X Bar Chart & R Chart


X Bar Chart & R Chart

X-bar Chart

X Bar chart is used to monitor the centering of process to control its accuracy. X-bar chart is a type of Shewhart control chart that is used to monitor the arithmetic means of successive samples of constant size, n. This type of control chart is used for characteristics that can be measured on a continuous scale, such as weight, temperature, thickness etc. For example, one might take a sample of 5 shafts from production every hour, measure the diameter of each, and then plot, for each sample, the average of the five diameter values on the chart.

R-Chart

R chart is used monitor the dispersion or precision of the process.

The construction of X-Bar chart & R Chart is-

1. Select the characteristics for applying a control chart.

2. Select the appropriate type of control chart.

3. Collect the data.

4. Choose the rational sub group i.e., Sample.

5. Calculate the average X-bar& Range (R) for each sample.
6. Calculate the average of the averages X & R-bar.

7. Calculate the control limits for X chart or R- Chart

For X-bar Chart-

Control Limit or Centre line = (X-Double bar)

Upper Control Limit = (X-Double bar) + ( A2 * R-bar)


Lower Control limit = (X-Double bar) - ( A2 * R-bar)

For R Chart-

Control Limit or Centre line = (R-bar)


Upper Control Limit = (D4 * R-bar)


Lower Control limit  = (D3 * R-bar)


8. Plot the CL, UCL & LCL on the chart.

9. Plot individual X-bar & R values on the chart.

10. Check whether is in control or not.

11. Revise the control limit if the points are out-of-control.

Problem- Following table contain the data on weight of a plastic components in grams. this component is manufactured using a plastic injection modeling process. Mean & range chart are required to established for this process.

Determine – 1.The trail central line & control limits2. Draw the mean & range chart & plot the values. 3.State whether the process is under statistical control. 4. It not assumes that the deviation occurred due to assignable causes which are rectified now. Revise the Centre line & control limits. 5. Draw the revised mean & range chart & plot the values. 6.State the process is now under statistical control.

Sample Number
Measurement
X1
X2
X3
X4
1
6.35
6.40
6.32
6.37
2
6.46
6.37
6.36
6.41
3
6.34
6.40
6.34
6.36
4
6.69
6.64
6.68
6.59
5
6.38
6.34
6.44
6.40
6
6.41
6.40
6.29
6.34
7
6.38
6.44
6.28
6.58
8
6.35
6.41
6.37
6.38
9
6.56
6.55
6.45
6.48
10
6.38
6.40
6.45
6.37

Solution- Subgroup size n=4 & N=10

Sample Number
Measurement
X bar =                (X1 +X2+X3+X4)/4
R =       Xmax-Xmin
X1
X2
X3
X4
1
6.35
6.40
6.32
6.37
6.36
0.08
2
6.46
6.37
6.36
6.41
6.40
0.10
3
6.34
6.40
6.34
6.36
6.36
0.06
4
6.69
6.64
6.68
6.59
6.65
0.10
5
6.38
6.34
6.44
6.40
6.39
0.10
6
6.41
6.40
6.29
6.34
6.36
0.12
7
6.38
6.44
6.28
6.58
6.42
0.30
8
6.35
6.41
6.37
6.38
6.37
0.06
9
6.56
6.55
6.45
6.48
6.51
0.11
10
6.38
6.40
6.45
6.37
6.40
0.08

 
= 64.22 / 10 = 6.422
= 1.11 / 10 = 0.111

For a subgroup size n=4 table give the following satisfactory result

A2 = 0.729,D3 = 0 & D4 = 2.282

1. To determine the trail Centre & control limits

For X-bar Chart-

Control Limit or Centre line CLX-bar = (X-Double bar)= 6.422

Upper Control Limit UCLX-bar = (X-Double bar) + (A2 * R-bar) = 6.422 + 0.729 × 0.111 = 6.503

Lower Control limit LCLX-bar = (X-Double bar) – (A2 * R-bar) = 6.422 - 0.729 × 0.111 = 6.341

For R Chart-

Control Limit or Centre line CLR = (R-bar) = 0.111

Upper Control Limit UCLR = D4 * (R-bar) = 2.282 × 0.111 = 0.253

Lower Control limit LCLR = D3 * (R-bar) = 0 × 0.111 = 0

2. To draw the X bar & R Chart-


 3. To check the process is statistical control-

 It may be observed that sample 7 is above the upper control limits on the R chart & sample 4 & 9 are above the upper control limits of the X-bar chart. Since those three points are upper control limits, therefore the process is statistically out of control.

 4. To revise the Centre line & control limits

 Assuming that the assignable causes for three samples (4,7 & 9) are identified& they are eliminated through appropriate remedial actions, the centre lines & control limits can be revised.

 With sample 4,7 & 9 deleted we get
Revised control limits for R- Chart

Control Limit or Centre line CLR = (R-bar) = 0.073

Upper Control Limit UCLR = D4 * (R-bar)  = 2.282 × 0.073= 0.166

Lower Control limit LCLR = D3 * (R-bar) =0 × 0.073 = 0

Revised control limits for X-bar Chart

Control Limit or Centre line CLX-bar = (X-Double bar) = 6.377


Upper Control Limit UCLX-bar = (X-Double bar) + (A2 *R-bar) = 6.377 + 0.729 × 0.073 = 6.324

5. To draw the revised X-bar & R chart-

6. To check the process for statistically control-

 Since all the remaining points are within the revised limits in both X-bar & R chart therefore it can be concluded that the process is in statistically controlled.

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