1. Errors and Uncertainties

2. Error and its types
Error – difference between your answer and the ‘true’ one. Generally, all errors are of one of three types.
Systematic – problem with the method, all errors are of the same magnitude and direction (affect accuracy)
Random – causes data to be scattered more or less symmetrically around a mean value. (affect precision)

3. UNCERTAINTIES No measurement is perfect
Our estimate of nearness to the true value is called the uncertainty (or error)
Uncertainty in data leads to uncertainty in calculated results
Uncertainty never decreases with calculations, only with better measurements
Reporting uncertainty is essential
The uncertainty is critical to decision-making
Estimating uncertainty is your responsibility

4. Accuracy and Precision
Accuracy is the degree of closeness of a measured or calculated quantity to its actual (true) value. Precision, the degree to which further measurements or calculations show the same or similar results.
High accuracy, but low precision
High precision, but low accuracy

5. Absolute and Relative Uncertainty
Absolute uncertainty expresses the margin of uncertainty associated with a measurement.
Relative uncertainty compares the size of the absolute uncertainty with its associated measurement.
Percentage uncertainty =
relative uncertainty x 100%

6. Example: If a measurement is written as 5.4 ± 0.2 g, then there is a,
Absolute Uncertainty = 0.2 g

7. Propagation of Uncertainties
We often do math with measurements
Density = (m ± m) / (V ± V)
What is the uncertainty on the density?
“Propagation of Error” estimates the uncertainty when we combine uncertain values mathematically.

8. Simple Rules
Rule 1: If a measured quantity is multiplied or divided by a constant then the absolute uncertainty is multiplied or divided by the same constant. (In other words the relative uncertainty stays the same.)
Example to illustrate rule 1 Suppose that you want to find the average thickness of a page of a book. We might find that 100 pages of the book have a total thickness of 9mm. If this measurement is made using an instrument having a precision of 0·1mm, we can write
Thickness of 100 pages, T = 9·0mm ± 0·1mm and, the average thickness of one page, t, is obviously given by t = T/100   therefore our result can be stated as t = 9/100mm ± 0·1/100mm or t = 0·090mm ± 0·001mm

9. Rule 2: If two measured quantities are added or subtracted then their absolute uncertainties are added.
To find a change in temperature, DT, we find an initial temperature, T1, a final temperature, T2, and then use DT = T2 - T1 If T1 is found to be 20°C ±1°C and if T2 is found to be 40°C ±1°C then DT = 20°C ± 2°C

10. Rule 3: If two (or more) measured quantities are multiplied or divided then their relative uncertainties are added. Rule 4: If a measured quantity is raised to a power then the relative uncertainty is multiplied by that power.

11. Example for rule 3
If we multiply 1.2 ± 0.5 cm and 3.0 ± 0.5 cm, then relative error of the first one is 0.5/ 1.2 = and for the second one is 0.5/3 =
Sum of the relative error is = .582 or 5.82%
The product is = 3.6 cm2
Absolute error = 5.82% x 3.6 cm2 = cm2
The errors are expressed to one significant figures = 2 cm2
The product = 3.6 ± 2 cm2.

12. Decay of charge in a capacitor

13. Comparison and Uncertainty
Significantly different from the accepted value.
Consistent with the accepted value.
May be significantly different from the accepted value.