**3. Results**

In this section, we provide the results of the methodologies discussed in Section 2. These results are presented in three folds, so is the discussion in the next section.

### *3.1. Practitioners' Estimation Variability*

Here, through the simulation results of the algorithm explained in Section 2.2, we observe the variability that appears in the Shewhart control chart due to different choices of sample size *m*, amongs<sup>t</sup> practitioners. Tables 1 and 2 depict the Shewhart chart whose parameters, both mean and variance, are estimated from *m* phase-I samples for both normal and non-normal distributions. It is evident from the result, the effect of parameter estimation on the performance of the chart. The ARL0s when δ = 0, are clustering around the target 370 with their respective *L*'s. However, when δ - 0, we observe that the smaller *m* becomes, the less effective the Shewhart chart performance. The ARL1's are expected to be sufficiently small in order to detect any drift in the ongoing process, but as *m* gets smaller, ARL1's ge<sup>t</sup> bigger. Which implies the chart is less sensitive in identifying the presence of shifts in the ongoing process early enough. Another noticeable effect of the parameter estimation on the Shewhart chart is the decrement in the limits *L*, as *m* reduces. This should be recorded as an edge if the corresponding phase-II charts detects shift earlier than when the parameters are known.

### *3.2. E*ff*ect of Outliers on the Shewhart Control Charts*

In Tables 3 and 4, we present the simulation results of environment (4) discussed in Section 2.3. From these results, the gross impact of outliers in the phase-I samples on the performance of the Shewhart chart cannot be over emphasized. Having seen the pattern of the IC and OoC RL properties in Tables 1 and 2, in order to save space, we restrict the performance evaluation to the IC RL properties. That is, considering the case when δ = 0 only. From Tables 3 and 4, when α = 0, in the absence of outlier, the ARL0's are clustering around its target 370, irrespective of the amount of phase-I

sample *m*. However, when α > 0, the ARL0's deviate from the target, vigorously. As the amount of phase-I samples *m* reduces, and the percentage of outliers present in the samples α increases, the more the ARL0's deviate from the target. Similarly the pattern of the SDRL, even more.

**Table 3.** ARL of the Shewhart chart in the presence of outliers with estimated parameters for standard normal and t (*v* = 100) distributions.


**Table 4.** SDRL of the Shewhart chart in the presence of outliers with estimated parameters for standard normal and t (*v* = 100) distributions.


*3.3. Improvement of Tukey and MAD Outlier Detection Models on Shewhart Chart Performance*

While incorporating the procedures in Sections 2.4.1 and 2.4.2, the simulation results are presented in Tables 5–8 respectively. Tables 5 and 7 represents the ARL result for Tukey and MAD outlier detection models respectively, as Tables 6 and 8 are the corresponding SDRL results. The effect of these detection models are noticed as ARLs and SDRLs are closer to when there is an absence of outliers or even better.


**Table 5.** ARL of the Shewhart chart with Tukey outlier detection for standard normal and t (*v* = 100) distributions.

**Table 6.** SDRL of the Shewhart chart with Tukey outlier detection for standard normal and t (*v* = 100) distributions.


**Table 7.** ARL of the Shewhart chart with median absolute deviation (MAD) outlier detection for standard normal and t (*v* = 100) distributions.



**Table 8.** SDRL of the Shewhart chart with MAD outlier detection for standard normal and t (*v* = 100) distributions.

For better visuals of the results, we depict the ARL results (Tables 3, 5 and 7) in Figures 2 and 3 and the SDRL results (Tables 4, 6 and 8) in Figures 4 and 5.

**Figure 2.** In-control ARL values for the Shewhart chart from standard normal distribution in the presence of outliers with and without outlier screening.

**Figure 3.** In-control ARL values for the Shewhart chart from t-(*v* = 100) distribution in the presence of outliers with and without outlier screening.

**Figure 4.** In-control SDRL values for the Shewhart chart from standard normal distribution in the presence of outliers with and without outlier screening.

**Figure 5.** In-control SDRL values for the Shewhart chart from t-(*v* = 100) distribution in the presence of outliers with and without outlier screening.
