© 2002 By CRC Press LLC Another common case is the difference of two averages, as in comparative t -tests. The variance of the difference is: If the measured quantities are multiplied by a fixed constant: the variance and standard deviation of y are: Table 10.1 gives the standard deviation for a few examples of algebraically combined data. Example 10.1 In a titration, the initial reading on the burette is 3.51 mL and the final reading is 15.67 mL both with standard deviation of 0.02 mL. The volume of titrant used is V = 15.67 − 3.51 = 12.16 mL. The variance of the difference between the two burette readings is the sum of the variances of each reading. The standard deviation of titrant volume is: The standard deviation for the final result is larger than the standard deviations of the individual burette readings, although the volume is calculated as the difference, but it is less than the sum of the standard deviations. Sometimes, calculations produce nonconstant variance from measurements that have constant variance. Another look at titration errors will show how this happens. Example 10.2 The concentration of a water specimen is measured by titration as C = 20( y 2 − y 1 ) where y 1 and y 2 are initial and final burette readings. The coefficient 20 converts milliliters of titrant used ( y 2 − y 1 ) into a concentration (mg / L). Assuming the variance of a burette reading is constant for all y , TABLE 10.1 Standard Deviations from Algebraically Combined Data x σ x x 2 2 x σ x y σ y 1 / xxy x + yx / y x − y exp( ax ) ln x log 10 x x σ x 2 x σ x x 2 σ x 2 y 2 σ y 2 x 2 + σ x 2 σ y 2 + x y σ x 2 y 2 σ y 2 x 2 +   σ x 2 σ y 2 + a 2 ax()exp σ x x e 10 σ x x log δ y 1 y 2 –= σ δ 2 σ y 1 σ y 2 += ykk a ak b bk c c … ++++= σ y 2 k a 2 σ a 2 k b 2 σ b 2 k c 2 σ c 2 … +++= σ y k a 2 σ a 2 k b 2 σ b 2 k c 2 σ c 2 … +++= σ V 0.02() 2 0.02() 2 + 0.03== σ y 2 L1592_Frame_C10 Page 88 Tuesday, December 18, 2001 1:46 PM © 2002 By CRC Press LLC the variance of the computed concentration is: Suppose that the standard deviation of a burette reading is σ y = 0.02 mL, giving = 0.0004. For y 1 = 38.2 and y 2 = 25.7, the concentration is: and the variance and standard deviation of concentration are: Notice that the variance and standard deviation are not functions of the actual burette readings. Therefore, this value of the standard deviation holds for any difference (y 2 − y 1 ). The approximate 95% confidence interval would be: Example 10.3 Suppose that a water specimen is diluted by a factor D before titration. D = 2 means that the specimen was diluted to double its original volume, or half its original concentration. This might be done, for example, so that no more than 15 mL of titrant is needed to reach the end point (so that y 2 − y 1 ≤ 15). The estimated concentration is C = 20D(y 2 − y 1 ) with variance: D = 1 (no dilution) gives the results just shown in Example 10.2. For D > 1, any variation in error in reading the burette is magnified by D 2 . Var(C) will be uniform over a narrow range of concentration where D is constant, but it will become roughly proportional to concentration over a wider range if D varies with concentration. It is not unusual for environmental data to have a variance that is proportional to concentration. Dilution or concentration during the laboratory processing will produce this characteristic. Multiplicative Expressions The propagation of error is different when variables are multiplied or divided. Variability may be magnified or suppressed. Suppose that y = ab. The variance of y is: and Var C() σ C 2 20() 2 σ y 2 2 σ y 1 2 + ()400 σ y 2 σ y 2 + ()800 σ y 2 == = = σ y 2 C 20 38.2 25.7–()250 mg/L== Var C() σ C 2 20() 2 0.0004 0.0004+()0.32== = σ C 0.6 mg/L= 250 2 0.6() mg/L± 250 1.2 mg/L±= σ C 2 20D() 2 σ y 2 σ y 2 + () 800D 2 σ y 2 == σ y 2 σ a 2 a 2 σ b 2 b 2 += σ y 2 y 2 σ a 2 a 2 σ b 2 b 2 += L1592_Frame_C10 Page 89 Tuesday, December 18, 2001 1:46 PM © 2002 By CRC Press LLC Likewise, if y = a/b, the variance is: and Notice that each term is the square of the relative standard deviation (RSD) of the variables. The RSDs are σ y /y, σ a /a, and σ b /b. These results can be generalized to any combination of multiplication and division. For: where a, b, c and d are measured and k is a constant, there is again a relationship between the squares of the relative standard deviations: Example 10.4 The sludge age of an activated sludge process is calculated from θ = , where X a is mixed- liquor suspended solids (mg/L), V is aeration basin volume, Q w is waste sludge flow (mgd), and X w is waste activated sludge suspended solids concentration (mg/L). Assume V = 10 million gallons is known, and the relative standard deviations for the other variables are 4% for X a , 5% for X w , and 2% for Q w . The relative standard deviation of sludge age is: The RSD of the final result is not so much different than the largest RSD used to calculate it. This is mainly a consequence of squaring the RSDs. Any efforts to improve the precision of the experiment need to be directed toward improving the precision of the least precise values. There is no point wasting time trying to increase the precision of the most precise values. That is not to say that small errors are unimportant. Small errors at many stages of an experiment can produce appreciable error in the final result. Error Suppression and Magnification A nonlinear function can either suppress or magnify error in measured quantities. This is especially true of the quadratic, cubic, and exponential functions that are used to calculate areas, volumes, and reaction rates in environmental engineering work. Figure 10.1 shows that the variance in the final result depends on the variance and the level of the inputs, according to the slope of the curve in the range of interest. σ y 2 σ a 2 a 2 σ b 2 a 2 b 2 += σ y 2 y 2 σ a 2 a 2 σ b 2 b 2 += ykab/cd= σ y y σ a a   2 σ b b   2 σ c c   2 σ d d   2 +++= X a V Q w X w σ θ θ 4 2 5 2 2 2 ++ 45 6.7%=== L1592_Frame_C10 Page 90 Tuesday, December 18, 2001 1:46 PM © 2002 By CRC Press LLC Example 10.5 Particle diameters are to be measured and used to calculate particle volumes. Assuming that the particles are spheres, V = π D 3 /6, the variance of the volume is: and The precision of the estimated volumes will depend upon the measured diameter of the particles. Suppose that σ D = 0.02 for all diameters of interest in a particular application. Table 10.2 shows the relation between the diameter and variance of the computed volumes. At D = 0.798, the variance and standard deviation of volume equal those of the diameter. For small D (<0.798), errors are suppressed. For larger diameters, errors in D are magnified. The distribution of V will be stretched or compressed according to the slope of the curve that covers the range of values of D. Preliminary investigations of error transmission can be a valuable part of experimental planning. If, as was assumed here, the magnitude of the measurement error is the same for all diameters, a greater number of particles should be measured and used to estimate V if the particles are large. FIGURE 10.1 Errors in the computed volume are suppressed for small diameter (D) and inflated for large D. TABLE 10.2 Propagation of Error in Measured Particle Diameter into Error in the Computed Particle Diameter D 0.5 0.75 0.798 1 1.25 1.5 V 0.065 0.221 0.266 0.524 1.023 1.767 0.00006 0.00031 0.00040 0.00099 0.00241 0.00500 σ V 0.008 0.018 0.020 0.031 0.049 0.071 σ V / σ D 0.393 0.884 1.000 1.571 2.454 3.534 0.154 0.781 1.000 2.467 6.024 12.491 1. 1. 0 0.5 1.0 1.5 0 5 0 Particle Volume, V Particle Diameter, D Error in y inflated Error in y Error in y suppressed Error in D Var V() 3 π 6 D 2   2 σ D 2 2.467D 4 σ D 2 == σ V 1.571D 2 σ D = σ V 2 σ V 2 / σ D 2 L1592_Frame_C10 Page 91 Tuesday, December 18, 2001 1:46 PM © 2002 By CRC Press LLC Case Study: Calcium Carbonate Scaling in Water Mains A small layer of calcium carbonate scale on water mains protects them from corrosion, but heavy scale reduces the hydraulic capacity. Finding the middle ground (protection without damage to pipes) is a matter of controlling the pH of the water. Two measures of the tendency to scale or corrode are the Langlier saturation index (LSI) and the Ryznar stability index (RSI). These are: where pH is the measured value and pH s the saturation value. pH is a calculated value that is a function of temperature (T), total dissolved solids concentration (TDS), alkalinity [Alk], and calcium concentration [Ca]. [Alk] and [Ca] are expressed as mg/L equivalent CaCO 3 . The saturation pH is pH s = A − log 10 [Ca] − log 10 [Alk], where A = 9.3 + log 10 (K s /K 2 ) + , in which µ is the ionic strength. K s , a solubility product, and K 2 , an ionization constant, depend on temperature and TDS. As a rule of thumb, it is desirable to have LSI = 0.25 ± 0.25 and RSI = 6.5 ± 0.3. If LSI > 0, CaCO 3 scale tends to deposit on pipes, if LSI < 0, pipes may corrode (Spencer, 1983). RSI < 6 indicates a tendency to form scale; at RSI > 7.0, there is a possibility of corrosion. This is a fairly narrow range of ideal conditions and one might like to know how errors in the measured pH, alkalinity, calcium, TDS, and temperature affect the calculated values of the LSI and RSI. The variances of the index numbers are: Var(LSI) = Var(pH s ) + Var(pH) Var(RSI) = 2 2 Var(pH s ) + Var(pH) Given equal errors in pH and pH s , the RSI value is more uncertain than the LSI value. Also, errors in estimating pH s are four times more critical in estimating RSI than in estimating LSI. Suppose that pH can be measured with a standard deviation σ = 0.1 units and pH s can be estimated with a standard deviation of 0.15 unit. This gives: Var(LSI) = (0.15) 2 + (0.1) 2 = 0.0325 σ LSI = 0.18 pH units Var(RSI) = 4(0.15) 2 + (0.1) 2 = 0.1000 σ RSI = 0.32 pH units Suppose further that the true index values for the water are RSI = 6.5 and LSI = 0.25. Repeated measure- ments of pH, [Ca], [Alk], and repeated calculation of RSI and LSI will generate values that we can expect, with 95% confidence, to fall in the ranges of: LSI = 0.25 ± 2(0.18) −0.11 < LSI < 0.61 RSI = 6.5 ± 2(0.32) 5.86 < RSI < 7.14 These ranges may seem surprisingly large given the reasonably accurate pH measurements and pH s estimates. Both indices will falsely indicate scaling or corrosive tendencies in roughly one out of ten calculations even when the water quality is exactly on target. A water utility that had this much variation in calculated values would find it difficult to tell whether water is scaling, stable, or corrosive until after many measurements have been made. Of course, in practice, real variations in water chemistry add to the “analytical uncertainty” we have just estimated. LSI pH pH s –= RSI 2pH s pH–= 2.5 µ µ 5.5+ L1592_Frame_C10 Page 92 Tuesday, December 18, 2001 1:46 PM © 2002 By CRC Press LLC In the example, we used a standard deviation of 0.15 pH units for pH s . Let us apply the same error propagation technique to see whether this was reasonable. To keep the calculations simple, assume that A, K s , K 2 , and µ are known exactly (in reality, they are not). Then: Var(pH s ) = (log 10 e) 2 {[Ca] −2 Var[Ca] + [Alk] −2 Var[Alk]} The variance of pH s depends on the level of the calcium and alkalinity as well as on their variances. Assuming [Ca] = 36 mg/L, σ [Ca] = 3 mg/L, [Alk] = 50 mg/L, and σ [Alk] = 3 mg/L gives: Var(pH s ) = 0.1886{[36] −2 (3) 2 + [50] −2 (3) 2 } = 0.002 which converts to a standard deviation of 0.045, much smaller than the value used in the earlier example. Using this estimate of Var(pH s ) gives approximate 95% confidence intervals of: 0.03 < LSI < 0.47 6.23 < RSI < 6.77 This example shows how errors that seem large do not always propagate into large errors in calculated values. But the reverse is also true. Our intuition is not very reliable for nonlinear functions, and it is useless when several equations are used. Whether the error is magnified or suppressed in the calculation depends on the function and on the level of the variables. That is, the final error is not solely a function of the measurement error. Random and Systematic Errors The titration example oversimplifies the accumulation of random errors in titrations. It is worth a more complete examination in order to clarify what is meant by multiple sources of variation and additive errors. Making a volumetric titration, as one does to measure alkalinity, involves a number of steps: 1. Making up a standard solution of one of the reactants. This involves (a) weighing some solid material, (b) transferring the solid material to a standard volumetric flask, (c) weighing the bottle again to obtain by subtraction the weight of solid transferred, and (d) filling the flask up to the mark with reagent-grade water. 2. Transferring an aliquot of the standard material to a titration flask with the aid of a pipette. This involves (a) filling the pipette to the appropriate mark, and (b) draining it in a specified manner into the flask. 3. Titrating the liquid in the flask with a solution of the other reactant, added from a burette. This involves filling the burette and allowing the liquid in it to drain until the meniscus is at a constant level, adding a few drops of indicator solution to the titration flask, reading the burette volume, adding liquid to the titration flask from the burette a little at a time until the end point is adjudged to have been reached, and measuring the final level of liquid in the burette. The ASTM tolerances for grade A glassware are ±0.12 mL for a 250-mL flask, ±0.03 mL for a 25-mL pipette, and ±0.05 mL for a 50-mL burette. If a piece of glassware is within the tolerance, but not exactly the correct weight or volume, there will be a systematic error. Thus, if the flask has a volume of 248.9 mL, this error will be reflected in the results of all the experiments done using this flask. Repetition will not reveal the error. If different glassware is used in making measurements on different specimens, random fluctuations in volume become a random error in the titration results. L1592_Frame_C10 Page 93 Tuesday, December 18, 2001 1:46 PM © 2002 By CRC Press LLC The random errors in filling a 250-mL flask might be ±0.05 mL, or only 0.02% of the total volume of the flask. The random error in filling a transfer pipette should not exceed 0.006 mL, giving an error of about 0.024% of the total volume (Miller and Miller, 1984). The error in reading a burette (of the conventional variety graduated in 0.1-mL divisions) is perhaps ±0.02 mL. Each titration involves two such readings (the errors of which are not simply additive). If the titration volume is about 25 mL, the percentage error is again very small. (The titration should be arranged so that the volume of titrant is not too small.) In skilled hands, with all precautions taken, volumetric analysis should have a relative standard deviation of not more than about 0.1%. (Until recently, such precision was not available in instrumental analysis.) Systematic errors can be due to calibration, temperature effects, errors in the glassware, drainage errors in using volumetric glassware, failure to allow a meniscus in a burette to stabilize, blowing out a pipette that is designed to drain, improper glassware cleaning methods, and “indicator errors.” These are not subject to prediction by the propagation of error formulas. Comments The general propagation of error model that applies exactly to all linear models z = f(x 1 , x 2 ,…, x n ) and approximately to nonlinear models (provided the relative standard deviations of the measured variables are less than about 15%) is: where the partial derivatives are evaluated at the expected value (or average) of the x i . This assumes that there is no correlation between the x’s. We shall look at this and some related ideas in Chapter 49. References Betz Laboratories (1980). Betz Handbook of Water Conditioning, 8th ed., Trevose, PA, Betz Laboratories. Langlier, W. F. (1936). “The Analytical Control of Anticorrosion in Water Treatment,” J. Am. Water Works Assoc., 28, 1500. Miller, J. C. and J. N. Miller (1984). Statistics for Analytical Chemistry, Chichester, England, Ellis Horwood Ltd. Ryznar, J. A. (1944). “A New Index for Determining the Amount of Calcium Carbonate Scale Formed by Water,” J. Am. Water Works Assoc., 36, 472. Spencer, G. R. (1983). “Program for Cooling-Water Corrosion and Scaling,” Chem. Eng., Sept. 19, pp. 61–65. Exercises 10.1 Titration. A titration analysis has routinely been done with a titrant strength such that con- centration is calculated from C = 20(y 2 − y 1 ), where (y 2 − y 1 ) is the difference between the final and initial burette readings. It is now proposed to change the titrant strength so that C = 40(y 2 − y 1 ). What effect will this have on the standard deviation of measured concentrations? 10.2 Flow Measurement. Two flows (Q 1 = 7.5 and Q 2 = 12.3) merge to form a larger flow. The standard deviation of measurement on flows 1 and 2 are 0.2 and 0.3, respectively. What is the standard deviation of the larger downstream flow? Does this standard deviation change when the upstream flows change? σ z 2 ∂ z/ ∂ x 1 () 2 σ 1 2 ∂ z/ ∂ x 2 () 2 σ 2 2 … ∂ z/ ∂ x n () 2 σ n 2 +++≈ L1592_Frame_C10 Page 94 Tuesday, December 18, 2001 1:46 PM © 2002 By CRC Press LLC 10.3 Sludge Age. In Example 10.4, reduce each relative standard deviation by 50% and recalculate the RSD of the sludge age. 10.4 Friction Factor. The Fanning equation for friction loss in turbulent flow is ∆ p = , where ∆ p is pressure drop, f is the friction factor, V is fluid velocity, L is pipe length, D is inner pipe diameter, ρ is liquid density, and g is a known conversion factor. f will be estimated from experiments. How does the precision of f depend on the precision of the other variables? 10.5 F/M Loading Ratio. Wastewater treatment plant operators often calculate the food to micro- organism ratio for an activated sludge process: where Q = influent flow rate, S 0 = influent substrate concentration, X = mixed liquor suspended solids concentration, and V = aeration tank volume. Use the values in the table below to calculate the F/M ratio and a statement of its precision. 10.6 TOC Measurements. A total organic carbon (TOC) analyzer is run by a computer that takes multiple readings of total carbon (TC) and inorganic carbon (IC) on a sample specimen and computes the average and standard deviation of those readings. The instrument also computes TOC = TC − IC using the average values, but it does not compute the standard deviation of the TOC value. Use the data in the table below to calculate the standard deviation for a sample of settled wastewater from the anaerobic reactor of a milk processing plant. 10.7 Flow Dilution. The wastewater flow in a drain is estimated by adding to the upstream flow a 40,000 mg/L solution of compound A at a constant rate of 1 L/min and measuring the diluted A concentration downstream. The upstream (background) concentration of A is 25 mg/L. Five downstream measurements of A, taken within a short time period, are 200, 230, 192, 224, and 207. What is the best estimate of the wastewater flow, and what is the variance of this estimate? 10.8 Surface Area. The surface area of spherical particles is estimated from measurements on particle diameter. The formula is A = π D 2 . Derive a formula for the variance of the estimated surface areas. Prepare a diagram that shows how measurement error expands or contracts as a function of diameter. 10.9 Lab Procedure. For some experiment you have done, identify the possible sources of random and systematic error and explain how they would propagate into calculated values. Variable Average Std. Error Q = Flow (m 3 /d) 35000 1500 S 0 = BOD 5 (mg/L) 152 15 X = MLSS (mg/L) 1725 150 V = Volume (m 3 ) 13000 600 Measurement Mean (mg/L) Number of Replicates Standard Deviation (mg/L) TC 390.6 3 5.09 IC 301.4 4 4.76 2 fV 2 L ρ gD F M QS 0 XV = L1592_Frame_C10 Page 95 Tuesday, December 18, 2001 1:46 PM © 2002 By CRC Press LLC 11 Laboratory Quality Assurance KEY WORDS bias, control limit, corrective action, precision, quality assurance, quality control, range, Range chart, Shewhart chart, (X-bar) chart, warning limit. Engineering rests on making measurements as much as it rests on making calculations. Soil, concrete, steel, and bituminous materials are tested. River flows are measured and water quality is monitored. Data are collected for quality control during construction and throughout the operational life of the system. These measurements need to be accurate. The measured value should be close to the true (but unknown) value of the density, compressive strength, velocity, concentration, or other quantity being measured. Measurements should be consistent from one laboratory to another, and from one time period to another. Engineering professional societies have invested millions of dollars to develop, validate, and standard- ize measurement methods. Government agencies have made similar investments. Universities, technical institutes, and industries train engineers, chemists, and technicians in correct measurement techniques. Even so, it is unrealistic to assume that all measurements produced are accurate and precise. Testing machines wear out, technicians come and go, and sometimes they modify the test procedure in small ways. Chemical reagents age and laboratory conditions change; some people who handle the test specimens are careful and others are not. These are just some of the reasons why systematic checks on data quality are needed. It is the laboratory’s burden to show that measurement accuracy and precision fall consistently within acceptable limits. It is the data user’s obligation to evaluate the quality of the data produced and to insist that the proper quality control checks are done. This chapter reviews how and Range charts are used to check the accuracy and precision of laboratory measurements. This process is called quality control or quality assurance . and Range charts are graphs that show the consistency of the measurement process. Part of their value and appeal is that they are graphical. Their value is enhanced if they can be seen by all lab workers. New data are plotted on the control chart and compared against recent past performance and against the expected (or desired) performance. Constructing X-Bar and Range Charts The scheme to be demonstrated is based on multiple copies of prepared control specimens being inserted into the routine work. As a minimum, duplicates (two replicates) are needed. Many labs will work with this minimum number. The first step in constructing a control chart is to get some typical data from the measurement process when it is in a state of good statistical control . Good statistical control means that the process is producing data that have negligible bias and high precision (small standard deviation). Table 11.1 shows measure- ments on 15 pairs of specimens that were collected when the system had a level and range of variation that were typical of good operation. Simple plots of data are always useful. In this case, one might plot each measured value, the average of paired values, and the absolute value of the range of the paired values, as in Figure 11.1. These plots X X X L1592_Frame_C11 Page 97 Tuesday, December 18, 2001 1:47 PM © 2002 By CRC Press LLC show the typical variation of the measurement process. Objectivity is increased by setting warning limits and action limits to define an unusual condition so all viewers will react in the same way to the same signal in the data. The two simplest control charts are the (pronounced X -bar) chart and the Range ( R ) chart. The chart (also called the Shewhart chart, after its inventor) provides a check on the process level and also gives some information about variation. The Range chart provides a check on precision (variability). The acceptable variation in level and precision is defined by control limits that bound a specified percentage of all results expected as long as the process remains in control. A common specification is 99.7% of values within the control limits. Values falling outside these limits are unusual enough to activate a review of procedures because the process may have gone wrong. These control limits are valid only when the variation is random above and below the average level. The equations for calculating the control limits are: where is the grand mean of sample means (the average of the values used to construct the chart), is the mean sample range (the average of the ranges [ R ] used to construct the chart), and n is the number of replicates used to compute the average and the range at each sampling interval. R is the absolute difference between the largest and smallest values in the subset of n measured values at a particular sampling interval. TABLE 11.1 Fifteen Pairs of Measurements on Duplicate Test Specimens Specimen 12345 6 7 8 9101112131415 X 1 5.2 3.1 2.5 3.8 4.3 3.1 4.5 3.8 4.3 5.3 3.6 5.0 3.0 4.7 3.7 X 2 4.4 4.6 5.3 3.7 4.4 3.3 3.8 3.2 4.5 3.7 4.4 4.8 3.6 3.5 5.2 4.8 3.8 3.9 3.8 4.3 3.2 4.2 3.5 4.4 4.5 4.0 4.9 3.3 4.1 4.4 0.8 1.5 2.8 0.1 0.1 0.2 0.7 0.6 0.2 1.6 0.8 0.2 0.6 1.2 1.5 Grand mean = Mean sample range = FIGURE 11.1 Three plots of the 15 pairs of quality control data with action and warning limits added to the charts for the average and range of X 1 and X 2 . X X 1 X 2 + 2 = R |X 1 X 2 |–= X 4.08,= R 0.86= 151050 X1 X2 & Ave. R Observation UCL UCL X = LCL 2 4 6 2 3 4 5 6 0 1 2 3 4 X X X chart Central line X= Control limits Xk 1 R±= R chart Central line R= Upper control limit (UCL) k 2 R= X X R L1592_Frame_C11 Page 98 Tuesday, December 18, 2001 1:47 PM [...]... 4 5 Time 1 11.8 11.7 13. 0 11.1 10.5 10.5 15.6 10.5 12 .3 11.6 11.7 13. 3 10.5 10.9 7.4 11.5 10.8 12.8 9.9 11.4 13. 6 12.7 12.1 10.6 9.1 11.8 11.7 13. 0 11.1 10.5 10.5 11 .3 10.5 12 .3 11.6 11.7 11.2 10.8 11.2 10.8 11.9 11.1 13. 1 10 .3 11.8 14.0 13. 1 12.5 11.1 9.6 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 11.4 13. 0 10.6 11 .3 10.9 13. 2 13. 4 13. 4 12.8 12 .3 11.1 11.8 12.7 12.8... Known Amounts of Lead Have Been Added Pb Added ( µg/L) Measured Lead Concentrations ( µg/L) Zero 1.25 2.5 2.8 4.0 3. 8 2.7 1.7 2.2 3. 4 2.2 2.2 2.4 2.4 3. 1 3. 0 3. 5 2.6 3. 7 2.2 4.6 2.7 3. 1 3. 2 3. 6 2.8 3. 1 4 .3 2.50 4.5 3. 3 3. 7 4.7 3. 8 4.4 4.4 5.4 3. 9 4.1 3. 7 3. 0 4.5 4.8 5.0 10 3. 9 12.2 5.0 13. 8 5.4 9.9 4.9 10.5 6.2 10.9 Note: The “zero added” specimens are not expected to be lead-free greater than zero... purpose and performance of the three charts Observ y1 y2 Observ y1 y2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5.88 5.64 5.09 6.04 4.66 5.58 6.07 5 .31 5.48 6. 63 5.28 5.97 5.82 5.74 5.97 5.61 5. 63 5.12 5 .36 5.24 4.50 5.41 6 .30 5. 83 5. 23 5.91 5.81 5.19 5.41 6.60 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 5.70 4.90 5.40 5 .32 4.86 6.01 5.55 5.44 5.05 6.04 5. 63 5.67 6 .33 5.94 6.68 5.96 5.65 6.71 5.67 4 .34 5.57 5.55... 12 13 14 15 16 17 18 19 20 21 22 23 24 25 11.8 11.7 13. 0 11.1 10.5 10.5 11 .3 10.5 12 .3 11.6 11.7 11.0 10.5 10.9 10.5 11.5 10.8 12.8 9.9 11.4 13. 6 12.7 12.1 10.6 9.1 Machine 2 3 11.8 11.7 13. 0 11.1 10.5 10.5 11 .3 10.5 12 .3 11.6 11.7 11.0 10.5 11.4 11.4 12.4 11 .3 12.8 9.4 10.6 12.6 11.8 11.6 10.6 9.6 11.8 11.7 13. 0 11.1 10.5 10.5 11 .3 13. 0 14.8 14.1 14.2 13. 5 10.5 10.9 10.5 11.5 10.8 12.8 9.9 11.4 13. 6... 11.4 12 .3 11.8 10.9 Machine 2 3 12 .3 14.0 11.5 11.8 10.9 12.7 12.6 12.4 12.0 11.8 11.1 11.8 12.7 12.8 10.9 11.7 12.0 11.7 10.4 9.8 10.2 11.4 12 .3 11.8 10.9 14.9 16.5 14.1 14.8 14.4 16.7 16.9 13. 4 12.8 12 .3 11.1 11.8 12.7 12.8 10.9 11.7 12.0 11.7 10.4 9.8 10.2 11.4 12 .3 11.8 10.9 4 5 11.4 13. 0 10.6 11 .3 10.9 13. 2 13. 4 13. 4 12.8 12 .3 11.1 11.8 12.7 12.8 10.9 11.7 12.0 11.7 10.4 9.8 10.2 11.4 12 .3 11.8... 2.5, 2.7, 2.2, 2.2, 3. 1, 2 2.6, 2.8 The estimated variance is s = 0.10, s = 0 .32 , t = 3. 1 43, and the MDL = 3. 1 43( 0 .32 ) = 1.0 Another set of seven replicate specimens was analyzed to get a second estimate of the 2 variance; the data were 1.6, 1.9, 1 .3, 1.7, 2.1, 0.9, 1.8 These data give s = 0.16, s = 0.40, and the MDL = 3. 1 43( 0.40) = 1 .3 A statistical F test (0.16/0.10 ≤ F6,6 = 4 .3) shows that the two... 13 y 12 11 MA (5) 10 13 3 σ limit 12 3 σ limit EWMA 11 13 λ = 0.5 3 σ limit 12 3 σ limit 11 0 30 60 90 Observation 120 150 FIGURE 12.2 Moving average (5-day) and exponentially weighted moving average (λ = 0.5) charts for the single observations shown in the top panel The mean level shifts up by 0.5 units from days 50–75, it is back to normal from days 76–92, it shifts down by 0.5 units from days 93 107,... where α is the area under the tail of the distribution that lies above za For example: α = 0.05 z a 1. 63 0.0 23 0.01 0.00 13 2.00 2 .33 3. 00 Using z = 3. 00 (α = 0.00 13) means that observing y > MDL justifies a conclusion that η is not zero at a confidence level of 99.87% Using this MDL repeatedly, just a little over one in one thousand (0. 13% ) of true blank specimens will be misjudged as positive determinations...L1592_Frame_C11 Page 99 Tuesday, December 18, 2001 1:47 PM TABLE 11.2 Coefficients for Calculating Action Lines on X and Range Charts n k1 k2 2 3 4 5 1.880 1.0 23 0.729 0.577 3. 267 2.575 2.282 2.115 Source: Johnson, R A (2000) Probability and Statistics for Engineers, 6th ed., Englewood Cliffs, NJ, Prentice-Hall The coefficients of k1 and k2 depend on the size of the subsample... Start the Control Charts for the Cusum, 5-Day Moving Average, and the Exponentially Weighted Moving Average (λ = 0.5) (1) yi (2) yI − 12 (3) Cusum 11.89 12.19 12.02 11.90 12.47 12.64 11.86 12.61 11.89 12.87 12.09 11.50 11.84 11.17 −0.11 0.19 0.02 −0.10 0.47 0.64 −0.14 0.61 −0.11 0.87 0.09 −0.50 −0.16 −0. 83 −0.11 0.08 0.10 0.00 0.47 1.11 0.97 1.57 1.47 2 .33 2.42 1. 93 1.76 0. 93 (4) MA(5) (5) EWMA (λ = . Specimen 1 234 5 6 7 8 9101112 131 415 X 1 5.2 3. 1 2.5 3. 8 4 .3 3.1 4.5 3. 8 4 .3 5 .3 3.6 5.0 3. 0 4.7 3. 7 X 2 4.4 4.6 5 .3 3.7 4.4 3. 3 3. 8 3. 2 4.5 3. 7 4.4 4.8 3. 6 3. 5 5.2 4.8 3. 8 3. 9 3. 8 4 .3 3.2. pieces. MA (5) y EWMA λ = 0.5 11 12 13 10 11 12 13 11 12 13 150120906 030 0 Observation 3 σ limit 3 σ limit 3 σ limit 3 σ limit 150120906 030 0 Cusum y 0 0 10 10 11 12 13 -1 Observation L1592_frame_C12.fm. Flow (m 3 /d) 35 000 1500 S 0 = BOD 5 (mg/L) 152 15 X = MLSS (mg/L) 1725 150 V = Volume (m 3 ) 130 00 600 Measurement Mean (mg/L) Number of Replicates Standard Deviation (mg/L) TC 39 0.6 3 5.09 IC