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## Statistical Process Control

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**Overview**• Variation • Control charts • R charts • X-bar charts • P charts**Statistical Quality Control (SPC)**• Measures performance of a process • Primary tool - statistics • Involves collecting, organizing, & interpreting data • Used to: • Control the process as products are produced • Inspect samples of finished products**Bottling Company**• Machine automatically fills a 20 oz bottle. • Problem with filling too much? Problems with filling to little? • So Monday the average is 20.2 ounces. • Tuesday the average is 19.6 ounces. • Is this normal? Do we need to be concerned? • Wed is 19.4 ounces.**Natural Variation**• Machine can not fill every bottle exactly the same amount – close but not exactly.**Assignable variation**• A cause for part of the variation**SPC**• Objective: provide statistical signal when assignable causes of variation are present**Control Chart Types**Continuous Numerical Data Categorical or Discrete Numerical Data Control Charts Variables Attributes Charts Charts R P C X Chart Chart Chart Chart**Measuring quality**• Characteristics for which you focus on defects • Classify products as either ‘good’ or ‘bad’, or count # defects • e.g., radio works or not • Categorical or discrete random variables Attributes Variables • Characteristics that you measure, e.g., weight, length • May be in whole or in fractional numbers • Continuous random variables**Control Chart Purposes**• Show changes in data pattern • e.g., trends • Make corrections before process is out of control • Show causes of changes in data • Assignable causes • Data outside control limits or trend in data • Natural causes • Random variations around average**Steps to Follow When Using Control Charts**TO SET CONTROL CHART LIMITS • Collect 20-25 samples of n=4 or n=5 a stable process • compute the mean of each sample. • Calculate control limits • Compute the overall means • Calculate the upper and lower control limits.**Steps to Follow When Using Control Charts - continued**TO MONITOR PROCESS USING THE CONTROL CHARTS: • Collect and graph data • Graph the sample means and ranges on their respective control charts • Determine whether they fall outside the acceptable limits. • Investigate points or patterns that indicate the process is out of control. Assign causes for the variations. • Collect additional samples and revalidate the control limits.**Control Charts for Variables**Glacier Bottling • Management at Glacier Bottling is concerned about their filling process. In particular, they want to know whether or not the machines are really filling the bottles with 16 ounces. • Create an Xbar chart that will be used to monitor the process. • Collected data for 25 days. Each day, pulled 4 bottles from the filling line and measured the amount in the bottle.**Glacier Bottling**Remember: There are 25 samples of size 4 to calculate the control limits. We are doing the first 5 right now…**Setting Control Limits for**R chart**R Chart**• Monitors variability in process • Variables control chart • Interval or ratio scaled numerical data • Shows sample ranges over time • Difference between smallest & largest values in inspection sample**R Chart Control Limits**From Table S6.1 Sample Range at Time i # Samples**Glacier Bottling**16.02 – 15.83 = 0.19**Glacier Bottling**16.02 – 15.83 = 0.19 16.12 – 15.85 = 0.27**R = 0.29**UCLR = D4R LCLR = D3R Glacier Bottling R-Charts**Control Chart Factors**Factor for UCL Factor for Factor Size of and LCL for LCL for UCL for Sample x-Charts R-Charts R-Charts (n) (A2) (D3) (D4) 2 1.880 0 3.267 3 1.023 0 2.575 4 0.729 0 2.282 5 0.577 0 2.115 6 0.483 0 2.004 7 0.419 0.076 1.924 This chart is in your text and will be provided for exams if needed.**R = 0.29**UCLR = D4R LCLR = D3R Glacier Bottling R-Charts D4 = 2.282 D3 = 0 UCLR = 2.282 (0.29) = 0.654 ounce LCLR = 0(0.29) = 0 ounce**R = 0.29**UCLR = D4R LCLR = D3R Glacier Bottling R-Charts D4 = 2.282 D3 = 0 UCLR = 2.282 (0.29) = 0.654 ounce LCLR = 0(0.29) = 0 ounce**Setting Control Limits for**Xbar chart**X Chart**• Monitors process average • Variables control chart • Interval or ratio scaled numerical data • Shows sample means over time**X Chart Control Limits**From Table S6.1 Sample Range at Time i Sample Mean at Time i # Samples**Glacier Bottling**(15.85+16.02+15.83+15.93)/4 = 15.908**Glacier Bottling**(16.12+16.00+15.85+16.01)/4 = 15.995**RBar = 0.29 ounce**XBarBar = 15.9469 ounces**Rbar = 0.29**xbarbar = 15.9469 Glacier Bottling:Setting Control Limits for XBar chart Xbar –Chart UCLx = x + A2R LCLx = x - A2R**Control Chart Factors**Factor for UCL Factor for Factor Size of and LCL for LCL for UCL for Sample x-Charts R-Charts R-Charts (n) (A2) (D3) (D4) 2 1.880 0 3.267 3 1.023 0 2.575 4 0.729 0 2.282 5 0.577 0 2.115 6 0.483 0 2.004 7 0.419 0.076 1.924 This chart is in your text and will be provided for exams if needed.**R = 0.29**x = 15.9469 A2 = 0.729 UCLx = 15.9469 + 0.729 (0.29) = 16.156 oz. Glacier Bottling:Setting Control Limits for XBar chart X –Chart = UCLx = x + A2R LCLx = x - A2R = =**R = 0.29**x = 15.9469 A2 = 0.729 UCLx = 15.9469 + 0.729 (0.29) = 16.156 oz. LCLx = 15.9469 – 0.729 (0.29) = 15.738 oz. Glacier Bottling:Setting Control Limits for XBar chart X –Chart = UCLx = x + A2R LCLx = x - A2R = =