1. ERT 207 Analytical Chemistry

2. Lecture 4
23rd July 2007

3. Titles to be covered: 2.3 The effect of errors towards data analysis
2.4 Significant Figures
2.5 Standard Deviation
2.6 Propagation of error

4. 2.3 THE EFFECT OF ERRORS TOWARDS DATA ANALYSIS
2.3.1 WAYS OF DESCRIBING ACCURACY
2.3.2 WAYS OF DESCRIBING PRECISION

5. 2.3.1 WAYS OF DESCRIBING ACCURACY
Recall: Accuracy is expressed as absolute error or relative error.
Errors will show the closeness of measurements to the accepted or correct values.
Absolute errors: E = O – A
E = absolute error
O = observed error
A = accepted value

6. Absolute errors is the difference between true value and measured value.
For example, if 2.62 g sample analyzed to be 2.52 g:
Therefore, the absolute error = 2.52 g g
= g.
The positive and negative sign is assigned to show whether the errors are high or low.

7. If the measured value is average of several measurements, the error is called the mean error.
Q1: How to calculate mean error?
A1: By taking the average difference wrt sign, of the individual test results from the true value.

8. Q1: What is relative error?
Relative errors

9. A1: Relative error is the absolute or mean error expressed as a percentage of the true value.

10. Consider the previous case,
Measured value = 2.52 g
True value = 2.62 g
The relative error = X 100%
2.62
= -3.8%

11. Now, Q2: What is relative accuracy?
A2: Relative accuracy is the measured value or mean expressed as a percentage of the true value.

12. Task: Calculate the relative accuracy of the above case.
Solution:
Relative accuracy = X 100%
2.62
= 96.2 %

13. Example (3.6 from Gary D. Christian):
The result of an analysis are g, compared with the accepted value of g. What is the relative error in parts per thousand?
Solution:
Absolute error = g – g
= g
Relative error = g X 100%o
37.06 g
= -2.4 ppt
100%o indicates parts per thousand.

14. Recall: Precision can be expressed as standard deviation,
2.3.2 WAYS OF DESCRIBING PRECISION
Recall: Precision can be expressed as standard deviation,
deviation from the mean, deviation from the median, and
range or relative precision.

15. Example 1: The analysis of chloride ion on samples A, B and C gives the following result
%Cl-
Deviation from mean
Deviation from median A
24.39
0.10
0.11 B
24.20
0.09
0.08 C
24.28
0.01
0.00
= 24.29
đ = 0.07
(d bar)
0.06

16. Deviation from mean
Deviation from mean is the difference between the values measured and the mean.
For example, deviation from mean for sample A= 0.10%
Q1: How to get X ?
A1: = 3
= 24.29

17. Deviation from median
Deviation from median is the difference between the values measured and the median.
Example, deviation from the median for sample A = 0.11%.
Now, identify median = 24.28
Therefore, deviation from median for sample A = = 0.11%

18. Range is the difference between the highest and the lowest values.
For example, range = = 0.19 %.

19. Sample standard deviation
For N (number of measurement) < 30

20. For N > 30

21. 2.4 SIGNIFICANT FIGURES
Q1 : What is
significant figures?

22. A1: The number of digits necessary to express the results of a measurement consistent with the measured precision.

23. A2: Digits that are known to be certain plus one digit that is uncertain.
-The zero value is significant if it is part of the numbers.
-The zero values is not significant figure when it is used to show magnitude or to locate decimal points.
-The position of decimal points have no relation with significant figures.

24. The number of significant figures = the number of digits necessary to express the results of a measurement consistent with the measured precision.

25. Since there is uncertainty (imprecision) in any measurement of at least ± 1 in the last significant figure, the number of significant figures includes all of the digits that are known, plus the first uncertain one.

26. Each digit denotes the actual quantity it specifies
Each digit denotes the actual quantity it specifies. For example, in the number 237, we have 2 hundreds, 3 tens, and 7 units.

27. The digit 0 can be a significant part of a measurement, or it can be used merely to place the decimal point.
The number of significant figures in a measurement is independent of the placement of the decimal point.

28. Take the number 92,067. This number has five significant figures, regardless of where the decimal point is placed.
For example μm, cm, dm and m all have the same number of significant figures.
They merely represent different ways (units) of expressing one measurement.

29. The zero between the decimal point and the 9 in the last number is used only to place the decimal point.
There is no doubt whether any zero that follows a decimal point is significant or is used to place the decimal point.
In the number 727.0, the zero is not used to locate the decimal point but is a significant part of the figure.

30. Ambiguity can arise if a zero precedes a decimal point
Ambiguity can arise if a zero precedes a decimal point. If it falls between two other nonzero integers, then it will be significant. Such was the case with 92,067.
In the number 936,600, it is impossible to determine whether one or both or neither of the zeros is used merely to place the decimal point or whether they are a part of the measurement.

31. It is best in cases like this to write only the significant figures you are sure about and then to locate the decimal point by scientific notation.
Thus, X 10 has five significant figures, but 936,600 contains six digits, one to place the decimal.

32. Example:
List the proper number of significant figures in the following numbers and indicate which zeros are significant.
0.216; 90.7; 800.0;

33. Solution
0.216 three significant figures
90.7 three significant figures; zero is significant
800.0 four significant figures; all zeros are significant
three significant figures; only the last zero is significant

34. Steps in Writing Significant Figures
Make sure that there is no digit after the first uncertainty figure.
For example:
uncertainty figure

35. Rounding Off
If the last digit to be removed is greater than 5, add one to the second last digit.
Example,
If the last digit to be removed is smaller than 5, then the second last digit does not change.
Example,

36. If the last digit is 5 and the second last digit is an even number, thus the second last digit does not change.
Example,
If the last digit is 5 and the second last digit is an odd number, thus add one to the last digit.
Example,

37. Addition and Subtraction Operation
The number of digits to the right of decimal point in the operation of addition/subtraction should remain.
The answer to this operation has a value with the least decimal point.

38. Example : Give the answer for the following operation to the maximum number of significant figures.

39. 43.7
4.941
answer is therefore 61.8 based on the key number (43.7).

40. Multiplication and Division Operation
The number of significant figures in this operation should be the same as the number with the least significant figure in the data.
Example : Give the correct answer for the following operation to the maximum number of significant figures.
x 2.07

41. Solution:
x 2.07 =
The correct answer is therefore 2.26 based on the key number (2.07).

42. Exponential
The exponential can be written as follows. Example, x 10-4

43. 2.5 STANDARD DEVIATION
The most important statictics.
Recall back: Sample standard deviation.
For N (number of measurement) < 30

44. For N > 30,

45. Try these questions on standard deviations.

46. Thank you