466 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES 8. The presence of an SQC scheme in any manufacturing concern has a very healthy effect as it creates quality consciousness among their personal. Such ways keep the staff and the worker on their alert they are by increasing their efficiency. 11.10.1 Causes of Variation In every manufacturing concern, it is intended that all the products produced should be exactly same quality and should confirm to same prescribed specification. However refined and accurate the manufacturing process is, some amount of variation among manufactured products is always noticed which is mainly due to two types of causes: (a) Chance Causes: Variation, which results from many minor causes, that behaves in a random manner. This type of variation is permissible and indeed inevitable, in manufacturing. There is no way in which it can completely be eliminated when the variability present in a production process is confined to chance variation only, the process is said to be in a state of statistical control. These type of causes are also known as random causes. These small variations, which are natural to and inherent in the manufacturing process, are also called allowable variations as they cannot be removed or prevented altogether in any way. The allowable variation is also sometimes known as natural variation, as it cannot be eliminated and one has to allow for such variation in the process. (b) Assignable Causes: These are some variations which are neither natural nor inherent in the manufacturing process and they can be assigned as well as prevented if the causes of such variations are detached. These variations are generally caused by the defects and faults in the production design and manufacturing process. 11.10.2 Types of Quality Control The control refers to action (or inaction) designed to change a present condition or causes it to remain unchanged; and quality refers to a level or standard which is turn, depends on manpower, materials, machines and management. The main purpose of any production process is to control and maintain a satisfactory quality level for produced product and also it should be ensured that the product conforms to specified quality standards i.e., it should not contain a large number of defective items. The quality of a product manufactured in any factory may be controlled by two ways. (a) Process Control: The first way for controlling the quality is process control which is concerned with controlling the quality during the process of production i.e., the control of a process during manufacture. Also, when statistical techniques are employed during manufacturing period for controlling the quality by detecting the systematic causes of variation as soon as they occur then it is called process control. Process control is achieved by the technique of control charts pioneered by W.A. Shewhart in 1924. (b) Product Control: This is concerned with the inspection of goods already produced whether these are fit to be dispatched. On the other hand by product control we mean controlling the quality of the product by critical examination at strategic points and this is achieved through ‘Sampling inspection plans’ pioneered by Dodge and Romig. Process Under Control: A production process is said to be under control when there is no evidence of the presence of assignable causes (or these causes have been detached and removed) and it is governed by the chance causes of variations alone. STATISTICAL QUALITY CONTROL 467 Tolerance of the Specification Limits: The manufactures of the manufactured goods often standards to which these product must confirm if they are to be considered of good quality. These standards generally specify the desirable process average together with the limits above and below this process average. These upper and lower limits are called the specification limits or the tolerance limits. 11.11 CONTROL CHARTS A control chart is a statistical device principally used for the study and control of repetitive process. A control chart is essentially a graphic device for presenting data so as to directly reveal the frequency and extent of variations from established standards of goals. Control charts are simple to construct and easy to interpret and they tell the user at a glance whether or not the process is in control i.e., with in the tolerance limits. Walter A. Shewhart of Bell Telephone laboratories made the discovery and development of the control charts in 1924. A control chart is an indispensable tool for bringing a process under statistical control. The Shewhart’s control charts provides a very simple but powerful graphic method of obtaining if a process is in statistical control or not. Its construction is based on 3– σ limits and a sequence of suitable sample statistics e.g., mean(x – ), Range(R – ), Standard deviation(S), fraction defective(p) etc. Computed from independent samples drawn at random from the product of the process. These sample points depict the frequency and extent of variations from specified standards. A control chart consist of three horizontal lines: (1) Upper Control Limit (2) Lower Control Limit (3) Central Limit together with a number of sample points. In the control chart UCL and LCL are usually plotted as dotted lines and the CL is plotted as a bold line. Sample (Subgroup) Numbers Q u a l i t y s c a l e UCL CL LCL 3– I 3– I 12 3 4 56 7 8910 FIG. 11.16 468 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Remarks: 1. A central limit representing the average value of the quality characteristics, or desired standard or level of the control process. 2. An upper control limits (UCL) and lower control limits (LCL) indicates the upper and lower tolerance limit. 3. Ordinarily UCL and LCL are at equal distance from central line, this common distance being equal to three items the standard deviation σ (called “standard error’ in sampling theory) of the sample characteristics for which the control chart is prepared. 4. If t is the underlying statistic, then UCL and LCL depends on the sampling distribution of t and are given by UCL = E(t) + 3S.E.(t) LCL = E(t) – 3S.E(t) CL = E(t) 11.12 3– σσ σσ σ CONTROL LIMITS Control chart is based on the fundamental property of area under the normal distribution. The standard normal probability curve is given by the equation () − =−∞≤≤∞ σπ 2 /2 1 ;, 2 t Pt e t where x t −µ = σ and x is normally distributed with mean xµ= and standard deviation σ . Therefore by the property of normal distribution µ− σ< <µ+ σ =[ 3 3 ] 0.9973Px (1) the probability statement in (1) states that if x is normal with µ and standard deviation σ , then the probability or chance that a randomly selected value of x will lie outside the limit µ + 3σ is 1.0 0.9973 0.0027,−= i.e. very small, only 27 out of 10,000. In veiw of this, 3-σ limits, are termed as LCL and UCL for quality characteristic x. In other words µ ± 3σ covers 99.73 percent of the sample. Hence if points fall outside 3-σ limits, they indicate the presence of some assignable cause all is not due to random causes. It should be noted that if points fall outside 3-σ limits, there is a good reason for believing that they point to some factor contributing to quality variation that can be identified. 11.13 TYPES OF CONTROL CHART There are two main types of control charts. 1. Control charts for variable (x – , R, σ chart) 2. Control charts for attributes (p, pn and C-chart) STATISTICAL QUALITY CONTROL 469 11.13.1 Control Chart for Variable The chart used for characteristics on which the actual measurements in numerical forms are possible to be made, i.e., whose samples are subjected to quantitative measurements such as weight, length, diameter, volume etc. are called control for variables. The charts used for qualitative characteristics as ‘defective’ or ‘non-defective’, as ‘good’ or ‘bad’, as ’better’ or ’worst‘, are called control charts for attributes. Control Chart for Mean (x – ) : The mean chart is used to show the quality averages of the samples drawn from a given process. Before a (x – ) chart is constructed the following values must be obtained. 1. Obtain the Mean of Each Sample: Let x – 1 , x – 2 , x – 3 , , x – i , be the means of sample on the 1st, 2nd, 3rd, , ith sample observations respectively and let R 1 , R 2 , R 3 , , R i be the values of corresponding ranges for the ith samples. Thus for the jth sample (j = 1, 2, 3, , i) j x = Mean of observations on the jth sample ⇒ 1 j x n = [Sum of observations on the jth sample] where n is the sample size. R j = R max − R min , (where R max is the largest and R min is the lowest observation in the jth sample.) 2. Obtain the Mean of Sample Mean: Now, x , the mean of i sample mean and R , the mean of the i sample ranges are given by () 12 1 i x xxx x ii =++ += ∑ () 12 1 i R RRR R ii =++ += ∑ 3. Setting of Control Limits for x : From sampling theory, we know that if µ be the process mean and σ be the process standard deviation then sample mean x ` is normally distributed with mean µ and stadnard deviation on n σ i.e., () Ex =µ and S.E. () x = , n σ where n is the sample size. Hence the 3-σ control limits for x chart are: () () ± 3 Ex SE x (1) or 3 A n σ µ± ⇒ µ± σ 470 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES UCL A=µ+ σ i.e., CL =µ (2) LCL A=µ− σ where 3 A n = is a constant and obtained for different values of n. Equation (2) are applicable in those situations in which values of µ and σ are known. Remark: If µ and σ are not known then we use their estimates provided by the i-given samples. Now x provides an unbiased estimate of the population mean ,µ while process standard deviation σ is estimate by 2 R d ; where d 2 is a constant depending on the given sample size n. Thus x µ= and 2 R d σ= (where d 2 is correlation factor) Substitute these values in (1) to obtain 3-σ control limits for x — -chart. = 2 2 13 3 R xxR d ndn ±×⇒± = 2 xAR =± where 2 2 3 A dn = is a constant depending on sample size n. UCL = 2 xAR + ∴ CL x = 2 LCL xAR =− 4. Construction of (x ) Chart: The control chart for mean is drawn by taking the sample number along the horizontal line (x-axis) and the statistic (x – ) along the vertical line (y- axis). The sample points are plotted as points or dots against the corresponding sample number. These points may or may not be joined. The central line is drawn as a bold and UCL or LCL are plotted as dotted horizontal lines at the compute values. 5. Control Chart for Range (R ): The R chart is used to show the variability or dispersion of the quality produced by a given process. The R chart is generally presented along with the (x – ) chart and procedure for constructing the R chart is similar to that for (x – ) chart. The required values for constructing R chart are 1. The range of each sample R 2. The mean of the sample ranges (R – ) 3. Setting of control limits for R STATISTICAL QUALITY CONTROL 471 The three sigma limits for R-chart if process standard deviation is known, are given by UCL 3 CL LCL 3 RR R RR =µ +σ =µ =µ −σ (1) Now if quality characteristics x is normally distributed with mean µ and standard deviation σ then () 2R ER d µ= =σ , where E(R) = expected mean for R () 3R vR d σ= =σ , where v(R) = Variance of R Therefore equation (1) becomes () () 2323 2 2 2323 1 UCL 3 3 CL LCL 3 3 dddd D d dddd D =σ+ σ= + σ=σ =σ =σ− σ= − σ=σ (2) where d 2 , D 1 and D 2 are constants depending on sample size n and have been computed for different values of n from 2 to 25. Since range can never be negative so if it comes out to be negative, it is taken as zero. Equation (2) is used when σ is known. Remark: When standard deviation are not known i.e., σ is unknown then in this case σ is obtained by 2 R d σ= Therefore from equation (2) 24 2 2 2 13 2 UCL CL LCL R DDR d R dR d R DDR d == == == i.e., 4 3 UCL CL LCL DR R DR = = = 3 D and 4 D also depends on sample size n and tabulated for different values of n from 2 to 25. 1. Construction of (R ) Control Chart: The control chart for mean is drawn by taking the sample number along the horizontal line (x-axis) and the statistic (R – ) along the vertical line (y-axis). The sample points are plotted as points against the corresponding 472 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES sample number. The central line is drawn as a bold and UCL or LCL are plotted as dotted horizontal lines at the computed values. Point to be noted that the use of R chart is recommended only for relatively small samples sizes (near 12 to 15 units). For the large sample sizes (n > 12) the σ chart is to be recommended. 2. Standard Deviation σ Chart: The variability in the quality characteristic is controlled by σ chart (when n > = 10). Control limits of σ charts are given by UCL = 3 Ss µ+σ CL = S µ LCL = S µ − 3 s σ (1) For normally distributed variable x , s µ and s σ is given by 2 () s ES C µ= = σ and () () 2 2 1 s n vS C n − σ= =σ − (2) Therefore from (1) control limits are: () () 2 22 2 22 1 UCL 3 1 LCL 3 n CC n n CC n − =σ+σ − − =σ−σ − ⇒ 2 2 UCL CL B C =σ =σ 1 LCL B=σ (3) These control limits are employed when standards ( σ is given) are given. If standards are not given then in that case σ is estimated by 2 , S C σ= where () 2 Sum of sample standard deviations Number of samples xx S n − == ∑ Using value of σ in equation (3) we get control limit as 2 4 2 2 2 1 3 2 UCL CL LCL B SBS C C SS C B SBS C == == == STATISTICAL QUALITY CONTROL 473 11.13.2 Control Chart for Attributes The chart used for qualitative characteristic are called control charts for attributes. When we deal with quantity characteristic which cannot be measured quantitatively, in such cases the inspection of units is accompanied by classifying them as acceptable or non-acceptable, defective or non-defective. Here we use two words ‘defect’ and ‘defective’. Any instance of a characteristic or unit not conforming to specification (required standards) is known as a defect. A defective is a unit which contains more than allowable number (usually one) of defects. Control chart for attributes are: 1. Control chart for fraction defectives, i.e., p-chart 2. Control chart for number of defectives, i.e., np-chart 3. Control chart for number of defects per unit, i.e., c-chart 1. Control Chart for Fraction Defectives (p-Chart): Control chart for fraction defective is used, when sample unit as a whole is classified as defective or non-defective, or good or bad. Total no. of defective units Fraction defective = Total no. of units i.e., d p n = therefore sampling distribution of the statistic ‘p’ is given by ()Ep p= () = pQ SE p n ; where Q = 1 – p (i ) 3- σσ σσ σ Control Limits for p -Chart: 3-σ control limits for p-chart are given by () () 3 .Ep SE p ± i.e., 3 pQ p n ± Therefore if p is known then UCL 3 CL LCL 3 pQ p n p pQ p n =+ = =− where Q = 1 − p Again if p is not given, then p is denoted by p – and is obtained as Total no. of defective in all samples Total no. of units in all samples p = (when k sample is used out of n) i.e., dd p n n == ∑∑ ∑ 474 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Therefore control limits of p-chart are () () 1 UCL 3 CL 1 LCL 3 pp p n p pp p n − =+ = − =− It can also be written as () () UCL 1 CL LCL 1 pAp p p pAp p =+ − = =− − Where 3 A n = is obtained for different values of n from the table. (ii ) Construction of p Chart: To construct p – -chart, take the sample number along the horizontal scale and the statistic ‘p’ along the vertical scale. Then the sample fraction defectives p 1 , p 2 , p 3 , p i are plotted against the corresponding sample numbers as points (dots). The central line as a dark horizontal line and UCL p and LCL p are plotted as dotted horizontal lines at the computed values. Since p cannot be negative so if LCL p obtained as negative, it is taken as zero for control chart. 2. Control Chart for Number of Defectives (np -Chart): If the sample size is constant for all the samples, say n then the sampling distribution of the statistic d = No. of defectives in the sample = np is given by E(d) = np and S.E.(d) = npQ Hence the 3- σ limits for np-chart are given by () 3 ()Ed SEd ± () 331np npQ np np p =± =± − ; where 1Qp=− Hence () () UCL 3 1 CL LCL 3 1 np np np np np p np np np p =+ − = =− − If p – is not known, then p is obtained by sample values and given by Total no. of defectives in all sample inspected Total no. of samples inspected p = 1 1 k i p p k = = ∑ STATISTICAL QUALITY CONTROL 475 Thus control limits for np-chart are () () UCL 3 1 CL LCL 3 1 np np p np np np p =+ − = =− − 3. Control Chart for Number of Defects per Unit (c-Chart): The statistical basis for the control c-chart is the poisson distribution. If we regard the statistic c distributed as a poisson variate with parameter λ then, () Ec =λ and () SE c =λ where λ is the average number of defects in all the inspection units. Hence the 3-σ control limits are given by () () 3 Ec SE c ± 3=λ± λ i.e., UCL 3 CL LCL 3 =λ+ λ =λ =λ− λ If λ is unknown then, 12 k c cc c c kk ++ + λ= = = ∑ where 12 , , , k cc c are the numbers of defects observed in kth sample observation. Hence, UCL 3 CL LCL 3 cc c cc =+ = =− Since ,c the number of defects per unit cannot be negative so if LCL c is obtained from above formula as negative then it is taken as zero. Construction of c -Chart: The sample points 12 , , , k cc c are plotted as dots by taking sample statistic c along the vertical scale and the sample number along the horizontal scale. The central line (CL) is drawn as bold horizontal line at λ or c and UCL c and LCL c are plotted as dotted lines at the computed values. Example 1. There are given the values of sample mean x — and range (R) for ten samples of size 5 each. Draw Mean and Range charts and comment on the state of control of the process. Sample No. 12345678910 x 43 49 37 44 45 37 51 46 43 47 R 5657748646 Given for n = 5, A 2 = 0.58, D 3 = 0, D 4 = 2.115 . a state of statistical control. These type of causes are also known as random causes. These small variations, which are natural to and inherent in the manufacturing process, are also called allowable. allowable variations as they cannot be removed or prevented altogether in any way. The allowable variation is also sometimes known as natural variation, as it cannot be eliminated and one has to allow. BASED NUMERICAL AND STATISTICAL TECHNIQUES Remarks: 1. A central limit representing the average value of the quality characteristics, or desired standard or level of the control process. 2. An