1. Chapters 3 Uncertainty
January 30, 2007
Lec_3

2. Outline Homework Chapter 1 Chapter 3 Start Chapter 4
Experimental Error
“keeping track of uncertainty”
Start Chapter 4
Statistics

3. Chapter 1 – “Solutions and Dilutions”
Homework
Chapter 1 – “Solutions and Dilutions”
Questions: 15, 16, 19, 20, 29, 31, 34

4. Experimental Error And propagation of uncertainty
Chapter 3
Experimental Error
And propagation of uncertainty

5. Keeping track of uncertainty
Significant Figures
Propagation of Error ml
35.21 (+ 0.04) ml

6. Suppose
You determine the density of some mineral by measuring its mass g
And then measured its volume ml

7. Significant Figures (cont’d)
The last measured digit always has some uncertainty.

8. 3-1 Significant Figures What is meant by significant figures?

9. Examples How many sig. figs in: 3.0130 meters 6.8 days 0.00104 pounds
350 miles
9 students

10. “Rules” All non-zero digits are significant Zeros:
Leading Zeros are not significant
Captive Zeros are significant
Trailing Zeros are significant
Exact numbers have no uncertainty
(e.g. counting numbers)

11. Reading a “scale”

12. What is the “value”?
When reading the scale of any apparatus, try to estimate to the nearest tenth of a division.

13. 3-2 Significant Figures in Arithmetic
We often need to estimate the uncertainty of a result that has been computed from two or more experimental data, each of which has a known sample uncertainty.
Significant figures can provide a marginally good way to express uncertainty!

14. 3-2 Significant Figures in Arithmetic
Summations:
When performing addition and subtraction report the answer to the same number of decimal places as the term with the fewest decimal places ?

15. Try this one
1.632 x 105
4.107 x 103
0.984 x 106
x 106
x 106
x 106 + +

16. 3-2 Significant Figures in Arithmetic
Multiplication/Division:
When performing multiplication or division report the answer to the same number of sig figs as the least precise term in the operation
x =
0.51 ?

17. 3-2 Logarithms and Antilogarithms
From math class:
log(100) = 2
Or log(102) = 2
But what about significant figures?

18. 3-2 Logarithms and Antilogarithms
Let’s consider the following:
An operation requires that you take the log of What is the log of this number?
log (3.39 x 10-5) =
log (3.39 x 10-5) =
log (3.39 x 10-5) =

19. 3-2 Logarithms and Antilogarithms
Try the following:
Antilog 4.37 =

20. “Rules” Logarithms and antilogs
1. In a logarithm, keep as many digits to the right of the decimal point as there are sig figs in the original number.
2. In an anti-log, keep as many digits are there are digits to the right of the decimal point in the original number.

21. 3-4. Types of error
Error – difference between your answer and the ‘true’ one. Generally, all errors are of one of three types.
Systematic (aka determinate) – problem with the method, all errors are of the same magnitude and direction (affect accuracy)
Random – (aka indeterminate) causes data to be scattered more or less symmetrically around a mean value. (affect precision)
Gross. – occur only occasionally, and are often large.

22. Absolute and Relative Uncertainty
Absolute uncertainty expresses the margin of uncertainty associated with a measurement.
Consider a calibrated buret which has an uncertainty ml. Then, we say that the absolute uncertainty is ml

23. Absolute and Relative Uncertainty
Relative uncertainty compares the size of the absolute uncertainty with its associated measurement.
Consider a calibrated buret which has an uncertainty is ml. Find the relative uncertainty is , we say that the relative uncertainty is

24. 3-5. Estimating Random Error (absolute uncertainty)
Consider the summation:
(+ 0.02)
(+ 0.03)
(+ 0.05)
(+ ?)

25. 3-5. Estimating Random Error
Consider the following operation:

26. Try this one

27. 3-5. Estimating Random Error
For exponents

28. 3-5. Estimating Random Error
Logarithms antilogs

29. Question
Calculate the absolute standard deviation for a the pH of a solutions whose hydronium ion concentration is
2.00 (+ 0.02) x 10-4

30. Question
Calculate the absolute value for the hydronium ion concentration for a solution that has a pH of 7.02 (+ 0.02)
[H+] = (+ ?) x 10-7

31. Suppose
You determine the density of some mineral by measuring its mass g
And then measured its volume ml
What is its uncertainty?
= g/ml

32. The minute paper Please answer each question in 1 or 2 sentences
What was the most useful or meaningful thing you learned during this session?
What question(s) remain uppermost in your mind as we end this session?

33. Chapter 4
Statistics

34. General Statistics Principles
Descriptive Statistics
Used to describe a data set.
Inductive Statistics
The use of descriptive statistics to accept or reject your hypothesis, or to make a statement or prediction
Descriptive statistics are commonly reported but BOTH are needed to interpret results.

35. Error and Uncertainty
Error – difference between your answer and the ‘true’ one. Generally, all errors are of one of three types.
Systematic (aka determinate) – problem with the method, all errors are of the same magnitude and direction (affect accuracy).
Random – (aka indeterminate) causes data to be scattered more or less symmetrically around a mean value. (affect precision)
Gross. – occur only occasionally, and are often large. Can be treated statistically.

36. The Nature of Random Errors
Random errors arise when a system of measurement is extended to its maximum sensitivity.
Caused by many uncontrollable variables that are an are an inevitable part of every physical or chemical measurement.
Many contributors – none can be positively identified or measured because most are so small that they cannot be measured.

37. Random Error
Precision describes the closeness of data obtained in exactly the same way.
Standard deviation is usually used to describe precision

38. Standard Deviation
Sample Standard deviation (for use with small samples n< ~25)
Population Standard deviation (for use with samples n > 25)
U = population mean
IN the absence of systematic error, the population mean approaches the true value for the measured quantity.

39. Example
The following results were obtained in the replicate analysis of a blood sample for its lead content: , 0.756, 0.752, ppm lead. Calculate the mean and standard deviation for the data set.

40. Standard deviation
0.752, 0.756, 0.752, ppm lead.

41. Distributions of Experimental Data
We find that the distribution of replicate data from most quantitative analytical measurements approaches a Gaussian curve.
Example – Consider the calibration of a pipet.

42. Replicate data on the calibration of a 10-ml pipet.

43. Frequency distribution

45. The minute paper Please answer each question in 1 or 2 sentences
What was the most useful or meaningful thing you learned during this session?
What question(s) remain uppermost in your mind as we end this session?