X bar and R control charts
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X-bar and R control charts are used to monitor the mean and variation of a process based on samples taken over time. An initial series of samples is used to estimate the mean and standard deviation of the process and establish control limits for subsequent X-bar and R charts. These control charts can then be used to monitor the process mean and variation and detect any points that are outside the control limits, indicating an out-of-control process that requires investigation. The document provides steps for constructing X-bar and R control charts using sample data and calculating control limits based on the sample mean and range.
X bar and R control charts
- 1. Industrial Engineering Presented By :- Dhruv Shah TOPIC : X-bar and R Control Charts
- 2. X-bar and R Control Charts It is used to monitor the mean and variation of a process based on samples taken from the process at given times (hours, shifts, days, weeks, months, etc.). The measurements of the samples at a given time constitute a subgroup. Typically, an initial series of subgroups is used to estimate the mean and standard deviation of a process. The mean and standard deviation are then used to produce control limits for the mean and range of each subgroup. During this initial phase, the process should be in control. If points are out- of-control during the initial (estimation) phase, the assignable cause should be determined and the subgroup should be removed from estimation. Determining the process capability may also be useful at this phase.
- 3. Once the control limits have been established of the X-bar and R charts, these limits may be used to monitor the mean and variation of the process going forward. When a point is outside these established control limits it indicates that the mean (or variation) of the process is out-of-control. An assignable cause is suspected whenever the control chart indicates an out-of-control process. X-bar and R Control Charts
- 4. Control Charts for Variables Mean (x-bar) charts Tracks the central tendency (the average value observed) over time Range (R) charts: Tracks the spread of the distribution over time (estimates the observed variation)
- 5. Constructing a X-bar Chart :- A quality control inspector at the Cocoa Fizz soft drink company has taken three samples with four observations each of the volume of bottles filled. If the standard deviation of the bottling operation is .2 ounces, use the data below to develop control charts with limits of 3 standard deviations for the 16 oz. bottling operation. Time 1 Time 2 Time 3 Observation 1 15.8 16.1 16.0 Observation 2 16.0 16.0 15.9 Observation 3 15.8 15.8 15.9 Observation 4 15.9 15.9 15.8
- 6. Step 1: Calculate the Mean of Each Sample Time 1 Time 2 Time 3 Observation 1 15.8 16.1 16.0 Observation 2 16.0 16.0 15.9 Observation 3 15.8 15.8 15.9 Observation 4 15.9 15.9 15.8 Sample means (X-bar) 15.875 15.975 15.9
- 7. Step 2: Calculate the Standard Deviation of the Sample Mean x σ .2 σ .1 n 4 = = = ÷
- 8. Step 3: Calculate CL, UCL, LCL Center line (x-double bar): Control limits for ±3σ limits (z = 3): 15.875 15.975 15.9 x 15.92 3 + + = = ( ) ( ) x x x x UCL x zσ 15.92 3 .1 16.22 LCL x zσ 15.92 3 .1 15.62 = + = + = = − = − =
- 9. Step 4: Draw the Chart
- 10. An Alternative Method for the X-bar Chart Using R-bar and the A2 Factor Use this method when sigma for the process distribution is not known. Use factor A2 from Table 6.1 Factor for x-Chart A2 D3 D4 2 1.88 0.00 3.27 3 1.02 0.00 2.57 4 0.73 0.00 2.28 5 0.58 0.00 2.11 6 0.48 0.00 2.00 7 0.42 0.08 1.92 8 0.37 0.14 1.86 9 0.34 0.18 1.82 10 0.31 0.22 1.78 11 0.29 0.26 1.74 12 0.27 0.28 1.72 13 0.25 0.31 1.69 14 0.24 0.33 1.67 15 0.22 0.35 1.65 Factors for R-Chart Sample Size (n)
- 11. Step 1: Calculate the Range of Each Sample and Average Range Time 1 Time 2 Time 3 Observation 1 15.8 16.1 16.0 Observation 2 16.0 16.0 15.9 Observation 3 15.8 15.8 15.9 Observation 4 15.9 15.9 15.8 Sample ranges (R) 0.2 0.3 0.2 0.2 0.3 0.2 R .233 3 + + = =
- 12. Step 2: Calculate CL, UCL, LCL Center line: Control limits for ±3σ limits: ( ) ( ) 2x 2x 15.875 15.975 15.9 CL x 15.92 3 UCL x A R 15.92 0.73 .233 16.09 LCL x A R 15.92 0.73 .233 15.75 + + = = = = + = + = = − = − =
- 13. Control Chart for Range (R-Chart) Center Line and Control Limit calculations: 4 3 0.2 0.3 0.2 CL R .233 3 UCL D R 2.28(.233) .53 LCL D R 0.0(.233) 0.0 + + = = = = = = = = = Factor for x-Chart A2 D3 D4 2 1.88 0.00 3.27 3 1.02 0.00 2.57 4 0.73 0.00 2.28 5 0.58 0.00 2.11 6 0.48 0.00 2.00 7 0.42 0.08 1.92 8 0.37 0.14 1.86 9 0.34 0.18 1.82 10 0.31 0.22 1.78 11 0.29 0.26 1.74 12 0.27 0.28 1.72 13 0.25 0.31 1.69 14 0.24 0.33 1.67 15 0.22 0.35 1.65 Factors for R-Chart Sample Size (n)