1. **Problem Statement:** Calculate the average ($\bar{x}$) and range ($R$) for each subgroup, then compute the overall mean ($\bar{x}$) and average range ($\bar{R}$), and finally determine the control limits for $\bar{x}$ and $R$ charts. 2. **Formulas:** - Subgroup average: $\bar{x}_i = \frac{\sum_{j=1}^n x_{ij}}{n}$ where $n=5$ is subgroup size. - Subgroup range: $R_i = X_{max} - X_{min}$. - Overall mean: $\bar{x} = \frac{\sum_{i=1}^k \bar{x}_i}{k}$ where $k=20$ is number of subgroups. - Average range: $\bar{R} = \frac{\sum_{i=1}^k R_i}{k}$. - Control limits for $\bar{x}$ chart: - Center Line (CL) = $\bar{x}$ - Upper Control Limit (UCL) = $\bar{x} + A_2 \bar{R}$ - Lower Control Limit (LCL) = $\bar{x} - A_2 \bar{R}$ - Control limits for $R$ chart: - Center Line (CL) = $\bar{R}$ - Upper Control Limit (UCL) = $D_4 \bar{R}$ - Lower Control Limit (LCL) = $D_3 \bar{R}$ 3. **Given constants:** - $A_2 = 0.577$ - $D_3 = 0$ - $D_4 = 2.115$ 4. **Subgroup calculations:** | Subgroup | $\bar{x}$ | $R$ | |---|---|---| | 1 | 230.0 | 20 | | 2 | 225.6 | 43 | | 3 | 222.2 | 14 | | 4 | 222.4 | 13 | | 5 | 215.2 | 25 | | 6 | 220.8 | 35 | | 7 | 224.6 | 10 | | 8 | 225.6 | 20 | | 9 | 222.4 | 24 | | 10 | 225.6 | 35 | | 11 | 229.8 | 19 | | 12 | 223.8 | 12 | | 13 | 224.2 | 9 | | 14 | 228.6 | 17 | | 15 | 225.6 | 7 | | 16 | 225.0 | 45 | | 17 | 223.0 | 37 | | 18 | 225.4 | 40 | | 19 | 232.0 | 26 | | 20 | 222.6 | 21 | 5. **Overall mean and average range:** $$\bar{x} = \frac{4498.4}{20} = 224.92$$ $$\bar{R} = \frac{472}{20} = 23.6$$ 6. **Control limits for $\bar{x}$ chart:** $$CL = 224.92$$ $$UCL = 224.92 + 0.577 \times 23.6 = 224.92 + 13.62 = 238.54$$ $$LCL = 224.92 - 0.577 \times 23.6 = 224.92 - 13.62 = 211.30$$ 7. **Control limits for $R$ chart:** $$CL = 23.6$$ $$UCL = 2.115 \times 23.6 = 49.91$$ $$LCL = 0 \times 23.6 = 0$$ **Final answers:** - Overall mean ($\bar{x}$): 224.92 - Average range ($\bar{R}$): 23.6 - $\bar{x}$ chart limits: CL = 224.92, UCL = 238.54, LCL = 211.30 - $R$ chart limits: CL = 23.6, UCL = 49.91, LCL = 0 This completes the control chart calculations based on the provided data and formulas.