P.G. DIPLOMA IN STATISTICAL METHODS AND APPLICATIONS DKSM4 : APPLIED STATISTICAL METHODS

Transcription

1 P.G. DIPLOMA IN STATISTICAL METHODS AND APPLICATIONS DKSM4 : APPLIED STATISTICAL METHODS SYLLABUS UNIT I Basic principles of experiments randomization, replication and local control. Size of experimental units. Basic Designs: Concept of Cochran s Theorem. Completely randomized design(crd)- Randomized block design(rbd) - Latin square design(lsd) and their analyses - Missing plot techniques in RBD and LSD. UNIT II Statistical Quality Control: Quality control and need for statistical quality control techniques in industries - causes of variation - process control and product control. Process control: specifications and tolerance limits-3σ limits, construction of Shewhart control charts - variable control charts - X, R and σ charts- simple problems. Control charts for attributes: control chart for fraction defectives (p chart), control chart for number of defectives (d chart) and control chart for number of defects per unit (c chart)-simple problems. UNIT III Time Series: Analysis of time series decomposition of time series. Measurement of secular trend fitting of equations, method of moving averages, variate difference method elimination of trend and its effects. Seasonal components and methods for eliminating them - moving average method, link relative method. Cyclic components and their determination. UNIT IV Index numbers calculation of indices ratio of simple aggregates (weighted and unweighted) average of price relatives calculation of index numbers Laspeyre s formula, Paasche s formula, Bowley s formula, Marshall - Edgeworth s formula and Fisher s formula - choice of period fixed base and chain base methods. UNIT V Vital Statistics: Measures of mortality crude death rate, specific death rate, infant mortality rate, fetal death rate Measures of fertility crude birth rate, general fertility rate, age-specific birth rate, total fertility rate. Life table and abridged life table. BOOKS FOR REFERENCE: 1. Goon, A.M., Gupta, M K. and Dasgupta (1982). B., Fundamentals of Statistics. Vol. 2., World Press Private Ltd, Kolkata. 2. Cooray, T.M.J.A. (2008).Applied Time Series: Analysis and Forecasting. Narosa Publishing House, New Delhi. 3. Gupta, S.C. and Kapoor, V.K. (2007) fundamental OF Applied Statistics, Sultan Chand & Sons. 1

2 Unit Introduction DESIGN OF EXPERIMENT Design of experiment means how to design an experiment in the sense that how the observations or measurements should be obtained to answer a query in a valid, efficient and economical way. The designing of experiment and the analysis of obtained data are inseparable. If the experiment is designed properly keeping in mind the question, then the data generated is valid and proper analysis of data provides the valid statistical inferences. If the experiment is not well designed, the validity of the statistical inferences is questionable and may be invalid. It is important to understand first the basic terminologies used in the experimental design. Experimental unit: For conducting an experiment, the experimental material is divided into smaller parts and each part is referred to as experimental unit. The experimental unit is randomly assigned to a treatment is the experimental unit. The phrase randomly assigned is very important in this definition. Experiment: A way of getting an answer to a question which the experimenter wants to know. Treatment Different objects or procedures which are to be compared in an experiment are called treatments. Sampling unit: The object that is measured in an experiment is called the sampling unit. This may be different from the experimental unit. Factor: A factor is a variable defining a categorization. A factor can be fixed or random in nature. A factor is termed as fixed factor if all the levels of interest are included in the experiment. A factor is termed as random factor if all the levels of interest are not included in the experiment and those that are can be considered to be randomly chosen from all the levels of interest. 2

3 Replication: It is the repetition of the experimental situation by replicating the experimental unit. Experimental error: The unexplained random part of variation in any experiment is termed as experimental error. An estimate of experimental error can be obtained by replication. Treatment design: A treatment design is the manner in which the levels of treatments are arranged in an experiment. Example: (Ref.: Statistical Design, G. Casella, Chapman and Hall, 2008) Suppose some varieties of fish food is to be investigated on some species of fishes. The food is placed in the water tanks containing the fishes. The response is the increase in the weight of fish. The experimental unit is the tank, as the treatment is applied to the tank, not to the fish. Note that if the experimenter had taken the fish in hand and placed the food in the mouth of fish, then the fish would have been the experimental unit as long as each of the fish got an independent scoop of food DESIGN OF EXPERIMENT: One of the main objectives of designing an experiment is how to verify the hypothesis in an efficient and economical way. In the contest of the null hypothesis of equality of several means of normal populations having same variances, the analysis of variance technique can be used. Note that such techniques are based on certain statistical assumptions. If these assumptions are violated, the outcome of the test of hypothesis then may also be faulty and the analysis of data may be meaningless. So the main question is how to obtain the data such that the assumptions are met and the data is readily available for the application of tools like analysis of variance. The designing of such mechanism to obtain such data is achieved by the design of experiment. After obtaining the sufficient experimental unit, the treatments are allocated to the experimental units in a random fashion. Design of experiment provides a method by which the treatments are placed at random on the experimental units in such a way that the responses are estimated with the utmost precision possible. 3

4 Principles of experimental design: There are three basic principles of design which were developed by Sir Ronald A. Fisher. (i) Randomization (ii) Replication (iii) Local control (i) Randomization The principle of randomization involves the allocation of treatment to experimental units at random to avoid any bias in the experiment resulting from the influence of some extraneous unknown factor that may affect the experiment. In the development of analysis of variance, we assume that the errors are random and independent. In turn, the observations also become random. The principle of randomization ensures this. The random assignment of experimental units to treatments results in the following outcomes. a) It eliminates the systematic bias. b) It is needed to obtain a representative sample from the population. c) It helps in distributing the unknown variation due to confounded variables throughout the experiment and breaks the confounding influence. (ii) Replication: In the replication principle, any treatment is repeated a number of times to obtain a valid and more reliable estimate than which is possible with one observation only. Replication provides an efficient way of increasing the precision of an experiment. The precision increases with the increase in the number of observations. Replication provides more observations when the same treatment is used, so it increases precision. (iii) Local control (error control) The replication is used with local control to reduce the experimental error. For example, if the experimental units are divided into different groups such that they are homogeneous within the blocks, than the variation among the blocks is eliminated and ideally the error component will contain the variation due to the treatments only. This will in turn increase the efficiency. 4

5 1.3. COMPLETED RANDOMIZED DESIGN: The simplest design using the two essential principles of replication and randomization is the CRD. Suppose that we have t treatments under comparison and ith treatment is to be replicated r i times, for i=1,2,3, t. Then the total number of experimental units necessary for this experiment is n t r i i1. In CRD, we allocate the t treatments completely at random to the n units, subject to the condition that the ith treatment appears in the r i units, for i-1,2,3,.t. A particular case of this is equal replication for different treatments, where r 1 =r 2 = =r t =r, so that n=rt. ANALYSIS We use the following model: Observation from the jth replicate of the ith treatment effect due to the random error = generaleff ect or, symbolically, ith treatment component y ij e i ij Where and are a set of constants with r 0 and eij are independently normally i t i1 i i 2 distributed each with mean zero and variance e. The least-square estimator of and i are obtained by minimizing 2 y i j ij i the normal equations being i j y n ij n i i y ni ni i i k j ij 1,2,...,, i Solving these equations, we have, since n 0, 5 ˆ y, ˆ 00 and i yio y00 ( i 1,2,..., k) y i0 be the mean of the i th class. The analysis of variance partitions the raw sum of squares of the observations, i ij j y 2, into three components sum of squares due to the general effect, sum of squares due to the class-effects and sum of squares due to errors. We may write i i i

6 y ij y y y y. ˆ ˆ i eij or yij y00 i0 00 ij i0 Squaring both the sides and summing overall the observations, we get i j y 2 ij i j y yi0 y00 yij yi0. i j (since the three sums of product terms on the right-hand side vanish) Or y ij ny 00 ni yi0 y00 yij yi0 i j i 2 2 Or ( yij y00 ) ni yi0 y00 yij yi0 i j i i i Or total sum of squares = Sum of squares due to class-effects + sum of squares due to errors, Or, in short, total SS = SSA+SSE Now, the total sum of squares is computed from n quantities like y ij y 00 of which only n-1 are independent, since ( y y00 ) 0 Hence it is said to carry out n-1 degrees of freedom. i j Similarly, sum of squares due to class-effects is obtained by squaring k quantities like y i 0 y 00 i satisfying one condition n y y 0 i i0 00. Thus it carries k-1 degrees of freedom. Lastly, the sum of squares due to error is calculated by squaring n quantities like yij y i0 i j Which satisfy k conditions: y 0 ij y i 0 Hence this sum of squares based on n-k degrees of freedom. Dividing an SS by its df, we get the corresponding variance or mean square (MS). Thus SSA/(k-1)=mean square due to class-effects = MSA SSE/(n-k) = mean square error due to error = MSE. j j ij i j 2 2 Now, SSA and SSE add up to the total SS and the corresponding df=(k-1) and (n-k) add up to the total df = (n-1), but the MSA and MSE will not add up to the total MS. So, though the procedure is called an analysis of variance, it is actually an analysis of SS. We are interested in testing equal. The analysis of variance table is given below: H 0 : r against the alternative that i are not all 6

7 Source of Variation Treatments Error Analysis of variance Table for a CRD Df Sum of Squares Mean Sum of Squares t-1 n-1 2 y y SST r i i i y y SSE i j ij i0 MST MSE F -Value F= MST/MSE Total n-1 y 00 i j ij y 2 MST We reject H0 at the level if F F ; : t1, nt otherwise Ho is accepted. MSE 1.4. RANDOMIZED BLOCK DESIGN: The CRD seldom be used if the experimental units are not homogeneous. For in that case the variation among the units will vitiate the test of significance of the treatment effects. The simplest design which enables as to take care of the variability among the units is the RBD, also called Randomized Completely Block design. Suppose we want to compare the effect t treatments, each treatment being replicated and equal number of times, say r times. Then we need n=rt experimental units, and these units are not perhaps homogenous. The RBD consist of two steps. The first is to divide the units in to are more or less homogenous groups. In each group or block, the take as many units as there are treatments. Thus the number of block is the same as the common replication number (r). The same technique should be applied to the units a block. Variation in techniques, if any, should be made between the blocks. The second step is to assign the treatments at random to the units of a block. This randomization has to be done afresh for each block. This is the the difference of an RBD from a CRD. In an RBD randomization is restricted within a homogenous block. ANALYSIS The analysis of this design is the same as that of two-way classified data with one observation per cell assuming additivity. We use the following model; Observation for ith treatment form the jth block 7

8 random error generaleff ith treatment effect or, component = ect jth block effect symbolically, y ij e Where, j and i are a set of constants with j i 0 and eij are independently 2 normally distributed each with mean zero and variance e. The least-square estimator of and i are obtained by minimizing are 2 yij i, i j j ˆ y 00 ˆ y i j io y, 0 j y 00 y 00., In the model, each observation is the sum of four components, and the analysis of variance partitions the raw sum of squares of the observations, i ij j y 2, also into four components sum of squares due to the general effect, sum of squares due to factor A, sum of squares due to factor B and sum of squares due to errors. We may write y ij y j y y y y y y y. 00 i j 00 ij i0 0 j y00 Squaring both the sides and summing overall the observations i and j, we get i j y 2 ij i j y 2 00 j i yi0 y00 y0 j y00 yij yi0 y0 j y00. j i Or ( y ) ij y y i y y j y yij yi y j y i j j i In words, total sum of squares = Sum of squares due to factor A + Sum of squares due to factor B + sum of squares due to errors, Or, in short, total SS = SSA+SSB+SSE The corresponding partitioning of the total df is as follows : r 1 t 1 r 1 t 1 rt 1 Dividing an SS by its df, we get the corresponding MS. ij i i i j j 8

9 We are interested in testing H 0 : t against the alternative that i s are not all equal. The analysis of variance table is given below: Analysis of variance Table for an RBD Source of Variation Blocks df Sum of Squares Mean Sum of Squares r-1 j t 2 y y SSB 0 j 00 MSB F -Value Treatments t-1 2 y y SST r i i i0 00 MST Error (r-1)(t-1) 2 y y y00 SSE i j ij i0 MSE F= MST/MSE Total rt-1 y 2 i j ij y - 00 MST We reject H0 at the level if F F : t1, r1 ( t1) ; otherwise Ho is accepted. MSE 1.5 LATIN SQUARED DESIGN (LSD) The principle of local control was used in RBD while grouping the units in one way, that is according to blocks. The grouping can be carried one step forward and we can group the units in two ways, each way corresponding to a source of variation among the units, and get the LSD. This design is used with advantage in agricultural field experiment where the fertility contours are not always known. Then the LSD eliminates the initial variability among the units in two orthogonal directions. The LSD has also being used successfully in industries and in the laboratory. In this design, the number of treatments equals the common replications number per treatment. So letting m stand for number of treatments as well as the number of replications for each treatment, the total number of experimental units needed for this design is m x m. This m 2, units are arranged in m rows and m columns. Then the m treatments are allotted to 9

10 these m 2 units at random, subject to the condition that each treatment occurs once and only once in each row and in each column. The LSD is actually an incomplete three-way layout, where all the three factors viz, row, column and treatment, are at the same number of levels(m). For a complete three-way layout with each factor at m levels. We need m 3 experiments units. But in the LSD we take observations on only m 2 of these m 3 units according to the plan stated above. ANALYSIS We shall denote by yijk the observation on the treatment combination where the factor A is at ith level(ith row), factor B is the jth level(jth column) and factor C is the kth level(kth treatment). The triplet (i, j, k) take only m 2 of the possible m 3 values that are dictated by the particular LSD. If we denote this set of m 2 possible values by D, then (i,j,k) takes values from D or symbolically, i, j, k D. Then our linear model is and the m 2 random variables eijk yijk i j k eijk, i, j, k D, with i j k 0, i j k are independently normally distributed each with mean zero and variance. We are interested in testing H : all 0. y ijk i, j, k D 2 e 0 k The least square estimates of the effects, obtained by minimizing i j k 2 Subject to the conditions in the model, are ˆ y ˆ ˆ 000, ˆ i yi00 y000, j y0 j0 y000 and k y 00k y 000 Analysis of variance Table for an m x m LSD Source of df Sum of Squares Mean Sum Variation of Squares Rows m-1 2 my y SSR i i MSR F -Value 10

11 Columns m-1 j m 2 y y SSC 0 j F= MST MST/MSE Treatments m-1 k m 2 y y SST 00 k 000 Error (m-1)(m- 2) Total m yijk yi00 y0 j0 y00k 2 y000 SSE i ijk y MSE y - MST We reject H0 at the level if F F : m1, m1 ( m2) ; otherwise Ho is accepted. MSE Question Bank: 1. Discuss the fundamental principles of Design of experiment. 2. How will you construct the ANOVA table in completely randomized design? 3. Estimate the single missing observation in RBD layout. 4. A manufacturer of television sets is interested in the effect on tube conductivity of four different types of coating for color picture tubes. The following conductivity data are obtained. Analyze the data using Completely Randomized block design. Coating Type Conductivity Soln: Dependent Variable: RESP Sum of Mean Source DF Squares Square F Value Pr > F Model Error Total

12 5. A manufacturer suspects that the batches of raw material furnished by her supplier differ significantly in calcium content. There is a large number of batches currently in the warehouse. Five of these are randomly selected for study. A chemist makes five determinations on each batch and obtains the following data. Batch 1 Batch 2 Batch 3 Batch 4 Batch Soln: Dependent Variable: RESP Sum of Mean Source DF Squares Square F Value Pr > F Model Error Total A chemist wishes to test the effect of four chemical agents on the strength of a particular type of cloth. Because there might be variability from one bolt to another, the chemist decides to use a randomized block design, with the bolts of cloth considered as blocks. She selects five bolts and applies all four chemicals in random order to each bolt. The resulting tensile strengths follow. Bolt Chemical

13 Soln: Dependent Variable: RESP Sum of Mean Source DF Squares Square F Value Pr > F Treat Block E-05 Error Total Three different washing solutions are being compared to study their effectiveness in retarding bacteria growth in 5-gallon milk containers. The analysis is done in a laboratory, and only three trials can be run on any day. Because days could represent a potential source of variability, the experimenter decides to use a randomized block design. Observations are taken for four days, and the data are shown here. Days Solution Solution: Dependent Variable: RESP Sum of Mean Source DF Squares Square F Value Pr > F Treat Block Error Total Analyze the data by using Latin square design and setup the analysis of variance A 12 C 18 C 19 B 12 B 10 D 6 13 D 8 A 11 B D A C

14 D A D B H.O.Dann obtains the body weights of calves at 8 weeks of age. There were three levels of feeding given to a random sample of 5 calves, each completely randomized design (CRD) was used. Obtain the ANOVA. Compute F and obtain the standard error of feeding treatment mean for the data on body weights. Levels of feeding Sub Normal Normal Super Normal Estimate the missing value and analyze the following experiment. Blocks Treatments

15 Unit -2 STATISTICAL QUALITY CONTROL 2.1. Introduction: Controlling and improving quality has become an important business strategy for many organizations; manufacturers, distributors, transportation companies, financial services organizations; health care providers, and government agencies. Quality is a competitive advantage. A business that can delight customers by improving and controlling quality can dominate its competitors. This book is about the technical methods for achieving success in quality control and improvement, and offers guidance on how to successfully implement these methods. There are two general aspects of fitness for use: quality of design and quality of conformance. All goods and services are produced in various grades or levels of quality. These variations in grades or levels of quality are intentional, and, consequently, the appropriate technical term is quality of design. For example, all automobiles have as their basic objective providing safe transportation for the consumer. However, automobiles differ with respect to size, appointments, appearance, and performance. These differences are the result of intentional design differences among the types of automobiles. These design differences include the types of materials used in construction, specifications on the components, reliability obtained through engineering development of engines and drive trains, and other accessories or equipment Causes of Variation Variation in the quality of manufactured product in the repetitive process in industry is inherent and inevitable. These variation are broadly classified as being due to two causes, 1. Chance causes 2. Assignable causes Chance causes Some stable pattern of variation or a constant cause system is inherent in any particular scheme of production and inspection. This pattern results from many minor causes that behave in a random manner. The variation due to these causes is beyond the control of 15

16 human hand and cannot be prevented or eliminated under any circumstances. One has got to allow for variation within these stable pattern, usually termed as allowable variation. The range of such variation is known as natural tolerance of the process. Assignable causes The second type of variation attributed to any production process is due to non random or the so-called assignable causes and termed as preventable variation. The assignable causes may greep in at stage of the process right from the arrival of the draw raw materials to the final delivery of goods. Some of the important factors of assignable causes of variation are substandard or defective raw material, new techniques or operations, negligence of the operators, wrong or improper handling of machines, faulty equipment, unskilled or inexperienced technical staff and so on. These causes can be identified and eliminated and are to be discovered in a production process before it goes wrong i.e., before the production become defective Process control and product control: The main objective in any production process is to control and maintain the quality of the manufactured product so that it conforms to specified quality standards. In other words, we want to ensure that the proportion of defective items in the manufactured product is not too large. This is called process control. On the other hand, by product control the mean controlling the quality of the product by critical examination at strategic points and this is achieved through sampling inspection plans Specification and Tolerance limits: The control chart may show that the process is in control at particular level. But it may also be interest to know whether the process can meet the specification limits set for the item. A decision of the point may be made by comparing what are called the natural tolerance limits of the process with the specification limits. If and are the process mean and process standard deviation., respectively, then the limits, which include on the average 9,973 out of 10,000 items, will be estimated by and or, and in this way we shall get estimates of the tolerances, which will be compared with the specification limits. 16

17 If the estimated natural tolerances are not included within the specification limits, then a readjustment of the process will be advisable, with respect to either the process average or process dispersion or both; or else a revision of specification limits will be called for. If the estimated tolerances lie well within the specification limits, this will signify that the process is too good. Then to a revision of the specification limits may be called for, or else it may mean that some relaxation of the conditions of production may be allowed, leading perhaps to lower costs. The ideal situation will be attained when the tolerance limits are approximately coincide with the specification limits CONTROL CHART FOR VARIABLES Control charts enhance the analysis of a process by showing how that process is performing over time. By combining control chart with an appropriate statistical summary, those studying a process can gain an understanding of what the process is capable of producing. Control charts describe where the process is in terms of current performance. To create a control chart, samples arranged into subgroups, are taken during the process. The average of the subgroup values are plotted on the control chart. The center line of this chart shows where the process average is centered, the central tendency of this data. The upper control limits (UCL) and lower control limit (LCL) calculated based on ± 3sigma, describe the spread of the process. Once the chart is constructed, it present the user with a picture of what the process is currently capable of producing. Variables are the measurable characteristics of a product or service. Example: height, weight or length of part. One of the most commonly used variables chart combinations in statistical process control is the Xbar and R charts. Xbar and R charts are used together to determine the distribution of the subgroup averages of sample measurements taken form a process. 17

18 State of process control is a process considered to be in a state of control or under control, whether performance of the process falls within the statistically calculated control limits and exhibits only chance or common causes. When a process is under control it considered stable and amount of future variation is predictable. A stable process does not necessarily meet the specifications set by the designer nor exhibit minimal variation: a stable process merely has a predictable amount of variation. Once an understanding of process performance is created, it is possible to interpret the variation associated with the process and make decisions based on this interpretation. When charted, the variation of a quality characteristic present in the process is an indicator of how the system is operating. When a system is subject to only chance causes of variation, 99.73% of the parts produced will fall within ± 3σ. This means that if 1,000 subgroups are sampled, on the normal curve, a control chart can be divided into 3 zones. (Figure 1). Zone A is ± 1 standard deviation from the center line and should contain approximately 68.3% of the calculated sample averages or ranges. Zone B is ± 2 standard deviation from the centerline and should contain 27.2% ( %) of the points. Zone C is ± 3 standard deviations from the centerline and should contain only approximately 4.2% of the points ( %). With these zones as a guide, a control chart exhibits a state of control when:- 1) 2/3 of the points are near the center value 2) A few of the points are in or near the center value 3) The points appear to float back and forth across the centerline 4) The points are balanced (in roughly equal numbers) on both sides of the centerline 5) There are no points beyond the control limits 6) There are no patterns or trends on the chart 18

19 UCL C B X A A B LCL C Figure 1: Zones on a control chart X and R Chart The X chart is used to monitor the variation of the subgroup averages that are calculated from the individual sampled data. Averages rather than individual observation are used on control chart because average values will indicate a change in the amount of variation much faster than will individual values. Control limits on this chart are used to evaluate the variation from one subgroup to another. In any situation it is necessary to determine what the goal of monitoring a particular quality characteristics or group of characteristics is. Variable control charts are based on measurements. Before creating control charts it is important to determine which product or service characteristics will be studied. Choice of a characteristic to be measured depends on what is being investigated. The characteristics selected should be ones that affect product or service performance. It is crucial to identify important characteristics, avoid the tendency to try to establish control charts for all measurements. 19

20 Subgroups and the samples composing them must be homogeneous. A homogeneous subgroup will have been produced under the same conditions, by the same machine, the same operator, the same mold and so on. Guidelines to be followed such as:- The larger the subgroup size, the more sensitive the chart becomes to small variation in the process average. While a larger subgroup makes for a more sensitive chart, it is also increases inspections costs. Destructive testing may make large subgroup sizes unfeasible. Subgroup sizes smaller than four do not create a representative distribution of subgroup averages. Subgroup averages are nearly formal for subgroups of four or more even when samples from a nonnormal population. When the subgroup size exceeds 10, the standard deviations (s) chart, rather than the range (R) chart, should be used. For large subgroup sizes, the s chart gives a better representation of the true dispersion or true differences between the individual sampled than does the R chart. To create a control chart, an amount of data sufficient to accurately reflect the statistical control of the process must be gathered. A minimum of 20 subgroups of sample n = 4 is suggested. Each time a subgroup of sample size n is taken, an average is calculated for the subgroup. To do this, the individual values are recorded, summed and then divided by the number of samples in the subgroups. This average, Xi, is then plotted on the control chart. X and s Chart The X and range charts are used together in order to show both the center of the process measurements (accuracy) and the spread of the data (precision). An alternative combination of charts to show both and central tendency and the dispersion of the data is the Xbar and standard deviation or s chart. When and R chart is compared with an s chart, the R chart stands out as being easer to compute. However, the s chart is more accurate than the R chart. This greater accuracy is a result of the manner in which the standard deviation is calculated. The subgroup sample standard deviation or s value is calculated using all of the data rather than just the high and low values in the sample. So, while the range chart is simple to construct, it is more effective when the sample size is less than 10. When the sample size 20

21 exceeds 10, the range does not truly represent the variation present in the process. Under these circumstances, the s chart is used with the X chart. The combination of X and s charts is created by the same methods as the X and R chart. The formulas are modified to reflect the use of s instead of R for the calculations. For the X chart, X = Xi m UCLx = X + A3s LCLx = X A3s For the s chart, s = si m UCLs = B4s LCLs = B3s Where si = standard deviation of the subgroup values si = average of the subgroup sample standard deviations A3, B3, B4 = factor used for calculating 3σ control limits for X and s charts using the average sample standard deviation and Appendix 1 Example 1: X and R chart Data is a key aspect of computer integrated manufacturing at Whisks Electronics Corporation. Automated data acquisition systems generate timely data about the product produced and the process producing it. Whisks believe that decision making processes can be enhanced by using valid data that have been organized in an effective manner. For this reason, Whisks uses an integrated system of automated statistical process control programming, data collection devices and programmable logic control (PLC s) to collect statistical information about its silicon wafer production. Utilizing the system relieves the 21

22 process engineers from the burden of number crunching, freeing time for critical analysis of the data. In order to monitor silicon wafer line B, X and R charts are being used. 1. Step 1 Define the Problem The production of integrated circuits by etching them onto silicon wafers requires silicon wafers of consistent thickness. Recently, Whisks customers have raised questions about whether or not Whisks process is capable of producing wafers of consistent thickness. 2. Step 2 Select the Product Characteristics to be Measured Wafer thickness is the critical characteristics. The customer has established a target value for wafer thickness at mm. 3. Step 3 Choose a Rational Subgroup Size to be Sampled Using micrometers linked to PLC s, every 15 minute four wafers are randomly selected and measured in the order they are produced. These values are stored in a database that is accessed by the company s statistical process control software. This subgroup size and sampling frequency were chosen based on the number of silicone wafers produced per hour. 4. Step 4 Collect the Data The data collected by the micrometer and PLC are shown in Table 1. Table 1: Silicon wafer thickness X R Subgroup Subgroup Subgroup Subgroup Subgroup Subgroup Subgroup Subgroup Subgroup Subgroup Subgroup

23 Subgroup Subgroup Subgroup Subgroup Step 5 Determine the Trial Center Line for the X Chart The center line of the X chart is created by using the following formula: X = Xi m Where X = ( ) 15 = mm X = the center line of the X chart X = the individual subgroup averages m = the number of subgroups 6. Step 6 Determine the Trial Control Limits for the X Chart The upper and lower control limits are based on the idea that 99.73% of the data should be located between ± 3 standard deviations of the mean. Rather than calculate the standard deviation for each subgroup, the A2R value allows us to estimate ± 3σ using the average range (R). A2 = when n = 4 ( see Appendix 1 ) R = Ri m R = ( ) 15 R = = mm 23

24 Upper control limit, X UCLx = X + A2R UCLx = (0.0037) UCLx = mm Lower control limit, X LCLx = X - A2R LCLx = (0.0037) LCLx = mm 7. Step 7 Determine the Trial Center Line for the R Chart R = Ri m R = ( ) 15 R = mm D4 = and D3 = 0 when n = 4 ( Appendix 1) Upper control limit, R UCLR = D4R UCLR = (0.0037) UCLR = mm Lower control limit, R 24

25 LCLR = D3R LCLR = 0 (0.0037) LCLR = 0 Figure 3: X and R chart for Silicon Wafer Thickness 8. Step 8 Interpret the Chart X and R charts provide information about process centering and process variation. If we study the X chart (Figure 3), it shows that the wafer thickness is centered around the target value of mm. Studying the R chart and noting the magnitude of the values (Figure 2) reveals that variation is present in the process. Not all the wafers are uniform in their thickness. The customer s target value is , while the process values have the potential to vary as mush as (UCL)Whisks will have to determine the best method of removing variation from their silicon wafer production process. 25

26 Example 2 (X and s Chart) The production from the machine making the shafts is consistent at 150 per hour. Since the process is currently exhibiting problems, your tam has decided to take a sample of 5 measurement every 10 minutes from the production. The values for the day s production run as shown below. A total of 21 subgroups of measurements is taken. Table 2: Shaft, Averages and Standard deviations. DEPT.: Roller PART NAME: Shaft PART NO: 1 MACHINE: 1 GROUP: 1 VARIABLE: Length Subgroup Time 07:30 07:40 07:50 08:00 08:10 Date 07/02/95 07/02/95 07/02/95 07/02/95 07/02/ X s Subgroup Time 08:20 08:30 08:40 08:50 09:00 Date 07/02/95 07/02/95 07/02/95 07/02/95 07/02/ X s Subgroup Time 09:10 09:20 09:30 09:40 09:50 26

27 Date 07/02/95 07/02/95 07/02/95 07/02/95 07/02/ X s Subgroup Time 10:00 10:10 10:20 10:30 10:40 Date 07/02/95 07/02/95 07/02/95 07/02/95 07/02/ X s Subgroup 21 Time 10:50 Date 07/02/ X S Each time a sample is taken, the individual values are recorded (1), summed and then divided by the number of samples taken to get the average (2). This average is then plotted on the control chart (1). Note that in this example, the values for X, s have been calculated to three decimal places for clarity. Example: from (2) 27

28 = Example: from (3) ( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 + ( ) = Using the 21 samples provided in Figure ----, we can calculate the X by summing all the averages from the individual samples taken and then dividing by the number of subgroups:- X = == This value is plotted as the center line of the X chart. Individual standard deviations are calculated for each of the subgroups by utilizing the formula for calculating standard deviations as presented in Figure 4 (3). Once calculated, these values are plotted on the s chart. s, the grand deviation average, is calculated by summing the value of the sample standard deviations (Figure 4 (3)) and dividing by the number of subgroup m: s = = = The A3 factor for sample size of five is selected from the table in Appendix 1. The values for the upper and lower control limits of the X chart are calculated as follows:- UCLx = X + A3s = (0.02) = UCLx = X - A3s = (0.02) = Once calculated, the upper and lower control limits (UCL and LCL, respectively) are placed on the chart. With n = 5, the values of B3 and B4 are found in the table in Appendix 1. The control limits for the s chart are calculated as follows:- UCLs = B4S = (0.02) =

29 LCLs = B3S = 0 (0.02) = 0 The control limits are placed on the s chart. The s chart (Figure 4) exhibits good control. The points are evenly spaced on both sides of the center line and there are no points beyond the control limits. There are no unusual patterns or trends in the data. Given these observations, it can be said that the process is producing parts of similar dimensions. Next the X chart is examined. Once again, an inspection of the X chart reveals the unusual pattern occurring at points 12, 13 and 14. These measurements are all below the lower control limit. When compared with other samples throughout the day s production, the parts produced during the time that samples 12, 13 and 14 were taken were much smaller than the parts produced during other times in the production of undersized parts needs to take place. Since a cause for the undersized parts was determined in the previous example, these values can be removed from the calculations for the X chart and s chart. The revised limits will be used to monitor future production. Figure 4: Xbar and s chart for roller shaft length 29

30 3.6. CONTROL CHART FOR ATTRIBUTES Attributes are characteristics associated with a product or service. These characteristics either do or do not exist, and they can be counted. Example, the number of leaking containers, of scratches on a surface, of on time deliveries or errors on an invoice. Attribute charts are used to study the stability of processes over time, provided that a count of nonconformities can be made. Attribute charts are used when measurements may not be possible or when measurements are not made because of time or cost issue. When nonconformities are investigated, two conditions may occur. The product or service may have a single nonconformity or several that prevent it from being used. These conditions are called nonconforming. In other situations, the nonconforming aspect may reduce the desirability of the product but not prevent its use; these are called nonconformities. When the interest is in studying the proportion of products rendered unusable by their conformities, a fraction nonconforming (p) chart, a number nonconforming (np) or a percent nonconforming chart should be used. When the situation calls for tracking the count of nonconformities, a number of nonconformities (c) chart or a number of nonconformities per unit (u) chart is appropriate. Charts for Fraction Nonconforming ( P-Chart) The fraction nonconforming chart is based on the binomial distribution and is used to study the proportion on nonconforming products or services being provided. This chart is also known as a fraction defective chart or p chart. The items may have several quality characteristics that are examined simultaneously by the inspector. If the item does not conform to standard on one or more of these characteristics, it is classified as nonconforming. When demonstrating or displaying the control chart to production, personnel or presenting results to management, the percent nonconforming is often used, as it has more intuitive appeal. Although it is customary to work with fraction nonconforming, we could also analyze the fraction conforming just as easily, resulting in a control chart on process yield. 30

31 P charts can be constructed using either a constant or variable sample size. The center line and control limits of the control chart for fraction nonconforming are computed as follows:- Fraction Nonconforming Control Chart UCL = p + 3 p ( 1 p ) n Center line = p LCL = p 3 p ( 1 p ) n A major application of P charts is in management systems in which various types of rejects are tracked on a routine basis. The control limits for the P chart allow a realistic assessment of whether a process is stable, improving or deteriorating. Overreaction to changes in various percentage indicators is not uncommon. As part of a management information system, the percentage of units failing a testing procedure was routinely discussed at weekly management meetings. Corrective action plans were required if any percentage was above the prior week s level. Number Nonconforming - (np) Chart A number nonconforming (np) chart tracks the number of nonconforming products or services produced by a process. A number nonconforming chart eliminates the calculation of p, the fraction nonconforming. The Number Nonconforming Control Chart UCL = np + 3 np ( 1 p ) Center line = np 31

32 LCL = np 3 np ( 1 p ) Number nonconforming (np) charts are interpreted in the same manner as are fraction nonconforming charts. The points on an np chart should flow smoothly back and forth across the center line with a random pattern of variation. The number of points on each side of the center line should be balanced, with the majority of the points near the center line. There should be no patterns in data such as trends, runs, cycles, points above the upper control limit, or sudden shift in level. Points approaching or going beyond the lower control limits are desirable. They mark an improvement in quality and should be studied to determine their cause and to ensure that it is repeated in the future. The process capability is np, the center line of the control chart. Charts for Numbers of Nonconformities (c) - Chart Charts for counts of nonconformities monitor the number of nonconformities found in a sample. Nonconformities represent problems that exist with the product or service. Nonconformities may or may not render the product or service unusable. Two types of charts recording nonconformities may be used: a count of nonconformities (c) chart or a count of nonconformities per unit (u) chart. As the names suggest, one chart is for total nonconformities in the sample, and the other chart is nonconformities per unit. The number of nonconformities chart, or c chart, is used to track the count of nonconformities observed in a single unit of product or single service experience, n = 1. Charts counting nonconformities are used when nonconformities are scattered through a continuous flow of product such as bubbles in a sheet of glass, flaws in a bolt of fabric, discolorations in a ream of paper, or services such as mistakes on an insurance form of errors on a bill. To be charted, the nonconformities must be expressed in terms of what is being inspected, such as four bubbles in a 3-square-foot pane of glass or two specks of dirt on an 8 ½ x 11 inch sheet of paper. Count of nonconformities charts can combine the counts of a variety of nonconformities, such as the number of mishandled, dented, missing or unidentified suitcases to reach a particular airport carousel. A constant sample size must be used when creating a count of nonconformities (c) chart. 32

33 The Numbers of Nonconformities Control Chart UCL = c + 3 c Center Line = c LCL = c 3 c Charts for counts of nonconformities are interpreted similarly to interpreting charts for the number of nonconforming occurrences. The charts are studied for changes in random patterns of variation. There should be no patterns in the data, such as trends, runs, cycles or sudden shifts in level. All of the points should fall between the upper and lower control limits. The data points on a c chart should flow smoothly back and forth across the center line and be balanced on either side of the center line. The majority of the points should be near the center line. Once again, it is desirable to have the points approach the lower control limits of zero, showing a reduction in the count of nonconformities. The process capability is c, the average count of nonconformities in a sample and the center line of the control chart. EXAMPLES OF CONTROL CHARTS FOR ATTRIBUTES EXAMPLE 1 (P chart) (1) With constant sample size Special Plastics, Inc. has been making the blanks for credit cards for a number of years. They use a p chart to keep track of the number of nonconforming cards that are created each time a batch of blank cards is run. Table 1: Data sheet: Credit Cards Sample Number n np p

34 , Step 1 Gather the data The characteristic that have been designated for study include blemishes on the card s front and back surfaces, colour inconsistencies, white spots or bumps caused by dirt, scratches, chips, indentations or other flaws. Several photographs are maintained at each operator s work station to provide a clear understanding of what constitutes a nonconforming product. Batches of 15,000 blank cards are run each day. Samples of size 500 are randomly selected and inspected. The number of nonconforming units (np) is recorded on data sheet. Step 2 Calculate p, the Fraction Nonconforming After each sample is taken and the inspections are complete, the fraction nonconforming (p) is calculated using n=500, the number of nonconforming items. For example, for the 1 st value:- P = np = 20 = n 500 Step 3 Plot the Fraction Nonconforming on the Control Chart As they are calculated, the values of p for each subgroup are plotted on the chart. The p chart in Figure 1 has been scaled to reflect the magnitude of data. 34