**4. Discussion**

We summarize the findings of the study under the following subsections: (a) parameter estimation effect on the Shewhart control chart, (b) effect of outliers on Shewhart *X* chart performance, and (c) improvement of outliers screening models on the Shewhart *X* chart performance. Through the discussion, we use the run length properties as a yardstick for measuring the performance of the charts.

### *4.1. Parameter Estimation E*ff*ect on the Shewhart X Control Chart*

Theoretically, when the Shewhart charts parameters are known, the limit *L* corresponding to the IC ARL0 = 370 is *L* = 3. When the parameters are estimated from phase-I samples, the first effect of the estimation is the change in *L*. The control limit *L* deviates from its theoretical value as much as the sample size *m* reduces. That implies, the smaller the sample size *m*, the farther the control limit from the theoretical value. This is noticeable in Tables 1 and 2, as *L* changes as the sample size does. We compute *L*s based on 100,000 iterations of simulation. Secondly, in the introduction of shifts, which makes the process OC, the RL properties values of the estimated parameters are bigger than the theoretical values. This indicates that the chart with estimated parameters are slower in detecting shifts in the process as compared to the chart with known parameters. For instance, (cf. Tables 1 and 2), with *m* = 1000, δ = 0.5 the resulting ARL1 and SDRL1 are 156.42 and 158.84 for normal distribution and 150.92 and 160.46 for *t*-distribution respectively. However, with *m* = 25, δ = 0.5 ARL1 and SDRL1 are 190.12 and 333.88 for normal distribution and 194.06 and 340.19 for *t*-distribution respectively.

### *4.2. E*ff*ect of Outliers on Shewhart X Control Chart performance*

Haven noticed the effect of parameter estimation on Shewhart chart performance; one major cause could be the presence of outliers in the dataset. The results in Tables 3 and 4 prove that extreme values in the sample causes grea<sup>t</sup> havoc to the performance of the process. As discussed earlier in Section 4, α = 0 indicates absence of outliers, and the presence of outliers if otherwise. We observe

jumps in the values of IC ARL and SDRL from Tables 3 and 4. With di fferent combinations of α and *m*, we say the bigger the value of α and the smaller the value of *m*, the gross the e ffect of the outliers on the chart. Take for instance, in the normal environment, the ARL and SDRL values of just 1% of outliers (α = 0.01) for when *m* = 1000 as against when *m* = 25. It shocks to see the ARL and SDRL jumped from 592.55 and 612.00 to 996.3 and 4012.72 respectively. However, in the *t*-distribution, ARL and SDRL values of 1% of outliers (α = 0.01) for when *m* = 1000 as against when *m* = 25, are 575.08 and 591.55 to 953.61 and 3823.24 respectively.

### *4.3. Improvement of Outliers Screening Models on Shewhart Chart Performance*

The proposed remedy for the e ffect of outliers on the Shewhart chart works perfectly. The incorporation of Tukey and MAD outlier-screening models in the Shewhart chart normalizes the outlier e ffects and restores the performance even much better than it was. To access the e ffect of these two screening methods, we present Figures 2–5, displaying the IC ARL values with *m* = 25, 50, 100, 250, 500 and 1000, and the magnitude *w* = 3, without outliers screening, alongside the IC ARL whose outliers are screened with the Tukey and MAD-based models. The IC ARL that are supposed to be around the target 370 has jumped to more than 250% increment due to the e ffect of outliers. However, with our proposed screening models; both Tukey and MAD-based models; the IC ARL is returned back to its target with less than 5% increment and decrement. The IC SDRL also exhibits the same pattern; in fact, its improvement is more appreciable as compared to the ARL's.
