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2.
Measurement Process Characterization
2.2. Statistical control of a measurement process 2.2.2. How are bias and variability controlled? 2.2.2.1. Shewhart control chart
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| Small changes only become obvious over time | Because it takes time for the patterns in the data to emerge, a permanent shift in the process may not immediately cause individual violations of the control limits on a Shewhart control chart. The Shewhart control chart is not powerful for detecting small changes, say of the order of 1 - 1/2 standard deviations. The EWMA (exponentially weighted moving average) control chart is better suited to this purpose. | ||
| Example of EWMA control chart for mass calibrations | The exponentially weighted moving average (EWMA) is a statistic for monitoring the process that averages the data in a way that gives less and less weight to data as they are further removed in time from the current measurement. The data $$ Y_1, Y_2, \, ..., \, Y_t$$ are the check standard measurements ordered in time. The EWMA statistic at time t is computed recursively from individual data points, with the first EWMA statistic, EWMA1, being the arithmetic average of historical data. $$ EWMA_{\, t+1} = \lambda Y_t + (1-\lambda)EWMA_{\,t}$$ | ||
| Control mechanism for EWMA | The EWMA control chart can be made sensitive to small changes or a gradual drift in the process by the choice of the weighting factor, \( \lambda \) . A weighting factor of 0.2 - 0.3 is usually suggested for this purpose (Hunter), and 0.15 is also a popular choice. | ||
| Limits for the control chart | The target or center line for the control chart is the average of historical data. The upper (UCL) and lower (LCL) limits are $$UCL = EWMA_{\,1} + k{\large s} \sqrt{\frac{\lambda}{(2-\lambda)}}$$ $$LCL = EWMA_{\,1} - k{\large s} \sqrt{\frac{\lambda}{(2-\lambda)}}$$ where s times the radical expression is a good approximation to the standard deviation of the EWMA statistic and the factor k is chosen in the same way as for the Shewhart control chart -- generally to be 2 or 3. | ||
| Procedure for implementing the EWMA control chart | The implementation of the EWMA control chart is the same as for any other type of control procedure. The procedure is built on the assumption that the "good" historical data are representative of the in-control process, with future data from the same process tested for agreement with the historical data. To start the procedure, a target (average) and process standard deviation are computed from historical check standard data. Then the procedure enters the monitoring stage with the EWMA statistics computed and tested against the control limits. The EWMA statistics are weighted averages, and thus their standard deviations are smaller than the standard deviations of the raw data and the corresponding control limits are narrower than the control limits for the Shewhart individual observations chart. | ||
