Chapter 5-7 Errors, Random Errors, and Statistical Data in Chemical Analyses

Transcription

1 Chapter 5-7 Errors, Random Errors, and Statistical Data in Chemical Analyses Impossible: The analytical results are free of errors or uncertainties. Possible: Minimize these errors and estimate their size with acceptable accuracy. Statistical calculations for use to judge the quality of experimental measurements are many. Error ( 偏差, 誤差 ): 1) The difference between a measured value and the true or known value. 2) the estimated uncertainty in a measurement or experiment

2 Figure 5-1 Results from six replicate determinations for iron in aqueous samples of a standard solution containing ppm of iron(iii). 3 3 Measurements are always accompanied by uncertainty. The true value always falls within a range due to uncertainty. The true value for any quantity is always unknown. The probable magnitude of the error can be evaluated. We can then defines the limit within which the true value of a measured quantity lies at a given level of probability. Data of unknown quality are worthless. (reliability of data) The true value of a measurement is never known exactly

3 Standards of known composition can be analyzed and the results compared with the known composition. Consult the literature. Calibrating equipment enhances the quality of data. Finally, statistical tests can be applied to the data. 3 5 Questions to answer before beginning an analysis: What is the maximum error that I can tolerate in the result? No one can afford to waste time generating data that are more reliable than is needed

4 Mean ( 均值 ), arithmetic mean, and average (x) are synonyms. (5-1) where x i represents the individual values of x making up a set of N replicate measurements. The symbol Σx i means to add all the values x i for the replicates. 3 7 The median ( 中值 ) is the middle result when replicate data are arranged in order of size. Equal numbers of results are larger and smaller than the median. For an odd number of data points, the median can be evaluated directly. For an even number, the mean of the middle pair is used

5 The mean of two or more measurements is their average value. The median is used advantageously when a set of data contains an outlier ( 離群值, 異常值 ), a result that differs significantly from others in the set. 3 9 Calculate the mean and the median for the data shown in Figure 5-1. Because the set contains an even number of measurements, the median is the average of the central pair:

6 Precision ( 精確度, 精密度 ) describes the reproducibility of measurements; the closeness of results to each other. Precision is determined by repeating the measurement on replicate samples. Three terms to describe the precision of a set of replicate data: standard deviation variance ( 2 ), and coefficient of variation (CV) Precision is a function of the deviation from the mean d i, or just the deviation, which is defined as (5-2) Precision is the closeness of results to others that have been obtained in exactly the same way

7 Accuracy ( 準確度 ) indicates the closeness of the measurement to its true or accepted value and is expressed by the error. Accuracy measures agreement between a result and its true value. Precision describes the agreement among several results that have been obtained in the same way Figure 5-2 Illustration of accuracy and precision using the pattern of darts on a dartboard

8 Absolute Error The absolute error E in the measurement of a quantity x i is given by the equation (5-3) where x t is the true, or accepted, value of the quantity. Note that we retain the sign in stating the error. Relative error Often, the relative error E r is a more useful quantity than the absolute error. The percent relative error is given by the expression (5-4) 3 15 Relative error Relative error is also expressed in parts per thousand (ppt). e.g

9 To determine accuracy, we have to know the true value, and this value is exactly what we are seeking in the analysis. Results can be precise without being accurate and accurate without being precise. The danger of assuming that precise results are also accurate is illustrated in Figure 5-3, which summarizes the results for the determination of nitrogen in two pure compounds: benzyl isothiourea hydrochloride ( 苄異硫脲鹽酸鹽 ) and nicotinic acid ( 菸鹼酸 ) NH S NH3 Cl O OH N benzyl isothiourea hydrochloride nicotinic acid Figure 5 3 Absolute error in the micro Kjeldahl determination of nitrogen ( 凱氏定氮法 ). Each dot represents the error associated with a single determination. Each vertical line labeled (x i x t ) is the absolute average deviation of the set from the true value. (Data from C. O. Willits and C. L. Ogg, J. Assoc. Anal. Chem., 1949, 32, 561. With permission.)

10 Figure 5-1 Results from six replicate determinations for iron in aqueous samples of a standard solution containing ppm of iron(iii) Figures 5-1 and 5-3 suggest that chemical analyses are affected by at least two types of errors. One type, called random (or indeterminate 不確定 ) error ( 隨機誤差 ), causes data to be scattered more or less symmetrically around a mean value. Refer to Figure 5-3: Notice that the scatter in the data, and thus the random error, for analysts 1 and 3 is significantly less than that for analysts 2 and

11 A second type of error, called systematic (or determinate) error ( 系統誤差 ), causes the mean of a set of data to differ from the true or accepted value. The results of analysts 1 and 2 in Figure 5-3 have little systematic error, but the data of analysts 3 and 4 show systematic errors of about 0.7 and 1.2% nitrogen Random (indeterminate) errors are errors that affect the precision of measurement. Systematic (determinate) errors affect the accuracy of results

12 A third type of error is gross error ( 過失誤差 ). Gross errors differ from indeterminate and determinate errors. They usually occur only occasionally, are often large, and may cause a result to be either high or low. Gross errors lead to outliers ( 離群值, 異常值 ), results that appear to differ markedly from all other data in a set of replicate measurements. (often the product of human errors) Various statistical tests can be done to determine if a data point is an outlier Systematic errors have a definite value, an assignable cause, and are of about the same magnitude for replicate measurements made in the same way. Systematic errors lead to bias ( 偏誤, 偏差 ) in measurement results. Note that bias affects all the data in a set in approximately the same way and that it bears a sign

13 Three types of systematic errors: (1) Instrument errors are caused by imperfections in measuring devices and instabilities in their components. (2) Method errors arise from non-ideal chemical or physical behavior of analytical systems. (3) Personal errors result from the carelessness, inattention, or personal limitations of the experimenter Instrument errors All measuring devices are sources of systematic errors. For example, pipets, burets, and volumetric flasks may hold or deliver volumes slightly different from those indicated by their graduations. These differences arise from using glassware at a temperature that differs significantly from the calibration temperature, from distortions in container walls due to heating while drying, from errors in the original calibration, or from contaminants or scratches on the inner surfaces of the containers. Calibration eliminates most systematic errors of this type

14 Instrument Errors Electronic instruments are subject to instrumental systematic errors. These uncertainties have many sources. For example, errors emerge as the voltage of a battery-operated power supply decreases with use. Instrument errors are detectable and correctable Method Errors The non-ideal chemical or physical behavior of the reagents and reactions upon which an analysis is based often introduce systematic method errors. Such sources of non-ideality include the slowness and incompleteness of reactions, the instability of species, non-specificity of most reagents, and possible interference

15 Method Errors In Figure 5-3, the results by analysts 3 and 4 show a negative bias that can be traced to the chemical nature of the sample, nicotinic acid. The compounds containing a pyridine ring are incompletely decomposed by the sulfuric acid; hence, the negative errors in Figure 5-3 are likely systematic errors from incomplete decomposition of the samples. Errors inherent in a method are often difficult to detect and are thus the most serious of the three types of systematic error NH S NH3 Cl O OH N benzyl isothiourea hydrochloride nicotinic acid Figure 3 3 Absolute error in the micro Kjeldahl determination of nitrogen ( 凱氏定氮法 ). Each dot represents the error associated with a single determination. Each vertical line labeled (x i x t ) is the absolute average deviation of the set from the true value. (Data from C. O. Willits and C. L. Ogg, J. Assoc. Anal. Chem., 1949, 32, 561. With permission.)

16 Kjeldahl method The Kjeldahl method or Kjeldahl digestion is a method for the quantitative determination of nitrogen in chemical substances. Degradation: Sample + H 2 SO 4 (NH 4 ) 2 SO 4 (aq) + CO 2 (g) + SO 2 (g) + H 2 O(g) Liberation of ammonia: (NH 4 ) 2 SO 4 (aq) + 2NaOH Na 2 SO 4 (aq) + 2H 2 O(l) + 2NH 3 (g) Capture of ammonia: B(OH) 3 + H 2 O + NH 3 NH 4+ + B(OH) 4 Back-titration: B(OH) 3 + H 2 O + Na 2 CO 3 NaHCO 3 (aq) + NaB(OH) 4 (aq) + CO 2 (g) + H 2 O NH S NH3 Cl O OH benzyl isothiourea hydrochloride N nicotinic acid 1-31 Personal Errors Measurements requiring personal judgments. Judgments of this type are often subject to systematic, unidirectional errors. An analyst who is insensitive to color changes tends to use excess reagent in a volumetric analysis. Physical disabilities are often sources of personal determinate errors. A universal source of personal error is prejudice

17 Personal Errors Number bias is another source of personal error that varies considerably from person to person. The most common number bias encountered in estimating the position of a needle on a scale involves a preference for the digits 0 and 5. Also prevalent is a prejudice favoring small digits over large and even numbers over odd. Color blindness amplifies personal errors in a volumetric analysis Systematic errors may be either constant or proportional. The magnitude of a constant error does not depend on the size of the quantity measured. Absolute error is constant, but relative error varies with sample size. (ex. solubility loss) Proportional errors increase or decrease in proportion to the size of the sample taken for analysis. Absolute error varies with sample size, but relative error stays constant. (interfering contaminants)

18 Systematic instrument errors are usually corrected by periodic calibration of equipment. The response of most instruments changes with time. Most personal errors can be minimized by care and selfdiscipline Bias ( 偏誤, 偏差 ) in an analytical method is particularly difficult to detect. One or more of the following steps can recognize and adjust for a systematic error in an analytical method

19 Analysis of Standard Samples Analyzing standard reference materials, SRM, is the best way to estimate the bias of an analytical method. The SRM Materials contain one or more analytes at wellknown or certified concentration levels. Standard materials can sometimes be prepared by synthesis. Standard reference material can be purchased from a number of governmental and industrial sources The concentration of one or more of the components in these materials has been determined in one of three ways: (1) by analysis with a previously validated reference method; (2) by analysis by two or more independent, reliable measurement methods; (3) by analysis by a network of cooperating laboratories that are technically competent and thoroughly knowledgeable with the material being tested

20 Standard reference materials from NIST. (Photo courtesy of the National Institute of Standards and Technology.) 3 39 Independent Analysis If standard samples are not available, a second independent and reliable analytical method can be used in parallel with the method being evaluated. A statistical test must be used to determine whether any difference is a result of random errors in the two methods or due to bias in the method under study

21 Performing Blank Determinations Blank determinations are useful for detecting certain types of constant errors. In a blank determination, or blank, all steps of the analysis are performed in the absence of a sample. The results from the blank are then applied as a correction to the sample measurements. Blank determinations reveal errors and correct data. Variation in Sample Size Constant errors can often be detected by varying the sample size All measurements contain random errors. Random, or indeterminate, errors occur whenever a measurement is made. Caused by many small but uncontrollable variables. The errors are accumulative

22 Analysts 2 & 4: larger random error. Figure 6-1 A three-dimensional plot showing absolute error in Kjeldahl nitrogen determinations for four different analysts Imagine a situation in which just four small random errors combine to give an overall error. We will assume that each error has an equal probability of occurring and that each can cause the final result to be high or low by a fixed amount ±U. Table 6-1 shows all the possible ways the four errors can combine to give the indicated deviations from the mean value

23 3 45 Figure 6-2 Frequency distribution for measurements containing (a) four random uncertainties, (b) ten random uncertainties, and (c) a very large number of random uncertainties

24 (c) For a very large number of individual errors, a bell-shaped curve like that shown in Figure 6-2c results. Such a plot is called a Gaussian curve ( 高斯曲線 ) or a normal error curve ( 正態誤差曲線 )

25 This ml spread of data, from a low of ml to a high of ml, results directly from an accumulation of all the random uncertainties in the experiment. Rearrange Table 6-2 into frequency distribution groups, as in Table % of the data reside in the cell containing the mean and median value of ml and that more than half the results are within ± ml of this mean

26 The frequency distribution data in Table 6-3 are plotted as a bar graph ( 條形圖 ), or histogram ( 直方圖 )(labeled A in Figure 6-5). As the number of measurements increases, the histogram approaches the shape of the continuous curve shown as plot B in Figure 6-5 (a Gaussian curve, or normal error curve) Figure 6-3 A histogram (A) showing distribution of the 50 results in Table 6-3 and a Gaussian curve (B) for data having the same mean and standard deviation as the data in the histogram

27 Many small and uncontrollable variables affect even the simple process of calibrating a pipet. The cumulative effect of random uncertainties is responsible for the scatter of data points around the mean. Statistics only reveal information that is already present in a data set (1) visual judgments, such as the level of the water with respect to the marking on the pipet and the mercury level in the thermometer (2) variations in the drainage time and in the angle of the pipet as it drains (3) temperature fluctuations, which affect the volume of the pipet, the viscosity of the liquid, and the performance of the balance (4) vibrations and drafts that cause small variations in the balance readings

28 The random, or indeterminate, errors in the results of an analysis can be evaluated by the methods of statistics. Ordinarily, statistical analysis of analytical data is based on the assumption that random errors follow a Gaussian, or normal, distribution. Sometimes analytical data depart seriously from Gaussian behavior, but the normal distribution is the most common. We base this discussion entirely on normally distributed random errors In statistics, a finite number of experimental observations is called a sample of data ( 數據樣本 ); this is different from the term used in chemical analysis. A population ( 母體 ) (or a universe): the collection of all measurements of interest to the experimenter [finite and real, or conceptual and infinite]. A sample is a subset of measurements selected from the population. Statistical laws must be modified when applied to a small sample because a few data points may not be representative of the population

29 Figure 6-4a shows two Gaussian curves in which the relative frequency y of occurrence of various deviations from the mean is plotted as a function of the deviation from the mean. The equation for a Gaussian curve has the form The equation contains just two parameters, the population mean μ and the population standard deviation σ Figure 6-4 Normal error curves. The standard deviation for curve B is twice that for curve A; that is, σ B = 2σ A. (a) The abscissa is the deviation from the mean in the units of measurement

30 (b) The abscissa is the deviation from the mean in units of σ. Thus, the two curves A and B are identical here The Population Mean μ and the Sample Mean x Sample mean = when N is small x, where Population mean = μ, where when N The difference between x and μ decreases rapidly as N reaches over 20 to 30. In the absence of systematic error, the population mean ( ) is also the true value (x t ) for the measured quantity

31 The Population Standard Deviation (σ) σ is a measure of the precision or scatter of a population of data, which is given by the equation (6-1) where N is the number of data points making up the population The Population Standard Deviation (σ) The two curves in Figure 6-4a are for two populations of data that differ only in their standard deviations. The standard deviation for the data set yielding the broader but lower curve B is twice that for the measurements yielding curve A. The precision of the data leading to curve A is twice as good as that of the data that are represented by curve B

32 The Population Standard Deviation (σ) Figure 6-4b shows another type of normal error curve in which the abscissa is now a new variable z, which is defined as (6 2) z is the deviation of a data point from the mean relative to one standard deviation. That is, when x μ = σ, z is equal to one; when x μ = 2σ, z is equal to two. y 2/2 x ) 2 e ( 2 /2 2 e z The Population Standard Deviation (σ) A plot of relative frequency versus this parameter (z) yields a single Gaussian curve that describes all populations of data regardless of standard deviation

33 The Population Standard Deviation (σ) Variance ( 變異數 ): The square of the standard deviation σ 2. A normal error curve has several general properties: (1) The mean occurs at the central point of maximum frequency, (2) there is a symmetrical distribution of positive and negative deviations about the maximum, (3) there is an exponential decrease in frequency as the magnitude of the deviations increases. Small random uncertainties are more common Areas under a Gaussian Curve Regardless of its width, 68.3% of the data making up the population will lie within the bounds bracketed by ±1σ. Approximately 95.4% of all data points are within ±2σ of the mean and 99.7% within ±3σ. Because of such area relationships, the standard deviation of a population of data is a useful predictive tool. (p. 101)

34 Equation 6-1 must be modified for a small sample of data. Thus, the sample standard deviation s is given by the equation (6 4) The quantity N - 1 is called the number of degrees of freedom (the number of independent results that enter into the computation of the s). Population standard deviation: 3 67 An Alternative Expression for Sample Standard Deviation (6 5)

35 The following results were obtained in the replicate determination of the lead content of a blood sample: 0.752, 0.756, 0.752, 0.751, and ppm Pb. Calculate the mean and the standard deviation of this set of data. To apply Equation 6-5, we calculate 2 xi and 2 ( )/ N xi 3 69 Substituting into Equation 6-5 leads to

36 Note in Example 6-1 that the difference between Σx 2 i and (Σx i ) 2 /N is very small. If we had rounded these numbers before subtracting them, a serious error would have appeared in the computed value of s. To avoid this source of error, never round a standard deviation calculation until the very end Standard Error of the Mean For replicate samples, each containing N measurements, are taken randomly from a population of data, the mean of each set will show less and less scatter as N increases. The standard deviation of each mean is known as the standard error of the mean (SEM, 平均值標準誤差 ) and is given the symbol s m. (6 6)

37 The rapid improvement in the reliability of s with increases in N makes it feasible to obtain a good approximation of σ when the method of measurement is not excessively time consuming and when an adequate supply of sample is available. When N is greater than about 20, s and can be assumed to be identical Pooled data ( 合併數據, 綜合數據 ) from a series of similar samples accumulated over time provide an estimate of s that is superior to the value for any individual subset. Assume the same sources of random error in all the measurements. To obtain a pooled estimate of the standard deviation, s pooled, deviations from the mean for each subset are squared; the squares of all subsets are then summed and divided by an appropriate number of degrees of freedom

38 The equation for computing a pooled standard deviation from several sets of data takes the form where N 1 is the number of results in set 1, N 2 is the number in set 2, and so forth. The term N t is the number of data sets that are pooled

39 3 77 Other than sample standard deviation, three other terms are often employ in reporting the precision. 1. The variance (s 2 ) is (6 8) People who do scientific work tend to use standard deviation rather than variance as a measure of precision

40 2. Relative standard deviation: RSD s r s x s RSDinppt 1000 ppt x 3. Coefficient of variation (CV) CV RSDinpercent s 100% x (6-9) 3 79 Spread or Range (w) Another term to describe the precision of a set of replicate results. It is the difference between the largest value in the set and the smallest. Example: The spread of the data is ( ) = 0.9 ppm Fe. Figure 3-1 Results from six replicate determinations for iron in aqueous samples of a standard solution containing ppm of iron(iii)

41 For the set of data in Example 6-3, calculate (a) the variance, (b) the relative standard deviation in parts per thousand, (c) the coefficient of variation, and (d) the spread

42 Consider the summation Absolute standard deviations 3 83 The summation could be as large as = +0.10, or as small as = -0.10, or any value lies between these two extremes, or even = 0, = 0 The variance of a sum or difference is equal to the sum of the individual variances. If y = a + b c For the computation The variance of y, s y2 is given by s y = ± ±

43 3 85 As shown in Table 6-4, the relative standard deviation of a product or quotient is determined by the relative standard deviations of the numbers forming the computed result

44 Applying this equation to the numerical example gives we can write the answer and its uncertainty as (±0.0003) One of the best ways of indicating reliability ( 信度 ) is to give a confidence interval at the 90% or 95% confidence level. Another method is to report the absolute standard deviation or the coefficient of variation (CV) of the data. A less satisfactory but more common indicator of the quality of data is the significant figure ( 有效數字 ) convention. 信度 (reliability) 是指測量結果的一致性 穩定性及可靠性

45 A simple way of indicating the probable uncertainty associated with an experimental measurement is to round the result so that it contains only significant figures ( 有效數字 ). The significant figures in a number are all the certain digits plus the first uncertain digit. A zero may or may not be significant depending on its location in a number. A zero that is surrounded by other digits is always significant (such as in ml). Zeros that only locate the decimal point for us are not Sums and Differences For addition and subtraction, the number of significant figures can be found by visual inspection. the second and third decimal places in the answer cannot be significant because 3.4 is uncertain in the first decimal place

46 Products and Quotients A rule of thumb sometimes suggested for multiplication and division is that the answer should be rounded so that it contains the same number of significant digits as the original number with the smallest number of significant digits. Ex The first answer would be rounded to 1.1 and the second to If we assume a unit uncertainty in the last digit of each number in the first quotient, however, the relative uncertainties associated with each of these numbers are 1/24, 1/452, and 1/

47 Because the first relative uncertainty is much larger than the other two, the relative uncertainty in the result is also the absolute uncertainty is then Therefore, the first result should be rounded to three significant figures or 1.08, but the second should be rounded to only two; that is Logarithms and Antilogarithms 1. 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. 2. 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

48 Round the following answers so that only significant digits are retained: (a) log = and (b) antilog 12.5 = (a) Following rule 1, we retain 4 digits to the right of the decimal point (b) Following rule 2, we may retain only 1 digit 3 95 A good guide to follow when rounding a 5 is always to round to the nearest even number. For example, rounds to 0.64 and rounds to We should note that it is seldom justifiable to keep more than one significant figure in the standard deviation because the standard deviation contains error as well

49 The uncertainty of the result is estimated using the techniques presented in Section 3E. Finally, the result is rounded so that it contains only significant digits. It is especially important to postpone rounding until the calculation is completed. At least one extra digit beyond the significant digits should be carried through all the computations to avoid a rounding error. This extra digit is sometimes called a guard digit Defining a numerical interval around the mean (x) of a set of replicate results within which the population mean ( ) can be expected to lie with a certain probability. This interval is called the confidence interval. 2. Determining the number of replicate measurements required to ensure at a given probability that an experimental mean falls within a certain confidence interval. 3. Using the least-squares method for constructing calibration curves

50 Confidence limits (CL, 信賴界限 ): define a numerical interval around a determined mean (x) that contains μ with a certain probability. A confidence interval (CI: 信賴區間 ) is the numerical magnitude of the confidence limit. The confidence level ( 信心水準 ) fixes the odds that the true mean ( ) will be within the defined limits. z = (x - )/ For a single measurement: CI for = x ± z For the mean of N measurements: CI for = x ± (z / N) 3 99 Confidence Interval (CI) CI for = x ± (z / N)

51 z = (x - )/ Table 3-6 CI for = x ± (ts/ N) A statistical technique called regression analysis provides the means for objectively obtaining such a line and also for specifying the uncertainties associated with its subsequent use. 迴歸分析 (Regression Analysis) 是一種統計學上分析數據的方法, 目的在於了解兩個或多個變數間是否相關 相關方向與強度, 並建立數學模型以便觀察特定變數來預測研究者感興趣的變數

52 Figure 8-9 Calibration curve of absorbance versus analyte concentration for a series of standards the method of least squares is used to generate a calibration curve (working curve), two assumptions are required. The first is that there is actually a linear relationship between the measured variable (y) and the analyte concentration (x). The mathematical relationship that describes this assumption is called the regression model, which may be represented as where b is the y intercept (the value of y when x is zero) and m is the slope of the line

53 The second assumption: any deviation of the individual points from the straight line arises from error in the measurement. That is: no error in x values of the points (concentration). The least-squares method finds the sum of the squares of the residuals SS resid and minimizes the sum using calculus. SS resid N 2 yi ( b mxi ) i the vertical deviation of each point from the straight line is called a residual ( 殘差 ). (8-10) (8-11) (8-12)

54 1. The slope of the line m: (8-13) 2. The intercept b: (8-14) 3. The standard deviation about regression s r : (3-24) (8-15) The standard deviation of the slope s m : (8-16) 5. The standard deviation of the intercept s b : (8-17) 6. The standard deviation for results obtained from the calibration curve s c : y c : mean of a set of M replicate analysis. y: mean for the N calibration points (8-18)

55 The standard deviation about regression s r (Equation 8-15) is the standard deviation for y when the deviations are measured not from the mean of y (as is usually the case) but from the straight line that results from the least-squares analysis: The standard deviation about regression is often called the standard error of the estimate or the standard error in y

56 56