1. ANALYTICAL PROPERTIES PART I ERT 207 ANALYTICAL CHEMISTRY SEMESTER 1, ACADEMIC SESSION 2015/16

2. Overview  INTRODUCTION  THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS  SYSTEMATIC ERRORS  RANDOM ERRORS  DISTRIBUTION OF EXPERIMENTAL RESULTS  STATISTICAL TREATMENT OF RANDOM ERRORS 2 bblee@unimap

3. Overview  THE SAMPLE STANDARD DEVIATION  STANDARD ERROR OF THE MEAN  VARIANCE AND OTHER MEASURES OF PRECISION  REPORTING COMPUTED DATA 3 bblee@unimap

4. INTRODUCTION bblee@unimap 4  Quality:  is defined as the totality of features (properties, attributes, capabilities) of an entity that make it equal to, better or worse than others of the same kind.  Figure1 provides an overview of analytical properties and classifies them into three hierarchical categories:  (i) capital properties, (ii) basic properties and (iii) accessory properties

5. INTRODUCTION bblee@unimap 5 Figure 1: Types of analytical properties and relationships among them and with analytical quality (results and processes).

6. INTRODUCTION bblee@unimap 6  Capital analytical properties are typical of results (analytical information).  The quality of the analytical process is obviously related to that of its results; consequently, basic properties support capital properties.

7. THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS bblee@unimap 7  Subjecting n aliquots of the same sample to an analytical process provides n results.  Obviously, the quality of the information obtained will increase with increase in n.  The lowest information level corresponds to an individual result (Xi) provided by a single sample aliquot.  The mean obtained with n > 30, μ ', will be of higher quality than the previous one.

8. bblee@unimap 8 Figure 2: The chemical metrological hierarchy and its relationships to analytical properties

9. THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS bblee@unimap 9  The theoretical upper quality limit for the series being the mean for a statistical population (n = α ), which is denoted by μ.  When n sample aliquots are analysed by different laboratories using various analytical processes and a consensus is reached from a thorough technical study, a result X' is obtained that is held as true.  Generic and specific uncertainty in Analytical Chemistry can be efficiently defined via the chemical metrological hierarchy as schematized in Figure 3.

10. THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS bblee@unimap 10 Figure 3: Graphical depiction of generic and specific uncertainty, and of their relationships to the chemical metrological hierarchy

11. THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS bblee@unimap 11  The maximum possible uncertainty about the relative proportion (concentration, content) of an analyte in an unknown sample corresponds to absolute specific uncertainty, which ranges from 0.00% to 100.00% in percentage terms.  The true value,, is subject to zero specific uncertainty, which coincides with the absolute absence of uncertainty in the analyte proportion in the sample.  This is intrinsic information that corresponds to ideal quality.

12. THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS bblee@unimap 12  Absolute trueness coincides with the sample's intrinsic information with the true value (X), and represents the maximum possible (ideal) quality level in Figure 4 and the top level in the metrological hierarchy of Figure 2.  Figure 5 illustrates different approaches to clarify the meaning of trueness in Analytical Chemistry.

13. THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS bblee@unimap 13 Figure 4: Analytical chemical information levels ranked according to quality and to their relationship to trueness, accuracy and uncertainty

14. THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS bblee@unimap 14  Intrinsic information:  It is subject to no uncertainty, represents the top level of analytical quality and is characterized via an ideal, unattainable property: trueness. At a lower level is  Referential information:  It is factual, reasonably accurate - nearly as much as intrinsic information - and scarcely uncertain.

15. THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS bblee@unimap 15  It is obtained under uncommon conditions (e.g. in intercomparison tests involving many participating laboratories using different analytical processes)  It is basically used as a reference to extract ordinary analytical information or assess its goodness.  Actual information:  it is also based on facts but is less accurate and more uncertain than referential information.

16. THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS bblee@unimap 16 Figure 5: Connotations of the word "trueness" in Analytical Chemistry

17. THE CHEMICAL METROLOGICAL HIERARCHY: UNCERTAINTY AND TRUENESS bblee@unimap 17 Figure 5 a: Expression of a result (and'its uncertainty) and relationships to the different types of error and to the analytical properties accuracy and precision

18. SYSTEMATIC ERRORS bblee@unimap 18  Error:  the uncertainty in a measurement.  Errors are often associated with mistakes or with something that is definitely wrong but scientists tend to think of them as either: (i) The discrepancy between a measured value and some generally accepted true value (the accuracy of the measurement), or (ii) The degree to which repeated measurement of a particular quantity gives the same result (precision).

19. SYSTEMATIC ERRORS bblee@unimap 19  An experimental method may be accurate but not especially precise if it suffers from random errors, but equally it may yield inaccurate results with high precision if it is subject to systematic errors.

20. SYSTEMATIC ERRORS bblee@unimap 20  In an experiment free from serious systematic errors, we can improve the precision considerably by increasing the sample size.  Systematic errors always affect the accuracy of the result and will tend to do so in the same direction  the answer will always tend to be too high or too low rather than fluctuating about some central value following successive measurements.

21. SYSTEMATIC ERRORS bblee@unimap 21  As a consequence of the fact that they derive from deficiencies of one sort or another in the experimental apparatus, they cannot be quantified by a statistical analysis of repeated observations. Figure 6: The frequency distribution of twenty boiling point measurements of an ester made with twenty different thermometers.

22. RANDOM ERRORS bblee@unimap 22  Random errors are present in every measurement no matter how careful the experimenter.  The somewhat uncomfortable reality of experimental measurement is that two measurements of the same physical property with identical apparatus, using a nominally identical procedure, will almost invariably yield slightly different results.  Repeating the measurement further will continue to yield slightly different values each time.

23. RANDOM ERRORS bblee@unimap 23  Such variations result from random fluctuations in the experimental conditions from one measurement to the next and from limitations associated with the precision of the apparatus or the technique of whoever is conducting the experiment.  The statistical nature of these fluctuations means that the discrepancies with respect to the ‘true’ value are equally likely to be positive or negative.

24. DISTRIBUTION OF EXPERIMENTAL RESULTS bblee@unimap 24  When a sufficiently large number of measurements, s frequency distribution like that shown in Figure 7 (a).  Theoretical distribution for ten equal-sized uncertainties is shown in Figure 7 (b).  When a same procedure is applied to a very large number of individual errors, a bell-shaped curve like that shown in Figure 7(c).  Such a plot is called a Gaussian curve or a normal error curve.

25. DISTRIBUTION OF EXPERIMENTAL RESULTS bblee@unimap 25 Figure 7: Frequency distribution for measurements containing (a) four random uncertainties (4U). (b) ten random uncertainties (10 U) (c) a very large number of random uncertainties. (a) (b) (c)

26. DISTRIBUTION OF EXPERIMENTAL RESULTS bblee@unimap 26  A Gaussian or normal error curve:  A curve that shows the symmetrical distribution of data around the mean of an infinite set of data.  A random uncertainties ( ± U):  An assumed fixed amount (high or low) of error where each error has an equal probability of occurring and that each can cause the final result.  The spread in a set of replicate measurements is the difference between the highest and lowest result.

27. DISTRIBUTION OF EXPERIMENTAL RESULTS bblee@unimap 27  A histogram is a bar graph. Figure 8: A histogram showing distribution of the results and a Gaussian curve for data having the same mean & standard deviation

28. STATISTICAL TREATMENT OF RANDOM ERRORS bblee@unimap 28  Statistical analysis:  Only reveals information that is already present in a data set.  No new information is created by statistical treatments.  Statistical methods do not allow us to categorize and characterize data in different ways and to make objective and intelligent decisions about data quality and interpretation.

29. STATISTICAL TREATMENT OF RANDOM ERRORS bblee@unimap 29  As a rule of thumb, if we have more than 30 results and the data are not heavily skewed, we can safely use a Gaussian distribution. (1) Samples and Populations  We gather information about a population or universe from observation made on a subset or sample.

30. STATISTICAL TREATMENT OF RANDOM ERRORS bblee@unimap 30  A population:  the collection of all measurements of interest to the experiment, while a sample is a subset of measurement selected from the population. (2) Properties of Gaussian curves  Figure 9 shows two Gaussian curves.  A normalized Gaussian curve can be described by: Population mean ( μ ) Population standard deviation ( σ )

31. STATISTICAL TREATMENT OF RANDOM ERRORS bblee@unimap 31 Figure 9: Normal error curves. The standard deviation for curve B is twice that for curve A, σ B = 2 σ A. (a)x-axis = deviation from mean (x- μ ), (b)X-axis = deviation from mean (in σ ).

32. STATISTICAL TREATMENT OF RANDOM ERRORS bblee@unimap 32  Sample mean:  The arithmetic average of a limited sample drawn from a population of data.  Population mean:  is the true mean for the population.  In the absence of systematic error, μ is the true value for the measured quantity. Number of measurement in the sample set. Total number of measurement in the population.

33. STATISTICAL TREATMENT OF RANDOM ERRORS bblee@unimap 33  Population standard deviation:  A measure of the precision of the population.  Precision of the data of curve A is twice as good as that of curve B. Number of data points making up the population.

34. STATISTICAL TREATMENT OF RANDOM ERRORS bblee@unimap 34  The quantity z represents the deviation of a result from the population mean relative to the standard deviation.  It is commonly given as a variable in statistical tables since it is a dimensionless quantity.  The equation for Gaussian error curve: σ 2 = variance

35. STATISTICAL TREATMENT OF RANDOM ERRORS bblee@unimap 35 (3) Areas under a Gaussian Curve  In reference to Figure 7, regardless of its width, 68.3% if the area beneath a Gaussian curve for a population lies within one standard deviation ( ± 1σ ) of the mean ( μ ).  Approximately 95.4% of all data points are within ± 2σ of the mean.  Approximately 99.7% data are within ± 3σ.

36. STATISTICAL TREATMENT OF RANDOM ERRORS bblee@unimap 36  Area under the Gaussian curve:

37. THE SAMPLE STANDARD DEVIATION bblee@unimap 37  It is a measure of precision.  The sample standard deviation:  The sample variance (s 2 ) is also of importance in statistical calculations. Deviation (d i ) of value x i from the mean Ẋ. Number of degree of freedom

38. STANDARD ERROR OF THE MEAN bblee@unimap 38  If a series of replicate results, each containing N measurements, are taken randomly from a population of results, each containing N measurements, are taken randomly from a population of results, the mean of each set will show less and less scatter as N increases.  Standard deviation of each mean is known as the standard error of the mean.

39. VARIANCE AND OTHER MEASURES OF PRECISION bblee@unimap 39 1) Variance  It is just the square of the standard deviation.  It is an estimate of the population variance σ 2.

40. VARIANCE AND OTHER MEASURES OF PRECISION bblee@unimap 40 2) Relative Standard Deviation (RSD) and Coefficient of Variation (CV)  Frequently standard deviations are given in relative rather than absolute terms.  The relative standard deviation is calculated by dividing the standard deviation by the mean value of the data set.

41. VARIANCE & OTHER MEASURES OF PRECISION bblee@unimap 41  RSD is often expressed in parts per thousand (ppt):  RSD multiplied by 100% is called the coefficient of variation (CV). 3) Spread or Range (w)  It is used to describe the precision of a set of replicate results.

42. VARIANCE AND OTHER MEASURES OF PRECISION bblee@unimap 42 Example 1:  From a set of data: 0.752, 0.756, 0.752, 0.751, 0.760  it was found that:, s = 0.0038 ppm Pb  Calculate: (a) the variance (b) The relative standard deviation (in ppt) (c) The coefficient of variation (d) The spread

43. REPORTING COMPUTED DATA bblee@unimap 43 1) Significant figures  The significant figures in a number are all of the certain digits plus the first uncertain digit.  Example:  The liquid level in a buret: 30.24 ml  Rules of determining the number of significant figures: i. Disregard all initial zeros. ii. Disregard all final zeros unless they follow a decimal point. CertainUncertain

44. REPORTING COMPUTED DATA bblee@unimap 44 iii. All remaining digits including zeros between nonzero digits are significant. 2) Significant figures in numerical computations  For addition & subtraction, the result should contain the same number of decimal places as the number with the smallest number of decimal places.

45. REPORTING COMPUTED DATA bblee@unimap 45  For multiplication and division is the number with the smallest number of significant figures.  For logarithms and antilogarithms: (i) In a logarithm of a number, keep as many digits to the right of the decimal point as there are significant figures in the original number. (ii) In an antilogarithm of a number, keep as many digits as there are digits to the right of the decimal point in the original number.

46. REPORTING COMPUTED DATA bblee@unimap 46 Example 2  Round the following calculated number so that only significant digits are retained: (a) Log 4.000 x 10 -5 = - 4.3979400 (b) Antilog 12.5 = 3.162277 x 10 12

47. REPORTING COMPUTED DATA bblee@unimap 47 3) Rounding Data  In rounding a number ending in 5, always round so that the result ends with an even number.  For example, 0.635 rounds to 0.64 0.625 rounds to 0.62

48. EXAMPLE 1 (a) s 2 = (0.0038) 2 = 1.4 x 10 -5 (b) RSD = 0.0038 / 0.754 x 1000 ppt = 5.0 ppt (c)CV = 0.0038 / 0.754 x 100% = 0.50% (d) W = 0.760 – 0.751 = 0.009 ppm Pb. bblee@unimap48

49. EXAMPLE 2 (a) Log 4.000 x 10 -5 = - 4.3979 [Rule 1] (b) Antilog 12.5 = 3 x 10 12 [Rule 2] bblee@unimap49