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

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2 Control Charts for x and R

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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

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4 Control Chart for The parameters for the x chart are: UCL = Center Line = LCL = Where A 2 is tabulated for various sample sizes in Table VI

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5 Control Chart for continued Estimate of , the process average, where Estimate of , the process standard deviation where d 2 is tabulated for various sample sizes in Table VI

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

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

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8 Control Chart for continued since where

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9 Control Chart for R Control Limits for the R Chart UCL Center Line LCL where D 3 and D 4 are tabulated for various values of n in Table VI

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10 Control Chart for R continued - Statistical Basis SinceUCL and the relative range is, and Thensince

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11 Control Chart for R continued and UCL Similarly, LCL

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

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13 Estimating Process Capability Process-Capability Ratio Percentage of the Specification Band that the Process Uses U P

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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 )

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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

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16 Control Chart for x and S

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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.

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

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

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