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STATISTICAL PROCESS CONTROL: CONTROL CHARTS Richard Mojena INTRODUCTION SPC is the application of statistical methodology to the control of processes. Also called Statistical Quality Control (SQC) Process is in statistical control… Variations due to natural causes (sampling error, “white” noise). Process is not in statistical control…. Variations due to both natural and assignable causes. Control charts are the most popular means of implementing SPC. TYPES OF CONTROL CHARTS Control charts for variables (continuous measure such as weight, volume, length, strength, …) track central tendency (mean) and dispersion or variation (range) to determine whether a process is in or out of statistical control. - x -charts… track mean. R-charts… track Range. (a) Mean and range in statistical control with natural causes Frequency Lower control limit Upper control limit (a) Mean but not range in control with natural causes (c) Out of control with assignable causes Size Weight, length, speed, etc. Control charts for attributes (binary category such as defective or not defective) track percent defectives or a count of defectives to determine whether a process is in or out of statistical control. p-charts… track percent defectives. c-charts… track count for number of defects. REVIEW OF DESCRIPTIVE STATISTICS IN SAMPLING Mean n Sample mean x is an estimate of xi population mean . x i 1 where xi is data item i and n is sample size. n m x x i1 where m is number of samples each with sample size n. Note that x is a mean of means or m grand mean. Range R = Max(xi) – Min(xi) Standard Deviation Sample standard deviation s is an estimate of n population standard deviation σ . ( xi x ) 2 s i 1 n-1 Six-sigma quality Normal curve fits 12 Standard Error of the Mean Sample standard error sx is an estimate of standard deviations or population standard error x. ±6σ within defined s sx upper and lower n tolerance limits (UTL, LTL). Central Limit Theorem (CLT) Amounts to about 2 defects per million. Confidence Interval (Limits) 6σ process capability 99.7% of all x fall ratio or Cp : 95.5% of all x fall within 3 within 2 x (UTL - LTL)/12σ x Six-sigma quality if Cp ≥ 1 where σ refers to measurement of interest. x x LTL UTL SPC Richard Mojena 9/27/10 Page 2 of 5 Example 1 Calculate mean, range, standard deviation, standard error of the mean, and 2sigma confidence limits for the following bar thicknesses in mm. Interpret the meaning of this confidence interval. xi 9 7 8 X-BAR-CHARTS USING STANDARD DEVIATION Uppper control limit UCL x z x x where z is the number of standard deviations (2 for 95.5% and 3 for 99.7%). LCL x z x x Lower control limit Example 2 Suppose we have 6 samples of size 3 for the application in the previous example, with sample means: 8.0, 8.5, 7.2, 8.8, 9.0, 10.0. Assume a robotic process set to a standard deviation of 1.0, but with unknown mean. Calculate the lower and upper 2-sigma control limits and plot the control chart. Conclusions? What if we were to use 3-sigma limits? What would change if the process is known to have a mean of 8.50? UCL mean LCL == 1 2 3 4 5 6 What’s the difference between confidence, control, and tolerance limits? SPC Richard Mojena 9/27/10 Page 3 of 5 X-BAR-CHARTS USING RANGE UCL xA R x 2 where A2 is a 3-sigma mean factor from TABLE 4.5, p.94. LCL xA R x 2 m Ri R i1 Note that R-bar is a mean of ranges for the m samples. m Example 3 Suppose in the previous example that the population standard deviation is not known. Ranges for the six samples are 2, 3, 2, 3, 4, 5. Recalculate 3-sigma control limits and draw conclusions. R-CHARTS UCLR D 4 R LCL R D 3 R where 3-sigma factors D3 and D4 are taken from TABLE 4.5, p.94. m Ri R i1 m Example 4 Find the R-chart control limits for the previous example. Conclusion? SPC Richard Mojena 9/27/10 Page 4 of 5 P-CHARTS If we define p as proportion defectives in a random sample of large (>30) size n, then the CLT says that the sampling distribution of p is normal with mean p (calculated on m samples) and standard error sp. p (1 p ) sp m n pi UCL p p zsp where p i1 or p = Average defects / n m LCL p p zsp Example 5 Suppose that 5 random samples of 1000 insurance claims are taken to audit for errors. The number of errors in each sample are 40, 30, 50, 35, and 45. Determine 3-sigma control limits. Conclusion? C-CHARTS Here the number of defects (count c) per unit (usually time, but could be other continuoums such as volume, distance, …) is determined across a sample of k units. Assuming a Poisson distribution, the mean is c and the standard deviation isc . Thus, UCL c c zsc k ci where c i 1 , sc = c. k LCL c c zsc Example 6 Consider a large corporation with many PCs. The event “PC requires repair” is assessed on a daily basis. Suppose that over a 5-day period the number of PC repairs is 20, 30, 10, 65, and 15. Calculate 3-sigma control limits and draw a conclusion. EXCEL IMPLEMENTATIONS Class software demo: SPC.xls. Sample commercial software as Excel add-ins (Googling Excel SPC yielded over 600k results): KnowWare International spcforexcel.com SPC Richard Mojena 9/27/10 Page 5 of 5