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Quality Control Part 2 By Anita Lee-Post © Anita Lee-Post Statistical process control methods • Control charts for variables: process characteristics are measured on a continuous scale, e.g., weight, volume, width • • Mean (X-bar) chart Range (R) chart • Control charts for attributes: process characteristics are counted on a discrete scale, e.g., number of defects, number of scratches • • Proportion (P) chart Count (C) chart • Process capability ratio and index © Anita Lee-Post Control charts • Use statistical limits to identify whether a sample of data falls within a normal range of variation or not © Anita Lee-Post Setting Limits Requires Balancing Risks • Control limits are based on a willingness to think that something is wrong when it’s actually not (Type I or alpha error), balanced against the sensitivity of the tool - the ability to quickly reveal a problem (failure is Type II or beta error) © Anita Lee-Post Control Charts for Variable Data • Mean (x-bar) charts • Tracks the central tendency (the average value observed) over time • Range (R) charts: • Tracks the spread of the distribution over time (estimates the observed variation) © Anita Lee-Post Mean (x-bar) charts CL process mean x x1 x2 ... xk k UCL x z x LCL x z x x ...xn where xi 1 , xi : observation i, n n : number of observations (sample size) k : number of samples z : number of normal standard deviation x : the standard deviation of the process mean n : the standard deviation of the process/population © Anita Lee-Post Mean (x-bar) charts continued • Use the x-bar chart established to monitor sample averages as the process continues: © Anita Lee-Post An example The diameters of five C&A bagels are sampled each hour during a 8-hour period. The data collected are shown as follows: © Anita Lee-Post An example continued a) Develop an x-bar chart with the control limits set to include 99.74% of the sample means and the standard deviation of the production process () is known to be 0.2 Inches. Step 1. Compute the sample mean x-bar: © Anita Lee-Post An example continued Step 2. Compute the process mean or center line of the control chart: 4.03 3.94 4.18 4.01 4.03 4.08 4.04 4.03 CL x 8 32.34 4.04 8 © Anita Lee-Post An example continued Step 3. Compute the upper and lower control limits: To include 99.74% of the sample means implies that the number of normal standard deviations is 3. i.e., z=3 UCL x z x 0. 2 4.3 4.04 3 5 LCL x z x 0.2 3.8 4.04 3 5 © Anita Lee-Post An example continued b. C&A collects the process characteristics (i.e., diameter) of their bagels in days 2 through 10. Is the process in control? Diameter of Sample Day 2 3 4 5 6 7 8 9 10 Average of 5 observations 4.21 3.99 3.89 4.05 4.22 4.28 3.55 3.78 5.00 x-bar Chart for Samples of 5 Bagels Average Diameter (in inches) 5.25 5.00 4.75 4.50 UCL = 4.25 4.25 4.00 3.75 LCL = 3.83 3.50 2 3 4 5 6 Sample © Anita Lee-Post 7 8 9 10 The process is not in control because the means of recent sample averages fall outside the upper and lower control limits Range (R) charts CL R R1 R2 ... Rk k UCL D4R LCL D3R where R : the average of the sample range k : number of samples Ri : the range of sample i max(xi) min(xi) xi : the observatio n of sample i D4 : the R - chart factor for UCL D3 : the R - chart factor for LCL © Anita Lee-Post An example The diameters of five C&A bagels are sampled each hour during a 8-hour period. The data collected are shown as follows: © Anita Lee-Post An example continued a) Develop a range chart. Step 1. Compute the average range or CL: 0.18 0.10 1.07 0.12 0.21 0.13 0.21 0.13 8 0.27 CL R © Anita Lee-Post An example continued Step 2. Compute the upper and lower control limits: Control Limit Factors for Range Charts Sample size, n D3 D4 2 0.00 3.27 3 0.00 2.57 4 0.00 2.28 5 0.00 2.11 6 0.00 2.00 7 0.08 1.92 8 0.14 1.86 © Anita Lee-Post UCL D4 R 2.11 0.27 0.57 LCL D3 R 0 0.27 0 An example continued b. C&A collects the process characteristics (i.e., diameter) of their bagels in days 2 through 10. Is the process in control? Day Range of 5 observations Diameter of Sample 2 3 4 5 6 7 8 9 10 0.30 0.20 0.33 0.20 0.14 0.11 0.05 0.35 0.20 Range Chart for Samples of 5 Bagels UCL = 0.57 0.60 Range 0.50 0.40 CL = 0.27 0.30 0.20 0.10 LCL = 0 0.00 2 3 4 5 6 Sample © Anita Lee-Post 7 8 9 10 The process is in control because the ranges of recent samples fall within the upper and lower control limits Using both mean & range charts • Mean (x-bar) chart: measures the central tendency of a process • Range (R) chart: measures the variance of a process Case 1: a process showing a drift in its mean but not its variance can be detected only by a mean (x-bar) chart © Anita Lee-Post Using both mean & range charts continued Case 2: a process showing a change in its variance but not its mean can be detected only by a range (R) chart © Anita Lee-Post Construct x-bar chart from sample range CL x x1 ... xk k UCL x A2R LCL x A2R R R2 ... Rk where R 1 k Ri : max(xi) min(xi ) xi : observation for sample i k : number of samples A 2 : Control limit factor for x - bar chart © Anita Lee-Post Control Charts for Attributes • p-Charts: • Track the proportion defective in a sample • c-Charts: • Track the average number of defects per unit of output © Anita Lee-Post Proportion (p) charts • Data requirements: • Sample size • Number of defects • Sample size is large enough so that the attributes will be counted twice in each sample, e.g., a defect rate of 1% will require a sample size of 200 units. © Anita Lee-Post Proportion (p) charts continued Total Number of defects from all samples CL p Number of samples Sample size UCL p z p LCL p z p where z : number of normal standard deviation p the sample standard deviation p (1 p ) n n : sample size © Anita Lee-Post Count (c) charts • Data requirements • Number of defects • Monitoring processes in which the items of interest (in this case, defects) are infrequent and/or occur in time or space, e.g., errors in newspaper, bad circuits in a microchip, complaints from customers. © Anita Lee-Post Count (c) charts continued CL c average number of defects n xi i 1 , where n : number of days/weeks/units n UCL c z c LCL c z c where z : number of standard deviation © Anita Lee-Post