1. **Problem Statement:** We have monthly data of the number of software file updates for 12 months. We need to create a variable control chart by finding the center line, upper control limit (UCL), and lower control limit (LCL), and then determine if the process is in control. 2. **Formulas and Rules:** - The center line (CL) for a variable control chart is the average number of updates: $$\bar{X} = \frac{\sum X_i}{n}$$ - The control limits are calculated as: $$UCL = \bar{X} + 3\sigma$$ $$LCL = \bar{X} - 3\sigma$$ where $\sigma$ is the standard deviation of the data. - If all points lie within the control limits and show no non-random patterns, the process is in control. 3. **Calculate the average (center line):** $$\bar{X} = \frac{323 + 268 + 290 + 405 + 383 + 368 + 249 + 181 + 92 + 80 + 30 + 75}{12}$$ $$= \frac{2744}{12} = 228.67$$ 4. **Calculate the standard deviation $\sigma$:** First, find each squared deviation: $$(323 - 228.67)^2 = 8896.11$$ $$(268 - 228.67)^2 = 1545.11$$ $$(290 - 228.67)^2 = 3780.11$$ $$(405 - 228.67)^2 = 31144.11$$ $$(383 - 228.67)^2 = 23944.11$$ $$(368 - 228.67)^2 = 19444.11$$ $$(249 - 228.67)^2 = 416.11$$ $$(181 - 228.67)^2 = 2254.11$$ $$(92 - 228.67)^2 = 18788.11$$ $$(80 - 228.67)^2 = 22288.11$$ $$(30 - 228.67)^2 = 39644.11$$ $$(75 - 228.67)^2 = 23744.11$$ Sum of squared deviations: $$= 189628.32$$ Variance: $$s^2 = \frac{189628.32}{12 - 1} = \frac{189628.32}{11} = 17239.85$$ Standard deviation: $$\sigma = \sqrt{17239.85} = 131.30$$ 5. **Calculate control limits:** $$UCL = 228.67 + 3 \times 131.30 = 228.67 + 393.90 = 622.57$$ $$LCL = 228.67 - 3 \times 131.30 = 228.67 - 393.90 = -165.23$$ Since LCL cannot be negative for number of updates, set: $$LCL = 0$$ 6. **Determine if process is in control:** All monthly updates are between 0 and 622.57, so all points lie within control limits. No obvious non-random patterns are given, so the process appears to be in control. **Final answers:** - Center line (CL): $228.67$ - Upper control limit (UCL): $622.57$ - Lower control limit (LCL): $0$ - Process status: In control