided into 2 main classes
Random errors
Systematic errors

3. Mistakes Mistakes on the part of an individual such as
misreading scales
poor arithmetic and computational skills
wrongly transferring raw data to the final report
using the wrong theory and equations
These are sources of error but are not considered as an experimental error

4. Systematic Errors
Cause a random set of measurements to be spread about a value rather than being spread about the accepted value
It is a system or instrument error

5. Systematic Errors result from
Badly made instruments
Poorly calibrated instruments
An instrument having a zero error, a form of calibration
Poorly timed actions
Instrument parallax error
Note that systematic errors are not reduced by multiple readings

6. Systematic Errors
Two types of systematic error can occur with instruments having a linear response:
Offset or zero setting error in which the instrument does not read zero when the quantity to be measured is zero.
Multiplier or scale factor error in which the instrument consistently reads changes in the quantity to be measured greater or less than the actual changes.

7. Systematic Errors
Examples of systematic errors caused by the wrong use of instruments are:
errors in measurements of temperature due to poor thermal contact between the thermometer and the substance whose temperature is to be found,
errors in measurements of solar radiation because trees or buildings shade the radiometer.

8. Random Errors
Are due to variations in performance of the instrument and the operator
Even when systematic errors have been allowed for, there exists error.

9. Random Errors result from
Vibrations and air convection
Misreading
Variation in thickness of surface being measured
Using less sensitive instrument when a more sensitive instrument is available
Human parallax error

10. Reducing Random Errors
Random errors can be reduced by
taking multiple readings, and
eliminating obviously erroneous result or
by averaging the range of results.

11. Example of a random error
You measure the mass of a ring three times using the same balance and get slightly different values: g, g, g

12. Practice Problem
Which of the following procedures would lead to systematic errors, and which would produce random errors?
(a) Using a 1-quart milk carton to measure 1-liter samples of milk.
(b) Using a balance that is sensitive to ±0.1 gram to obtain 250 milligrams of vitamin C.
(c) Using a 100-milliliter graduated cylinder to measure 2.5 milliliters of solution.

13. Answer
Procedure (a) would result in a systematic error. The volume would always be too small because a quart is slightly smaller than a liter.
Procedures (b) and (c) would lead to random errors because the equipment used to make the measurements is not sensitive enough.

14. Precision
Precision is how close the measured values are to each other.
A precise experiment has a low random error

15. Accuracy
Accuracy is how close a measured value is to the actual (true) value.
An accurate experiment has a low systematic error.
Accuracy tells us something about the quality or correctness of the result.

16. Examples of Precision and Accuracy:
Low Accuracy High Precision
High Accuracy Low Precision

17. High Accuracy High Precision
So, if you are playing soccer and you always hit the left goal post instead of scoring, then you are not accurate, but you are precise!

18. Limit of Reading and Uncertainty
The Limit of Reading of a measurement is equal to the smallest graduation of the scale of an instrument
The Degree of Uncertainty of a measurement is equal to half the limit of reading
e.g. If the limit of reading is 0.1cm then the uncertainty range is 0.05cm
This is the absolute uncertainty or absolute error.

19. Example
The measurement is taken below using a ruler. The limit of reading is 0.05 cm and the uncertainty is ½ of the limit of reading, or ± cm. Since uncertainties are only given to one significant figure, the length is stated as 0.44 ± 0.02 cm.