How to Measure Roll Quality

W«b Handling Research Cente Oklahoma State University

How to measure roll quality David R. Roisum* The best (est method is the one that can detect a small change in roll structure with the fewest measurements.

The winding process takes a strip of flexible material and turns it into a wound roll. We can avoid for a time the description of the complex details of the winding process by using a black-box approach, as seen in Fig. 1. The system can be divided into three categories: input parameters, process laws, and output results. Input parameters are variables that we can (sometimes) control to optimize the output results, which can be loosely described as roll quality. The process laws are the constant and inviolate behavior of physical systems that we seek to describe using engineering mechanics and other methods. Our task as quality control engineers is to select values for the input parameters so that we maximize roll quality.

between drum and paper (usually about 0.35) and the normal force (determined by roll weight, geometry, and nip loading at the rider roll). Exceeding the friction limit will simply cause slipping and, possibly, instability and sheet marking. Nip cannot be negative, nor can it be so high that it kneads the winding roll, increasing interlayer slippage to the point of creping or shear burst. If tension is too low, the propensity for wrinkling is increased, and the sheet run may flutter. Additionally, web tension cannot be increased too much because sheet breaks are an exponential function of web tension. The point to remember is that although the TNTs are the most easily controlled inputs, they often have narrow ranges of useful adjustment. The operator must find a delicate balance among all of these parameters. Additionally, the task Winding process—input parameters is made even more difficult because the material properties Torque, nip, and tension change, either intentionally (grade changes) or The input parameters that are the easiest to control are known unintentionally (manufacturing process instabilities). These as the TNTs of winding: torque, nip, and tension. Machine property changes cause the optimum winding profiles to vary operators can usually adjust TNTs via benchboard controls. with time. As seen in Fig. 2, torque can be applied as a differential between the front and back drum on a two-drum winder, or Paper properties through a center-shaft on a duplex winder. Nip is the lineal Some paper properties have a strong influence on roll pressure between the winding roll and a roller or drum. structure. These include caliper, density, MD (machine Tension is the lineal load applied on the draw immediately direction) yield and tensile strength, coefficient of friction, coefficient of hygroscopic expansion, porosity, as well as the upstream of the winding roll. As seen in Fig. 3, the TNTs of winding are set point elastic moduli in both the MD and the ZD ( z direction). functions of wound-roll diameter. The TNTs of winding are Other paper properties that affect web quality—including often linearly decreasing from start to finish to give the roll a optical properties (brightness and opacity) and hygroscopic structure that is hard near the core and then decreases properties (freeness and water retention)—have essentially smoothly to a softer finish at the outside. Rider-roll nip on a no effect on the quality of roll structure. If one were to two-drum winder is perhaps the most complicated of the choose an optimum web for winding, it would probably have controls because the total back-drum nip is the sum of rider- (a) high caliper, strength, friction, and porosity and (b) low anisotropy (MD/ZD modulus) and coefficient of hygroscopic roll nip, roll weight, and winding-angle geometry (/). Increasing any one of the TNTs will usually make the roll expansion. Two points must be made concerning paper properties as harder at the point of increase. Decreasing them will make the roll softer. However, we cannot specify the TNTs on an inputs to the winding process. First, though some paper unlimited range. Torque is limited to the available friction, properties have a profound influence on winding, they cannot usually be considered an input variable for the which depends on the coefficient of friction purpose- of o p t i m i z i n g the w i n d i n g process. Paper •Roisum is currently a senior research engineer. Web Handling Research properties are generally specified by the paper mill's Center. Oklahoma State University. 218 Engineering North, Stillwater, Okla. customers. Therefore, average paper properties are not an 74078. i n p u t variable at our disposal. In addition, paper properties are not a modifiable output from the winding system. Paper Roisum is a winder consultant for Beloit Corp. R&D, properties are seldom measurably changed by the winding 1165 Prairie Hill Road, Rockton, 111. 61072. process. As finishing room superintendents would say, "You cannot make paper on a winder." October 1988 Tappi Journal 91

1. Knowledge of the winding process can be divided into three categories: input parameters, process laws, and output results

=3 0 The process r

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Second, the mechanical properties of paper vary with MD position (or time) and CD (cross-machine direction) position. These variations exert a great deal of influence on the winding process. Web breaks, for example, occur at rare local weaknesses in the web. It can be shown that the webbreak problem is influenced more by the variations of strength than by the average strength (2, 3). Another example is the ropes and corrugations that are almost entirely related to variations of caliper across the web. Reducing paper property variations to optimize winding can be considered a controllable input parameter only if the finishing room has influence over the paper machine. Usually, we must wind the paper that is given to us.

3. Typical set points for torque, nip, and tension across the diameter of a wound roll £

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0 10 20 30 40 Machine configuration Winding machines can be classified as centerwinders, center— --------------------------------------------------surfacewinders, surfacewinders. and two-drum winders. These basic machine classes determine to a large extent the range over which the TNTs can be controlled. The centerwinder can vary only tension as a TNT parameter; 0 10 40 center-surfacewinders and two-drum winders can vary any of 20 30 ROLL the TNTs, and surfacewinders can vary only nip and tension. DIAMETER, in. Centerwinders and center-surfacewinders usually give the highest roll quality, but these machines are often more expensive and less productive. Additionally, centerwinders can be speed limited by air entrainment on nonporous webs. 4. Optimum profile for roll-structure parameters (hardness, wound-in tension, interlayer pressure, density, and stress or strain) across the However, winding-machine type can only be considered an diameter of a wound roll input variable when purchasing a new winder. For any given machine, there are design details that affect UJ wound-roll quality. These include the web run, roll diameters ^__ DC Z> and grooving, spreaders, as well as the type of TNT controls HUJ O-l and drives. But unless the winder is being rebuilt, these oc< design details are not considered to be inputs that can be •-E «?< _j> varied to optimize the winding process. _i O The winding process can be improved, however, by OS controlling various maintenance parameters. Winder rolls

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Input parameters

2. Torque, nip, and tension are the easiest parameters to control on a winder Torque can be applied as a differential force between the front and back drums on rwo-drum winders or through a center shaft on duplex winders. Nip is the lineal pressure between the paper roll and a roller or drum Tension is the lineal load applied on the draw immediately upstream of the paper roll.

92

October 1988 Tappl Journal"

ROLL DIAMETER

5. The Rhometer is an impact-based device tor measuring roll hardness The instrument measures the peak deceleration of a small hammer as it strikes the paper roll

must be accurately aligned (0.005 in./lOO in.) to avoid wrinkling of the web and roll. Drives and brakes must be tuned so that speed changes result in less than a 5% change in tension. Rider-roll loading systems and core-chuck slides must move freely without binding or stick-slip. Finally, the framework must be rigid, and there must be no looseness or play in any of the moving parts U). Operator technique In addition to setting the TNTs for each run, the operator also has subtle but important influences on roll-structure quality while setting up and running a set of rolls. Successful operators can be identified by objective measurement of productivity and quality. The techniques of these operators can be passed to others through training. Thus, operator technique is another variable that can be used to optimize roll quality.

Winding process—physical laws The winding process is the connection between input variables and output results. The process is not a variable but, rather, a system of inviolate physical laws that describe the paper stresses in a roll resulting from the various input parameters. The importance of these physical laws is that they quantitatively describe certain aspects of roll quality, allowing us to predict roll-structure results for any set of input parameters. These laws, which are usually solid mechanics formulations, can be coded into a computer. One can then optimize the winding process by running "what if scenarios on a computer, much like the business use of spreadsheets for modeling a product or a company. The advantage of computer modeling is that numerous combinations of inputs can be run in a short period of time at no risk to the product. The user then scans the results and chooses the best combination of inputs. The disadvantage of modeling is that stresses are difficult to verify by measurement. In addition, defect

6. The Schmidt hammer is an impact-based device for measuring roll hardness The instrument measures the rebound height of a small plunger after it strikes the paper roll.

models are only now beginning to emerge. Mechanics equations were used in the first analytical model of roll winding, which appeared in the late 1950s. This was an accretion model based on linear, isotropic, hoopstress formulas. The model superimposed the stresses resulting from the addition of a single wrap upon the existing stress distribution as each wrap was added from the core to the finish diameter. Subsequent works removed model restrictions by allowing for nonlinear anisotropy. Current winding models provide a close description of centerwinding. However, the effects of the nip, air entrainment, interlayer slippage, and CD variations remain to be incorporated into a single description of winding. Briefly reviewing the history of winding models, we find numerous works in the last 25 years, each improving on its predecessors. In 1962, Catlow and Walls used a linear isotropic model to derive formulas for the stress distribution of rolls during winding (5). In the mid '60s, Tramposch analytically described the isotropic and anisotropic stress relaxation resulting from creep and thermally induced stresses in wound rolls of magnetic tape (6, 7). In 1968, Altmann extended the winding model to include linear anisotropy (8). In 1974, Blaedel described an optimization procedure for winding (9). In the early '80s, Yagoda improved the winding model near the core and accounted for centrifugal effects during winding (10-18). In 1986, Hakiel extended the anisotropic winding model to include a nonlinear radial modulus as a function of interlayer pressure (H). In 1987, Wu demonstrated that the parameter having the greatest influence on wound-roll stresses was the radial or ZD modulus (15).

Measurement of output roll structure Now that we have covered winding inputs and briefly reviewed winding process models, we now move to the winding process outputs, which are the primary topic of October 1988 Tappl Journal 93

7. The Cameron gap test is a strain-based technique for measunng the tensile force on the outer layer of a paper roll. The layer is severed, and strain is calculated based on the width of the gap and the diameter of the roll.

8. The J-line test is a strain-based technique for measunng the magnitude of mterlayer slippage as a function of winding or unwinding cycles c ■ maximum circumferential movement; a = depth to tangent c a = slope of tip; r * depth; r, - radius when line struck; r2 ■ radius after winding

this article. Returning to Fig. 1, we see that the output results are loosely described as roll structure, which is some measure related to winding stress as a function of roll diameter. Hardness was the first description of roll structure. Since then, wound-in-tension, interlayer pressure, density, and stress have been added as measurable roll-structure quality variables. Figure 4 illustrates an ideal roll-structure profile from the core to the outer layer. The shape of the profile, which shows a hard (tight) start with a smooth transition to a softer (looser) finish, is typical for all grades of paper, regardless of the parameter being measured. Different grades of paper simply have different values for the starting and finish hardnesses. This widely used figure is based on the cumulative experiences of many winding experts, but it is not easily quantifiable. This means that our roll-structure qualitycontrol efforts are limited to measuring roll-structure profiles and then comparing them with the ideal profile. Other techniques, such as judgment or statistics, are used to set starting and finish magnitudes. Roll structure is just one measure of roll quality. Paper properties (tensile strength, burst, caliper, brightness, printability) must also be tested to meet the customer's particular needs. Roll quality also can be quantified by measuring roll geometry (diameter, length, width) against the target size, as well as the magnitude of dishing or offsets. Finally, roll quality can be defined qualitatively as the absence of defects (bursts, corrugations, stars, crushed cores) (16). In this article, however, we restrict our discussion to general testing considerations and roll-structure evaluation.

tightness or hardness. Though roll hardness as measured by the billy club is not a fundamental roll-structure property and lacks quantitative definition, it is nonetheless a useful device that can be found at many winder stations. Quantifying roll hardness became possible with the invention of the Rhometer and the Schmidt hammer. The Rhometer, illustrated in Fig. 5. was invented in 1965 and measures the peak deceleration of a small hammer striking the paper roll (17). The Schmidt hammer, illustrated in Fig. 6, was originally developed to measure the hardness of concrete but later applied to paper rolls. The Schmidt hammer measures the rebound height of a plunger striking the paper roll and is related to the impact's coefficient of restitution. The Rhometer is used extensively for quality control in both the paper and film industries. The Schmidt hammer is widely used in European paper mills.

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MD tension, stress, and strain Another class of roll-structure measurements is based on web stress or strain. The Cameron gap test, illustrated in Fig. 7, is a TAPPI test method for calculating the tensile strain on the outer layer of a roll. Tensile strain is calculated by severing the outer layer and then measuring the resulting gap and the roll circumference (18). The J-line test, illustrated in Fig. 8, indicates the magnitude of interlayer paper slippage as a function of winding or unwinding cycles (19). This test involves striking chalk lines on the roll edge. Interlayer slippage can be measured by the extent of deformation of the J-line tip after winding. Since both the Cameron gap and J-line tests are labor intensive and destructive, they are used infrequently in Hardness impacters production testing. The earliest device used to measure roll-structure quality was Finally, strain gauges can be bonded to the paper web to the backtender's stick or "billy club," a short wooden stick measure MD stress (20-22). Strain-gauge measurement, though potentially more accurate, is also a destructive test that the operator struck against the roll to sound its and d i f f i c u l t to set up. Consequently, strain gauges are

94 October 1988 Tappi Journal

9. The core-torque test is a fnction-based technique tor measuring radial stress The pressure al the roll s core is measured by fitting a core chuck to a torque wrench and measuring the amount of torque required to cause core slippage

strictly limited to research testing.

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Interlayer pressure A third class of roll-structure measurements is interlayer (radial) pressure or stress. These friction-based measurements of pressure include the core-torque test, the pull-tab test, and the Smith needle. Core pressure can be measured indirectly by fitting a core chuck to a torque wrench and measuring the amount of torque required to cause the core to slip {23). The core-torque test is illustrated in Fig. 9. The Smith rolltightness tester (Smith needle), illustrated in Fig. 10, is a handheld instrument that measures the force required to insert a needle between adjacent layers on the roll end. The pull-tab test, illustrated in Fig. 11, measures the force required to withdraw a tab that is inserted into the roll end during winding. Each of these methods measures interlayer pressure indirectly. Pressure is inferred through the use of simple mechanics equations based on measurements of force, friction, and area of contact. The pull-tab technique is not suitable for production testing, and the core-torque test measures only the pressure at the core. In addition, the friction coefficients of paper vary widely, so the resulting measurements usually have considerable scatter or uncertainty. The friction-based techniques are not the only way of measuring interlayer pressure. Time-of-flight acoustic measurements can infer interlayer pressure (24) based on the speed of acoustic waves passing through a stack or roll. Unfortunately, this method is extremely difficult, and the resulting measurements of radial pressure have too much scatter to accurately compute tangential (MD) stresses, which are the derivative of radial stresses. Another method utilizes thin pressure gauges, such as capacitance gauges. These are wound into a roll and directly measure interlayer pressure (9). Both of these methods are suitable only for research testing.

Rewinders The WIT-WOT (wound-in tension-wound-off tension) winder, illustrated in Fig. 12, is a single-drum duplex laboratory winder designed and built in late '60s. Pfeiffer used this instrument to measure the tension wound into a roll as a function of web tension and nip load (25. 26). The WITWOT winder is able to resolve changes in tension occurring over less than one wrap while either winding or unwinding. Many of the tests used to measure roll structure must be performed on stationary rolls. Such tests may not be suitable for on-line production testing. However, a rewinder is an ideal test stand for occasional in-depth testing that would not be practical on a production winder. A roll is typically unwound and then stopped. The roll is profiled at this diameter. The rewinder is restarted and runs until the roll diameter reaches the next test interval. The roll is profiled again. This process is repeated through the entire depth of the roll. Using this technique, hardness profiling can be extended from a typical one-dimensional function of CD position to a two-dimensional function of CD position and diameter. Density The density analyzer, illustrated in Fig. 13, consists of a winder or rewinder, two incremental rotary encoders, signal conditioning, and a microcomputer. One encoder measures web length and is mounted to a roll or wheel that is in contact with the free web or paper roll. The other encoder (or proximity switch or photoelectric eye) counts roll revolutions and is connected to the paper roll's core through its core chuck or center shaft. The pulses generated by the roller and core encoders are counted over some sample interval and passed to the microcomputer for calculation. The ratio of the pulses measured by both encoders during a sample interval provides a measure of the paper roll's diameter. The density analyzer, the only truly automated rollstructure measurement method, is both sensitive and repeatable. However, it can only profile as a function of diameter and not across the width. In addition, the density analyzer has no absolute reference or calibration, as do many of the other devices. Thus comparisons between rolls or grades are not reliable. Despite these limitations, the density analyzer remains an efficient and sensitive roll-structure test method. The importance of density as a roll-structure parameter was recognized in the late '60s, when various researchers noted that winding parameters had an effect on roll density (27-29). In 1980, Eriksson et al. (30) invented a computerized roll-density analyzer. Eriksson showed the relationship between wound-in and wound-off density as well as the effect of torque, nip, and tension on wound-in density. McDonald showed the relationship between Rho hardness and density, the effect of calendering (caliper) on density, and the effect of torque on density {HI). Odell's study was the most complete, providing details al>out how the density profile at the winder was affected by the source paper machine, winder torque, nip, speed, acceleration, splices, and the set location on the parent log (32). Similar studies were performed by Granlund (33), Holmer (34), and Komulainen (35). This list of articles indicates that the density analyzer is the most carefully investigated roll-

Octobcr 1988 Tappi Journal 95

structure test method. 10. The Smith needle is a friction-based technique for measuring radial stress Roll tightness is measured by determining the amount of force required to insert a needle between adjacent layers on the roll end. c/5

-1

s i o x

Evaluation of measurement methods General considerations There are a number of considerations in selecting a particular method for testing roll structure. These include the ability to profile across the width, the ability to profile through the diameter, accuracy, the ability to automatically record data, ease of use. whether or not the test is destructive, and cost. The evaluation chart in Fig. 14 summarizes these criteria. The relative importance of these criteria varies with application. Thus the choice of the appropriate test method depends on the type of problems to be diagnosed, frequency of testing, production demands, and the level of roll quality needed to satisfy the customer. Most of the methods can profile either across the width or through the diameter, but not both. For example, the impact testers can easily profile hardness across the width by taking readings at various points along the width of the roll. On the other hand, the impact testers cannot profile through the diameter unless a set of readings is taken as a roll is wound or unwound. This requires frequent stopping of the winder, making online testing impractical. The pull-tab and Smith needle friction testers can only profile with diameter because readings can only be taken at the roll ends. The core-torque test cannot be used to develop a profile of any sort. This test provides only a single reading, which represents the average core pressure across the width of the role. The strain-based tests, such as the Cameron gap and J-line, can only profile through diameter, and then only with some difficulty. To profile diameter with the J-line, it is necessary to strike many J-lines during the winding or unwinding of the set. This is potentially dangerous because of the amount of time an operator must spend in proximity to rotating machinery. To profile diameter with the Cameron gap, the roll must be completely slabbed down and destroyed. The density and WIT-WOT analyzers also can only profile through diameter. However, they are considerably more convenient than the other diameter-profiling methods because they are automated and continuous. The trend toward increased automation of testing procedures is the result of the escalating demand for higher quality rolls and the high cost of labor. An ideal system acquires data through sensors connected to a computer, which processes the data into statistical reports. Of all the roll-structure test methods, only the density analyzer and the WIT-WOT winder can operate unattended from setup through report generation. The next level of test automation includes the Rhometer and Schmidt hammer. These devices must be manually operated, although data recording and report generation can be automated. The remaining methods are completely manual and labor intensive. Another consideration is whether the test can be performed without damaging the product. Potential test damage to the product is application dependent. The most destructive is the Cameron gap, which completely destroys the roll. The Smith needle can be destructive to lightweight

96 October 1988 Tappi Journal

grades, since the needle can sever a layer on penetration. The J-line can be destructive if the resultant chalk lines on the edge of the roll prevent its sale. The impact testers and the density analyzer can be destructive on carbonizing and other mark-sensitive grades. Ease of use is important to reduce both labor costs and the noisy data that is caused when the results are sensitive to operator technique. Of all the methods, the density analyzer is the easiest to use because it can run nearly unattended after a brief setup. Though the WIT-WOT winder is similarly capable, it is not easy to use simply because it is not widely available. At the other end of the spectrum, the strain-based Cameron gap and J-line tests, the use of strain gauges, and the pull-tab test are all fairly involved procedures and thus impractical for use in production testing.

The cost of testing The cost of testing is a primary consideration in evaluating a test procedure. The initial costs for the test equipment and setup are heavily outweighed by the ongoing cost of test labor. Thus the analysis of testing costs can be simplified to a discussion of data gathering and recording capabilities quality control (QC) or statistical evaluation, and report generation. Additionally, once the report is generated, it must be reviewed by QC personnel and/or management for action if it is to be of use. As we review the elements of testing costs, it is apparent that the frequency of testing, the ease of use, and automation of testing and reporting are key parameters. In addition to the cost of testing, there are the costs of no testing. These costs can be in the form of reduced throughput, high reject and rewind rates, loss of customers or compensation for dissatisfied customers. To maximize profits, we must minimize the total costs of testing and no testing. This approach is illustrated in Fig. 15. Costs are presented as a function of the amount of testing. As the amount of testing increases, the cost of testing increases. As the amount of testing increases, the cost of not testing decreases as the winding process becomes more efficient and produces higher quality rolls. The optimal amount of testing is where the total cost

11. The pull-tab is a friction-based technique for measuring radial stress The pressure between layers is determined by measuring the force required to withdraw a tab inserted into the roil end dunng the winding process

of testing and not testing is at its minimum. Intuition says that the optimum point occurs at the intersection of the two curves. However, the true optimum is the point at which the slopes of the cost-of-testing and cost-of-not-testing curves are equal in magnitude but opposite in sign. This optimum test frequency varies considerably with application. Lesscritical heavy grades may require almost no roll-structure testing to achieve optimum payback, while lighter specialty grades might require testing of every roll that is shipped. Figure 15 is widely used in economic analysis and applies in situations that are much broader in scope than the one presented here. The figure is useful as a visual aid for the concept of minimizing total cosLs as a function of some parameter. However, it is difficult to use this concept to determine precise magnitudes for optimum testing. Reliable estimates of the costs of testing and not testing are difficult to obtain. In addition, these costs depend on both the application and the chosen test procedure. Different test methods have different potentials to diagnose different types of problems. Nonetheless, it is evident that test methods must be carefully evaluated if we hope to choose the best procedure for our particular applications and then choose a test frequency that minimizes the total cost of testing and not testing. Accuracy

Testing accuracy is a function of many parameters, including repeatability, sensitivity to the measured variable, and sensitivity to noise, drift, or operator technique. The testing device also must be capable of being calibrated against a known standard. If an instrument has no calibration procedure, it cannot be accurate, since accuracy has no meaning without an independent standard. The roll-structure test methods based on the fundamental properties of stress and strain (2, 3) can be independently checked against each other. Calibratable test methods include the friction-based radial-stress measurements (gauge-based tests, core-torque and pull-tab tests), the strain-based measurements (strain gauges, Cameron gap, J-line), and the WIT-WOT winder. The Smith needle and the Rhometer, which have

arbitrary scales, are not truly calibratable. The density analyzer cannot be calibrated because there is no independent measure of density with sufficient accuracy against which it can be checked. Nonquantifiable methods such as the billy club also cannot be calibrated. A noncalibratable method is not totally useless. However, conclusions based on the results of such tests are subject to restrictions. For example, if a method provides consistent and repeatable results, it can be used to analyze structural trends in a single roll. However, it would be risky to compare results for two rolls, especially if the rolls were of different grades or measured under different conditions. Without calibration, accuracy cannot be defined, and magnitudes or values are suspect. Repeatability If a measurement method is to be accurate, it must also be repeatable. After making two nominally identical measurements of tension, we expect nearly identical values. This would be easy to check for the methods that do not alter the roll in the process of testing. However, if the density analyzer is used to profile a roll, the roll cannot be rewound identically to make another run to check for repeatability. Similarly, the Cameron gap cannot be checked for repeatability because the tension on each layer is measured by severing the layer. The process cannot be repeated on the same layer, and subsequent layers may be under different tensions. The impact-based hardness tests also cannot be checked for repeatability because the impact compacts the paper at that location, making subsequent readings higher. Adjacent positions may be at different levels of hardness. The factors that affect repeatability are sensitivity to operator technique and measurement noise. Handheld devices are particularly sensitive to operator technique. An experienced operator can use the Rhometer, the Schmidt hammer, or the Smith needle and produce consistent, repeatable results. However, these results can differ from those obtained by another experienced operator. People with less experience operating a handheld tester will find considerable scatter in their results as they alter their grips and motions from one reading to another. If a short-term study of roll structure requires high-quality data, a single experienced operator should perform all of the tests. Noise Measurement noise increases data uncertainty and reduces the quantity and quality of conclusions based on that data. The total noise comprises the noise from each element in the test system. Sensitivity to operator technique is one source of noise. Noise also is generated by worn mechanical components, which can cause backlash, hysteresis, slip, friction, or drag. If instrument electronics are in proximity to motors and other electrical equipment, noise can be picked up on low-voltage lines or sensors. Electronic equipment can also experience thermal d r i f t from powerup, grounding noise, or frequency-response limitations. Digital equipment always has a minimum one-count uncertainty for pulses and one-bit uncertainty for analog-to-digital converters. Test measurements are often sensitive to the orientation of the testing device and to gravity. If handheld testers

October 1988 Tappl Journal 97

DRQ. DRQ is determined by looking up the value in a table. The deviation ratio (DR) is determined using Eq. 1. DR = (A, - X)/c

(1)

Correlation coefficient The correlation coefficient is a measure of the degree of association between two variables. How much effect does X have on Yl The effect is large if r is close to +1 or -1.

r^inlXiYi-lXilYO/lnlXMlXifHnXYi'-aYif?

(2)

Markov's inequality Markov's inequality is used to calculate the probability that a sample data point X is greater than k />(A>k| 401< (40/50) P\ A>40|