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SMU EMIS 7364

SMU EMIS 7364. NTU TO-570-N. Statistical Quality Control Dr. Jerrell T. Stracener, SAE Fellow. Control Charts for Variables x-bar and R & x-bar and S charts Updated: 3/17/04. Control Charts for x and R. Control Charts Based on: X ~ N(, ) at least 20 to 25 samples

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SMU EMIS 7364

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  1. SMU EMIS 7364 NTU TO-570-N Statistical Quality Control Dr. Jerrell T. Stracener, SAE Fellow Control Charts for Variables x-bar and R & x-bar and S charts Updated: 3/17/04

  2. Control Charts for x and R

  3. Control Charts Based on: • X ~ N(, ) • at least 20 to 25 samples • sample size of usually 4, 5 or 6 • process is thought to be controlled

  4. Control Chart for • The parameters for the x chart are: • UCL = • Center Line = • LCL = • Where A2 is tabulated for various sample sizes in Table VI

  5. Control Chart for continued • Estimate of , the process average, • where • Estimate of , the process standard deviation • where d2 is tabulated for various sample sizes in • Table VI

  6. Control Chart for continued where and the range of the ith sample

  7. Control Chart for continued - Statistical Basis UCL = LCL = Let

  8. Control Chart for continued since where

  9. Control Chart for R • Control Limits for the R Chart • UCL • Center Line • LCL • where D3 and D4 are tabulated for various values • of n in Table VI

  10. Control Chart for R continued - Statistical Basis Since UCL and the relative range is , and Then since

  11. Control Chart for R continued and UCL Similarly, LCL

  12. Control Chart for R continued Let Therefore, UCL and LCL

  13. Estimating Process Capability • Process-Capability Ratio • Percentage of the Specification Band that the Process Uses UP

  14. The Operating-Characteristic Function • The ability of the x and R charts to detect shifts in process quality is described by their operating-characteristic (OC) curves. • The OC curve for an x chart with the standard deviation  known and constant. If the mean shifts from the in-control value, say 0, to another value 1 = 0 + k, the probability of not detecting this shift on the first subsequent sample or the -risk is •  = P[LCL  x  UCL| =1 =0 + k] • or •  = (L - k(n)1/2) - (-L - k(n)1/2)

  15. The Average Run Length for the x Chart For any Shewhart control chart, the Average Run Length is For the in control process and for an out of control process

  16. Control Chart for x and S

  17. Control Charts for x and S • Although x and R charts are widely used, it is occasionally desirable to estimate the process standard deviation directly instead of indirectly through the use of the range R. • This leads to control charts for x and S, where S is the sample standard deviation. • Generally, x and S charts are preferable to x and R charts when either • 1. The sample size n is moderately large, • say n > 10 or 12. • 2. The sample size n is variable.

  18. Control Charts for x • Parameters for the x Chart • UCL = • Center Line = • LCL = • where • and

  19. Control Chart for S • Control limits for the S Chart • UCL = • Center Line = • LCL =

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