Absolutely! Let’s **break down the problem** into simple steps and explain what an **X-bar and R chart** is, and how you’d construct one using your data. --- ### **1. Background Explanation** In semiconductor manufacturing, photolithography uses light to transfer a pattern onto a wafer. The photoresist expands during baking, and this expansion is measured. The target dimension after baking should be **1.50 microns ± 0.15 microns**. To **monitor the process stability**, you use statistical process control (SPC) tools like the **X-bar and R chart**, which help track the average (X-bar) and the range (R) of measurements over time. --- ### **2. What is an X-bar and R Chart?** - **X-bar Chart**: Shows how the average value of your samples changes over time. - **R Chart**: Shows how the range (difference between the largest and smallest observation in a sample) of your samples changes over time. - **n = 4**: Each sample/group contains 4 measurements. --- ### **3. The Data** You are given **20 subgroups** (each with 4 measurements). For each subgroup, you have: - The **average** (X-bar) - The **range** (R) --- ### **4. Steps to Construct the Charts** #### **Step 1: Calculate the Averages** **Overall Average of X-bar (\(\overline{\bar{X}}\))**: Add up all the averages and divide by the number of subgroups (20). \[ \overline{\bar{X}} = \frac{\sum_{i=1}^{20} \text{Average}_i}{20} \] **Overall Average of Range (\(\bar{R}\))**: Add up all the ranges and divide by 20. \[ \bar{R} = \frac{\sum_{i=1}^{20} \text{Range}_i}{20} \] --- #### **Step 2: Find Control Chart Constants** For **n = 4**: - \(A_2 = 0.729\) - \(D_3 = 0\) - \(D_4 = 2.282\) --- #### **Step 3: Calculate Control Limits** ##### **R Chart Limits:** \[ \text{UCL}_R = D_4 \times \bar{R} \] \[ \text{LCL}_R = D_3 \times \bar{R} \] ##### **X-bar Chart Limits:** \[ \text{UCL}_{\bar{X}} = \overline{\bar{X}} + A_2 \times \bar{R} \] \[ \text{LCL}_{\bar{X}} = \overline{\bar{X}} - A_2 \times \bar{R} \] --- ### **5. Let’s Calculate!** #### **Sum up the values:** **Averages:** 1.6, 1.45, 1.65, 1.55, 1.58, 1.46, 1.52, 1.44, 1.57, 1.6, 1.48, 1.36, 1.53, 1.55, 1.6, 1.56, 1.48, 1.5, 1.41, 1.59 Sum = 1.6 + 1.45 + 1.65 + 1.55 + 1.58 + 1.46 + 1.52 + 1.44 + 1.57 + 1.6 + 1.48 + 1.36 + 1.53 + 1.55 + 1.6 + 1.56 + 1.48 + 1.5 + 1.41 + 1.59 = **30.88** \[ \overline{\bar{X}} = \frac{30.88}{20} = 1.544 \] **Ranges:** 0.05, 0.1, 0.08, 0.07, 0.04, 0.09, 0.08, 0.06, 0.06, 0.05, 0.07, 0.04, 0.06, 0.06, 0.09, 0.08, 0.1, 0.07, 0.07, 0.04 Sum = 0.05 + 0.1 + 0.08 + 0.07 + 0.04 + 0.09 + 0.08 + 0.06 + 0.06 + 0.05 + 0.07 + 0.04 + 0.06 + 0.06 + 0.09 + 0.08 + 0.1 + 0.07 + 0.07 + 0.04 = **1.16** \[ \bar{R} = \frac{1.16}{20} = 0.058 \] --- #### **Plug into Control Limit Formulas** **R Chart:** - UCL = \(2.282 \times 0.058 = 0.132\) - LCL = \(0 \times 0.058 = 0\) **X-bar Chart:** - UCL = \(1.544 + 0.729 \times 0.058 = 1.544 + 0.042 = 1.586\) - LCL = \(1.544 - 0.042 = 1.502\) --- ### **6. Final Control Chart Limits** **R Chart:** - **UCL = 0.132** - **LCL = 0** **X-bar Chart:** - **UCL = 1.586** - **LCL = 1.502** - **Center line = 1.544** --- ### **Summary Table** | Chart | Center Line | UCL | LCL | |------------|-------------|--------|-------| | X-bar | 1.544 | 1.586 | 1.502 | | R | 0.058 | 0.132 | 0 | --- ### **How to Use These Charts** - Plot each subgroup’s average and range on the respective charts. - If all points stay between the control limits, the process is in control. - If any point is outside, investigate for special causes. --- **Let me know if you want to see how to plot the points or need further explanation!**