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Statistical Quality Control

- 2. W. Edward Deming advocated the implementation of a statistical quality management approach. His philosophy behind this approach is ‘reduce variation’- fundamental to the principle of continuous improvement and the achievement of consistency, reliability, and uniformity. It helps in trustworthiness, competitive position, and success. Statistical Quality Control Statistics: data sufficient enough to obtain a reliable result. Quality: relative term and can be defined as totality of features and characteristics of a product or service that bear on its ability to satisfy stated or implied need (ISO). Control: The operational techniques and activities (a system for measuring and checking) used to fulfil the requirements for quality. It incorporates a feedback mechanism system to explore the causes of poor quality or unsatisfactory performance and takes corrective actions. also suggests when to inspect, how often to inspect, and how much to inspect. Basic Concept
- 3. Statistical Quality Control A quality control system using statistical techniques to control quality by performing inspection, testing and analysis to conclude whether the product is as stated or designed quality standard. Relying on the probability theory, SQC evaluates batch quality and controls the quality of processes or products It makes the inspection more reliable and less costly. The basis of the measurement is the performance indicator, either individual, group or departmental calculated over time (hourly, daily, or weekly). These performance measures are plotted on a chart. Pattern obtained from plotting these measures are basis of taking appropriate actions so that The process variation in minimized and Major problems are prevented in future. The timing and type of, and responsibility for, these actions depends on whether the causes of variation is controlled or uncontrolled Basic Concept ….Cont’d
- 4. Statistical Quality Control In repetitive manufacture of a product, even with refined machinery, skilled operator, and selected material, variations are inevitable in the quality of units produced due to interactions of various causes. Variation may be due to Common or random causes of variation (as a result of normal variation in material, method, and so on that causes natural variation in product or process quality) resulting in stable pattern of variation. Special causes (changes in men, machine, materials or tools, jigs and fixture and so on) resulting in a shift from the stable pattern of variation. SQC assists in timely identification and elimination of the problem with an object of reducing variations in process or product. The application of statistical method of collecting and analyzing inspection and other data for setting the economic standards of product quality and maintaining adherence to the standards so that variation in product quality can be controlled Basic Concept ….Cont’d
- 5. Statistical quality control (SQC) is the term used to describe the set of statistical tools used by quality professionals SQC encompasses three broad categories of; Descriptive statistics used to describe quality characteristics and relationships. the mean, standard deviation, and range. Acceptance sampling used to randomly inspect a batch of products to determine acceptance or rejection of entire lot based on the results. Does not help to identify and catch the in-process problems Statistical process control (SPC) Involves inspecting the output from a process Quality characteristics are measured and charted Helpful in identifying in-process variations Three SQC Categories
- 6. Variability: Sources of Variation Variation exists in all processes. Variation can be categorized as either; Common or Random causes of variation, or Random causes that cannot be identified Unavoidable: inherent in the process Normal variation in process variables such as material, environment, method and so on. Can be reduced almost to zero only through improvements in the process variables. Assignable causes of variation Causes can be identified and eliminated e.g. poor employee training, worn tool, machine needing repair Can be controlled by operator but it needs attention of management.
- 7. Traditional Statistical Tools Descriptive Statistics include Measure of accuracy (centering) Measure of central tendency indicating the central position of the series. A measure of the central value is necessary to estimate the accuracy or centering of a process. The Mean- simply the average of a set of data Sum of all the measurements/data divided by the number of observations. The Median- simply the value of middle item if the data are arranged in ascending or descending order. Applies directly if the number in the series is odd. It lies between two middle numbers if the number of the series is even. The Mode- value that repeat itself maximum number of times in the series. Shape of Distribution of Observed Data A measure of distribution of data Normal or bell shaped Skewed n x x n 1i i 1 K j j X K
- 8. Distribution of Data Also a measure of quality characteristics. Symmetric distribution - same number of data are observed above and below the mean. This is what we see only when normal variation is present in the data Skewed distribution – a disproportionate number of data are observed either above or below the mean. Mean and median fall at different points in the distribution Centre of gravity is shifted to oneside or other.
- 9. Traditional Statistical Tools …cont’d Measure of Precision or Spread Reveals the extent to which numerical data tend to spread about the mean value. The Range- the simplest possible measure of dispersion. Difference between largest and smallest observations in a set of data. o Depends on sample size and it tends to increase as sample size increases. o Remains the same despite changes in values lying between two extreme values. Standard Deviation- a measure deviation of the values from the mean. Small values >> data are closely clustered around the mean Large values >> data are spread out around the mean. 1n Xx σ n 1i 2 i
- 10. Statistical Process Control Process Control Refers to procedures or techniques adopted to evaluate, maintain and improve the quality standard in various stages of manufacture. A process is considered satisfactory as long as it produces items within designed specification. Process should be continuously monitored to ensure that the process behaves as it is expected. Salient features of process control Controling the process at the right level and variability. Detecting the deviation as quickly as possible so as to take immediate corrective actions. Ultimate aim is not only to detect trouble, but also to find out the cause. Developing an efficient information system in order to establish an efficient system of process control.
- 11. Statistical Process Control Statistical Process Control (SPC) Statistical evaluation of the output of a process during production. Goal is to make the process stable over time and then keep it stable unless the planned changes are made. Statistical description of stability requires that ‘pattern of variation’ remains stable over time, not that there be no variation in the variable measured. In statistical process control language: A process that is in control has only common or random cause variation - an inherent variability of the system. When the normal functioning of the prosess is disturbed by some unpredictable events, special cause variation is added to common cause variation. Applying SPC to service Nature of defect is different in services Service defect is a failure to meet customer requirements One way to deal with service quality is to devise quantifiable measurement of service elements Number of complaints received per month, Number of telephone rings before call is answered
- 12. Hospitals timeliness and quickness of care, staff responses to requests, accuracy of lab tests, cleanliness, courtesy, accuracy of paperwork, speed of admittance and checkouts Grocery Stores waiting time to check out, frequency of out-of-stock items, quality of food items, cleanliness, customer complaints, checkout register errors Airlines flight delays, lost luggage and luggage handling, waiting time at ticket counters and check-in, agent and flight attendant courtesy, accurate flight information, passenger cabin cleanliness and maintenance Fast-Food Restaurants waiting time for service, customer complaints, cleanliness, food quality, order accuracy, employee courtesy Catalogue-Order Companies order accuracy, operator knowledge and courtesy, packaging, delivery time, phone order waiting time Insurance Companies billing accuracy, timeliness of claims processing, agent availability and response time Statistical Process Control
- 13. Statistical Process Control: Control Chart Control Chart A graphical display of data over time (data are displayed in time sequence in which they occurred/measured) used to differentiate common cause variation from special cause variation. Control charts combine numerical and graphical description of data with the use of sampling distribution normal distribution is basis for control chart. Goal of using this chart is to achieve and mainatain process stability A state in which a process has displayed a certain degree of consistency Consistency is characterized by a stream of data falling within the control limits. Basic Components of a Control Chart A control chart always has a central line usually mathematical average of all the samples plotted; upper control and lower control limits defining the constraints of common variations or range of acceptable variation; Performance data plotted over time. Lines are determined from historical data.
- 14. Control Chart …Cont’d When to use a control chart? Controlling ongoing processes by finding and correcting problems as they occur. Predicting the expected range of outcomes from a process. Determining whether a process is stable (in statistical control). Analyzing patterns of process variation from special causes (non-routine events) or common causes (built into the process). Determining whether the quality improvement project should aim to prevent specific problems or to make fundamental changes to the process. Control Chart Basic Procedure Choose the appropriate control chart for the data. Determine the appropriate time period for collecting and plotting data. Collect data, construct the chart and analyze the data. Look for “out-of-control signals” on the control chart. When one is identified, mark it on the chart and investigate the cause. Document how you investigated, what you learned, the cause and how it was corrected. Continue to plot data as they are generated. As each new data point is plotted, check for new out-of-control signals
- 15. Control Chart …Cont’d Interpretation of control chart Points between control limits are due to random chance variation One or more data points above an UCL or below a LCL mark statistically significant changes in the process A process is in control if No sample points outside limits Most points near process average About equal number of points above and below centerline Points appear randomly distributed A process is assumed to be out of control if Rule 1: A single point plots outside the control limits; Rule 2: Two out of three consecutive points fall outside the two sigma warning limits on the same side of the center line; Rule 3: Four out of five consecutive points fall beyond the 1 sigma limit on the same side of the center line; Rule 4: Nine or more consecutive points fall to one side of the center line; Rule 5: There is a run of six or more consecutive points steadily increasing or decreasing Time period Measured characteristics
- 16. Control Chart …Cont’d Setting Control Limits Type I error Concluding a process is not in control when it actually is. Type II error Concluding a process is in control when it is not. In control Out of control In control No Error Type I error (producers risk) Out of control Type II Error (consumers risk) No error Mean LCL UCL /2 /2 Probability of Type I error Mean LCL UCL /2 /2 Probability of Type I error
- 17. General model for a control chart UCL = μ + kσ CL = μ LCL = μ – kσ where μ is the mean of the variable σ is the standard deviation of the variable UCL=upper control limit; LCL = lower control limit; CL = center line. k is the distance of the control limits from the center line, expressed in terms of standard deviation units. When k is set to 3, we speak of 3-sigma control charts. Historically, k = 3 has become an accepted standard in industry. Control Chart …Cont’d
- 18. Control Chart …Cont’d Suggested Number of Data Points More data points means more delay Fewer data points means less precision, wider limits A tradeoff needs to be made between more delay and less precision Generally 25 data points judged sufficient Use smaller time periods to have more data points Fewer cases may be used as approximation Sample Size Attribute charts require larger sample sizes 50 to 100 parts in a sample Variable charts require smaller sample sizes 2 to 10 parts in a sample
- 19. Control Chart …Cont’d Types of the control charts Variables control charts Variable data are measured on a continuous scale. For example: time, weight, distance or temperature can be measured in fractions or decimals. Applied to data following continuous distribution Attributes control charts Attribute data are counted and cannot have fractions or decimals. Attribute data arise when you are determining only the presence or absence of something: success or failure, accept or reject, correct or not correct. For example, a report can have four errors or five errors, but it cannot have four and a half errors. Applied to data following discrete distribution
- 20. Variable control charts R chart (range chart) X-bar (mean chart) S chart (sigma chart) Individual or run chart i-chart Moving range chart Median chart EWMA (exponentially weighted moving average chart) General formulae for a control chart UCL or UAL = μ + kσx k = 3 ; Accepted Standard UWL = μ + 2/3 kσx CL = μ LWL = μ – 2/3 kσx LCL or LAL = μ – kσx Control Chart …Cont’d m i i X X m X n m: # of sample mean n: # of observations in each sample
- 21. Control Chart …Cont’d Mean control charts Used to detect the variations in mean of a process. X-bar chart Range control charts Used to detect the changes in dispersion or variability of a process R chart Use X-bar and R charts together Sample size : 2 ~ 10 Use X-bar and S charts together Sample size : > 10 Use i-chart and Moving range chart together Sample size : 1 or one-at-a-time data System can show acceptable central tendencies but unacceptable variability or System can show acceptable variability but unacceptable central tendencies Interpret the R-chart first: If R-chart is in control -> interpret the X-bar chart -> (i) if in control: the process is in control; (ii) if out of control: the process average is out of control If R-chart is out of control: the process variation is out of control -> Investigate the cause; no need to interpret the X-bar chart
- 22. Control Chart …Cont’d Constructing a X-bar Chart: A quality control inspector at the Cocoa Fizz soft drink company has taken three samples with four observations each of the volume of bottles filled. If the standard deviation of the bottling operation is 0.2 ounces, use the below data to develop control charts with limits of 3 standard deviations for the 16 oz. bottling operation. Centerline and 3-sigma control limit formulas Time 1 Time 2 Time 3 Observation 1 15.8 16.1 16.0 Observation 2 16.0 16.1 15.9 Observation 3 15.8 15.8 15.9 Observation 4 15.9 15.9 15.8 Sample means (X- bar) 15.875 15.975 15.9 Sample ranges (R) 0.2 0.3 0.2 3X X UCL X 3X X LCL X X CL X m i i X X m X n Where, m: # of sample mean n: # of observations in each sample
- 23. Control Chart …Cont’d Centerline (x-double bar): Control limits for±3σ limits: Control Chart Plot the sample mean in the sequence from which it was generated and interpret the pattern in the control chart. 15.875 15.975 15.9 x 15.92 3 x x x x .2 UCL x zσ 15.92 3 16.22 4 .2 LCL x zσ 15.92 3 15.62 4
- 24. Control Chart …Cont’d Second Method for X-bar Chart using Range and A2 factor Use this method when standard deviation for the process distribution is unknown. Control limits solution: Center line and 3-sigma control Fomulas: 1 2 k i i x n R R k R R or d d n ; ;& 2 2 2 2 3 3 x x x CL X R UCL X X A R d n R LCL X X A R d n
- 25. Control Chart …Cont’d OBSERVATIONS(SLIP-RINGDIAMETER,CM) SAMPLEk 1 2 3 4 5 x R 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08 4 5.03 4.91 5.01 4.98 4.89 4.96 0.14 5 4.95 4.92 5.03 5.05 5.01 4.99 0.13 6 4.97 5.06 5.06 4.96 5.03 5.01 0.10 7 5.05 5.01 5.10 4.96 4.99 5.02 0.14 8 5.09 5.10 5.00 4.99 5.08 5.05 0.11 9 5.14 5.10 4.99 5.08 5.09 5.08 0.15 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15 OBSERVATIONS(SLIP-RINGDIAMETER,CM) SAMPLEk 1 2 3 4 5 x R 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08 4 5.03 4.91 5.01 4.98 4.89 4.96 0.14 5 4.95 4.92 5.03 5.05 5.01 4.99 0.13 6 4.97 5.06 5.06 4.96 5.03 5.01 0.10 7 5.05 5.01 5.10 4.96 4.99 5.02 0.14 8 5.09 5.10 5.00 4.99 5.08 5.05 0.11 9 5.14 5.10 4.99 5.08 5.09 5.08 0.15 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15 Calculate limits for X-bar chart using sample range
- 26. Control Chart …Cont’d R-Chart: Always look at the Range chart first. The control limits on the X-bar chart are derived from the average range, so if the Range chart is out of control, then the control limits on the X-bar chart are meaningless. Look for out of control signal. If there are any, then the special causes must be eliminated. There should be more than five distinct values plotted, and no one value should appear more than 25% of the time. If there are values repeated too often, then you have inadequate resolution of your measurements, which will adversely affect your control limit calculations. Once the effect of the out of control points from the Range chart is removed, look at the X-bar Chart. Standard Deviation of Range and Standard Deviation of the process is related as: Centerline and 3-sigma Control Limit Formulas: Where 3 3 2 R d d R d 3 3 4 2 2 3 3 3 2 2 3 1 3 3 1 3 R R R CL R d d UCL R R R D R d d d d LCL R R R D R d d ( ) ( ) d D d 3 4 2 1 3 max( , ) d D d 3 3 2 0 1 3
- 27. Control Chart …Cont’d OBSERVATIONS(SLIP-RINGDIAMETER,CM) SAMPLEk 1 2 3 4 5 x R 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08 4 5.03 4.91 5.01 4.98 4.89 4.96 0.14 5 4.95 4.92 5.03 5.05 5.01 4.99 0.13 6 4.97 5.06 5.06 4.96 5.03 5.01 0.10 7 5.05 5.01 5.10 4.96 4.99 5.02 0.14 8 5.09 5.10 5.00 4.99 5.08 5.05 0.11 9 5.14 5.10 4.99 5.08 5.09 5.08 0.15 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15 OBSERVATIONS(SLIP-RINGDIAMETER,CM) SAMPLEk 1 2 3 4 5 x R 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08 4 5.03 4.91 5.01 4.98 4.89 4.96 0.14 5 4.95 4.92 5.03 5.05 5.01 4.99 0.13 6 4.97 5.06 5.06 4.96 5.03 5.01 0.10 7 5.05 5.01 5.10 4.96 4.99 5.02 0.14 8 5.09 5.10 5.00 4.99 5.08 5.05 0.11 9 5.14 5.10 4.99 5.08 5.09 5.08 0.15 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15 Calculate limits for R-chart
- 28. Control Chart …Cont’d S-Chart The sample standard deviations are plotted in order to control the process variability. For sample size (n>12), With larger samples, the resulting mean range does not give a good estimate of standard deviation the S-chart is more efficient than R-chart. For situations where sample size exceeds 12, the X-bar chart and the S-chart should be used to check the process stability. Centerline and 3-sigma Control Limit Formulas: Where s s s CL S c UCL S S B S c c LCL S S B S c 2 4 4 4 2 4 3 4 1 3 1 3 max( , ) c B c c B c 2 4 4 4 2 4 3 4 1 1 3 1 0 1 3 ( ) & kn j ji ji j Sx x S S n k 2 11 1
- 29. Changing Sample Size on the X-bar and R Charts In some situations, it may be of interest to know the effect of changing the sample size on the X-bar and R charts. Needed information: = average range for the old sample size = average range for the new sample size nold = old sample size nnew = new sample size d2(old) = factor d2 for the old sample size d2(new) = factor d2 for the new sample size Centerline and 3-sigma Control Limit Formulas: oldR newR Control Chart …Cont’d ( ) ( ) ( ) ( ) old old x chart d new UCL x A R d old d new LCL x A R d old 2 2 2 2 2 2 ( ) ( ) ( ) ( ) ( ) max , ( ) old new old old R chart d new UCL D R d old d new CL R R d old d new LCL D R d old 2 4 2 2 2 2 3 2 0
- 33. Control Chart: Interpreting the Patterns Patterns A nonrandom identifiable arrangement of plotted points on the chart. Provides sufficient reasons to look for special causes. Causes that affect the process intermittently and can be due to periodic and persistent disturbances Natural pattern No identifiable arrangement of the plotted points exists No point falls outside the control limit; Majority of the points are near the centerline; and Few points are close to the control limits These patterns are indicative of a process that is in control. One point outside the control limits Also known as freaks and are caused by external disturbance Not difficult to identify the special causes for freaks. However, make sure that no measurement or calculation error is associated with it, Sudden, very short lived power failure, Use of new tool for a brief test period or a broken tool, incomplete operation, failure of components
- 34. Interpreting the Patterns …cont’d Sudden shift in process mean A sudden change or jump in process mean or average service level. Afterward, the process becomes stable. This sudden change can occur due to changes- intentional or otherwise in Process settings e.g. temperature, pressure or depth of cut Number of tellers at the Bank, New operator, new equipment, new measurement instruments, new vendor or new method of processing. Gradual shift in the process mean Such shift occurs when the process parameters change gradually over a period of time. Afterward, the process stabilizes X-bar chart might exhibit such shift due to change in incoming quality of raw materials or components over time, maintenance program or style of supervision. R-chart might exhibit such shift due to a new operator, decrease in worker skill due to fatigue or monotoy, or improvement in incoming quality of raw materials.
- 35. Interpreting the Patterns …cont’d Trending pattern Trend represents changes that steadily increases or decreases. Trends do not stabilize or settle down X-bar chart may exhibit a trend because of tool wear, dirt or chip buildup, aging of equipment. R-chart may exhibit trend because of gradual improvement of skill resulting from on-the-job-training or a decrease in operator skill due to fatigue. Cyclic pattern A repetitive periodic behavior in the system. A high and low points will appear on the control chart X-bar chart may exhibit a cyclic behavior because of a rotation of operator, periodic changes in temperature and humidity, seasonal variation of incoming components, periodicity in mechanical or chemical properties of the material R-chart might exhibit cyclic pattern because of operator fatigue and subsequent energization following breaks, a difference between shifts, or periodic maintenance of equipment. Graph will not show cyclic pattern, if the samples are taken too infrequently
- 36. Interpreting the Patterns …cont’d Zones for Pattern Test UCL LCL Zone A Zone B Zone C Zone C Zone B Zone A Process average 3 sigma = x + A2R = 3 sigma = x - A2R = 2 sigma = x + (A2R) = 2 3 2 sigma = x - (A2R) = 2 3 1 sigma = x + (A2R) = 1 3 1 sigma = x - (A2R) = 1 3 x = Sample number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 UCL LCL Zone A Zone B Zone C Zone C Zone B Zone A Zone A Zone B Zone C Zone C Zone B Zone A Process average 3 sigma = x + A2R = 3 sigma = x - A2R = 2 sigma = x + (A2R) = 2 3 2 sigma = x - (A2R) = 2 3 1 sigma = x + (A2R) = 1 3 1 sigma = x - (A2R) = 1 3 x = Sample number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13
- 37. Interpreting the Patterns …cont’d Performing a Pattern Test OBSERVATIONS (SLIP- RING DIAMETER, CM) SAMPLE k 1 2 3 4 5 x R 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08 4 5.03 4.91 5.01 4.98 4.89 4.96 0.14 5 4.95 4.92 5.03 5.05 5.01 4.99 0.13 6 4.97 5.06 5.06 4.96 5.03 5.01 0.10 7 5.05 5.01 5.10 4.96 4.99 5.02 0.14 8 5.09 5.10 5.00 4.99 5.08 5.05 0.11 9 5.14 5.10 4.99 5.08 5.09 5.08 0.15 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15
- 38. Interpreting the Patterns …cont’d Performing a Pattern Test xx-- barbar ChartChart ExampleExample (cont.)(cont.) UCL = 5.08 LCL = 4.94 Mean Sample number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 5.10 – 5.08 – 5.06 – 5.04 – 5.02 – 5.00 – 4.98 – 4.96 – 4.94 – 4.92 – x = 5.01= UCL = 5.08 LCL = 4.94 Mean Sample number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 5.10 – 5.08 – 5.06 – 5.04 – 5.02 – 5.00 – 4.98 – 4.96 – 4.94 – 4.92 – x = 5.01=x = 5.01=
- 39. Control Chart …Cont’d A process is assumed to be out of control if Rule 1: A single point plots outside the control limits; Rule 2: Two out of three consecutive points fall outside the two sigma warning limits on the same side of the center line; Rule 3: Four out of five consecutive points fall beyond the 1 sigma limit on the same side of the center line; Rule 4: Nine or more consecutive points fall to one side of the center line; Rule 5: There is a run of six or more consecutive points steadily increasing or decreasing
- 40. Interpreting the Patterns …cont’d Performing a Pattern Test 11 4.984.98 BB —— BB 22 5.005.00 BB UU CC 33 4.954.95 BB DD AA 44 4.964.96 BB DD AA 55 4.994.99 BB UU CC 66 5.015.01 —— UU CC 77 5.025.02 AA UU CC 88 5.055.05 AA UU BB 99 5.085.08 AA UU AA 1010 5.035.03 AA DD BB SAMPLESAMPLE xx ABOVE/BELOWABOVE/BELOW UP/DOWNUP/DOWN ZONEZONE 11 4.984.98 BB —— BB 22 5.005.00 BB UU CC 33 4.954.95 BB DD AA 44 4.964.96 BB DD AA 55 4.994.99 BB UU CC 66 5.015.01 —— UU CC 77 5.025.02 AA UU CC 88 5.055.05 AA UU BB 99 5.085.08 AA UU AA 1010 5.035.03 AA DD BB SAMPLESAMPLE xx ABOVE/BELOWABOVE/BELOW UP/DOWNUP/DOWN ZONEZONE 11 4.984.98 BB —— BB 22 5.005.00 BB UU CC 33 4.954.95 BB DD AA 44 4.964.96 BB DD AA 55 4.994.99 BB UU CC 66 5.015.01 —— UU CC 77 5.025.02 AA UU CC 88 5.055.05 AA UU BB 99 5.085.08 AA UU AA 1010 5.035.03 AA DD BB SAMPLESAMPLE xx ABOVE/BELOWABOVE/BELOW UP/DOWNUP/DOWN ZONEZONESAMPLESAMPLE xx ABOVE/BELOWABOVE/BELOW UP/DOWNUP/DOWN ZONEZONE
- 41. Control Chart …Cont’d A process is assumed to be out of control if Rule 1: A single point plots outside the control limits; Rule 2: Two out of three consecutive points fall outside the two sigma warning limits on the same side of the center line; Rule 3: Four out of five consecutive points fall beyond the 1 sigma limit on the same side of the center line; Rule 4: Nine or more consecutive points fall to one side of the center line; Rule 5: There is a run of six or more consecutive points steadily increasing or decreasing
- 42. Control Chart for Attributes Attributes are discrete events: yes/no or pass/fail Construction and interpretation are same as that of variable control charts. Attributes control charts p chart Uses proportion nonconforming (defective) items in a sample. Based on a binomial distribution. Can be used for varying sample size. np chart Uses number of nonconforming items in a sample. Should not be used when sample size varies. c chart Uses total number of nonconformities or defects in samples of constant size. Occurence of nonconformities follows poisson distribution. u chart when the sample size varies, the number of nonconformities per unit is used as a basis for this control chart.
- 43. Control Chart: p chart Proportion nonconforming or defectives for each sample are plotted on the p-chart The chart is examined to determine whether the process is in control. Means to calculate center line and control limits No standard or target value of proportion nonconforming is specified It must be estimated from sample infromation and For each sample, proportion of nonconforming items are determined as The average of these individual sample proportion of nonconforming items is used as the center line (CLp): As true value of p is not known, p-bar is used as an estimate x p n m m i i i p p x CL p m nm ( ) ( ) p p p p UCL p n p p LCL p n 1 3 1 3
- 44. 20 samples of 100 pairs of jeans20 samples of 100 pairs of jeans NUMBER OFNUMBER OF PROPORTIONPROPORTION SAMPLESAMPLE DEFECTIVESDEFECTIVES DEFECTIVEDEFECTIVE 11 66 .06.06 22 00 .00.00 33 44 .04.04 :: :: :: :: :: :: 2020 1818 .18.18 200200 20 samples of 100 pairs of jeans20 samples of 100 pairs of jeans NUMBER OFNUMBER OF PROPORTIONPROPORTION SAMPLESAMPLE DEFECTIVESDEFECTIVES DEFECTIVEDEFECTIVE 11 66 .06.06 22 00 .00.00 33 44 .04.04 :: :: :: :: :: :: 2020 1818 .18.18 200200 UCL = p + z = 0.10 + 3 p(1 - p) n 0.10(1 - 0.10) 100 UCL = 0.190 LCL = 0.010 LCL = p - z = 0.10 - 3 p(1 - p) n 0.10(1 - 0.10) 100 = 200 / 20(100) = 0.10 total defectives total sample observations p = UCL = p + z = 0.10 + 3 p(1 - p) n 0.10(1 - 0.10) 100 UCL = 0.190 UCL = p + z = 0.10 + 3 p(1 - p) n 0.10(1 - 0.10) 100 UCL = 0.190 LCL = 0.010 LCL = p - z = 0.10 - 3 p(1 - p) n 0.10(1 - 0.10) 100 LCL = 0.010 LCL = p - z = 0.10 - 3 p(1 - p) n 0.10(1 - 0.10) 100 LCL = p - z = 0.10 - 3 p(1 - p) n 0.10(1 - 0.10) 100 = 200 / 20(100) = 0.10 total defectives total sample observations p = = 200 / 20(100) = 0.10 total defectives total sample observations p = = 200 / 20(100) = 0.10 total defectives total sample observations p = If the target or standard value is specified Center line is selected as that target value i.e. CLp= po where, po represent a standard value Control limits are also based on the target velue. If the lower control limit for p is turned out to be negative, LCL is simply counted as zero. Lowest possible value for proportion of nonconformng item is zero Control Chart: p chart …Cont’d
- 45. Control Chart: p chart …Cont’d Variable sample size Changes in sample size casues the control limits to change, although the center line remained fixed. Control limits can be constructed: For individual samples If no standard value is given and sample mean proportion nonconforming is p-bar, control limit for sample i with size ni are Using average sample size Where ( ) ( ) i i p p UCL p n p p LCL p n 1 3 1 3 ( ) ( ) p p UCL p n p p LCL p n 1 3 1 3 m i i n n m 1
- 46. Control Chart: c chart No standard given Average number of nonconformities per sample unit is found from the sample observation and is denoted by c-bar. The center line and control limits are: If lower control limit is found to be less than zero, it is converted to zero. Standard given if the specified target for the number of nonconformities per sample unit be co.. The center line and control limits are then calculated from: c c c CL c UCL c c LCL c c 3 3 c o c o o o o o CL c UCL c c LCL c c 3 3
- 47. Control Chart: c chart …Cont’d Number of defects in 15 sample roomsNumber of defects in 15 sample rooms 1 121 12 2 82 8 3 163 16 : :: : : :: : 15 1515 15 190190 SAMPLESAMPLE cc = = 12.67= = 12.67 190190 1515 UCLUCL == cc ++ zzcc = 12.67 + 3 12.67= 12.67 + 3 12.67 = 23.35= 23.35 LCLLCL == cc ++ zzcc = 12.67= 12.67 -- 3 12.673 12.67 = 1.99= 1.99 NUMBER OF DEFECTS Number of defects in 15 sample roomsNumber of defects in 15 sample rooms 1 121 12 2 82 8 3 163 16 : :: : : :: : 15 1515 15 190190 SAMPLESAMPLE cc = = 12.67= = 12.67 190190 1515 cc = = 12.67= = 12.67 190190 1515 cc = = 12.67= = 12.67 190190 1515 190190 1515 UCLUCL == cc ++ zzcc = 12.67 + 3 12.67= 12.67 + 3 12.67 = 23.35= 23.35 UCLUCL == cc ++ zzcc = 12.67 + 3 12.67= 12.67 + 3 12.67 = 23.35= 23.35 LCLLCL == cc ++ zzcc = 12.67= 12.67 -- 3 12.673 12.67 = 1.99= 1.99 NUMBER OF DEFECTS 33 66 99 1212 1515 1818 2121 2424 NumberofdefectsNumberofdefects Sample numberSample number 22 44 66 88 1010 1212 1414 1616 UCL = 23.35 LCL = 1.99 c = 12.67 33 66 99 1212 1515 1818 2121 2424 NumberofdefectsNumberofdefects Sample numberSample number 22 44 66 88 1010 1212 1414 1616 UCL = 23.35 LCL = 1.99 c = 12.67 33 66 99 1212 1515 1818 2121 2424 NumberofdefectsNumberofdefects Sample numberSample number 22 44 66 88 1010 1212 1414 161622 44 66 88 1010 1212 1414 1616 UCL = 23.35 LCL = 1.99 c = 12.67