Xbar and R Control Charts: Statistical Process Control Charts for Competitive Exams for 2019
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An XBar and RChart is a type of statistical process control chart for use with continuous data collected in
subgroups at set time intervals  usually between 3 to 5 Continuous Data pieces per subgroup. The Mean (XBar) of each subgroup  Subgroups is charted on the top graph and the Range (R) of the subgroup is charted on the bottom graph. Out of Control points or patterns can occur on either the Xbar or R chart.

Like all control charts, an XBar and RChart is used to answer the following questions:

Is the process stable over time?

What is the effect of a process change on the output characteristics?

How will I know if the process becomes unstable, or the performance changes over time?

When is it used?
Constructed throughout the DMAIC process, particularly in the Measure, Analyze and Control phases of the cycle.
Used to understand process behavior, evaluate different treatments or methods, and to control a process.
Recommended for subgroup sizes of 10 or less. If the subgroup size exceeds 10, the range chart is replaced by a chart of the subgroup standard deviation, or S chart.
NOTE: It has been estimated that 98% of all processes can be effectively represented by using either the XmR charts or & R charts.
How to Construct an XBar and R Control Chart
and Range (or S) charts are always constructed and viewed as a pair.
To construct an XBar and R Chart, follow the process steps below. For subgroup sizes greater than 10, substitute the subgroup standard deviation (S) for range (R), and use constants for S from the table located after the instructional steps.
1. Record subgroup observations.
2. Calculate the average () and range (R) for each subgroup.
3. Plot the and R values for each subgroup in time series. You can create a meaningful control chart from as few as 67 data points, although a larger sample size (20+ subgroups) will provide much more reliability. In most cases, control limits are not calculated until at least 20 subgroups of data are collected.
4. Calculate the average R value, or , and plot this value as the centerline on the R chart.
5. Based on the subgroup size, select the appropriate constant, called D4, and multiply by to determine the Upper Control Limit for the Range Chart.
6. All constants are available from the reference table.
UCL (R) =
Plot the Upper Control Limit on the R chart.
If the subgroup size is between 7 and 10, select the appropriate constant, called D3. and multiple by Rbar to determine the Lower Control Limit for
the Range Chart. There is no Lower Control Limit for the Range Chart if the subgroup size is 6 or less.
Plot the Lower Control Limit on the R chart.
7. Using the values for each subgroup, compute the average of all , or (also called the Grand Average). Plot the Xbarbar value as the centerline on the X Chart.
8. Calculate the Xbar Chart Upper Control Limit, or upper natural process limit, by multiplying Rbar by the appropriate A2 factor (based on subgroup size) and adding that value to the average (Xbarbar).
Plot the Upper Control Limit on the chart.
9. Calculate the Xbar Chart Lower Control Limit, or lower natural process limit, for the Xbar chart by multiplying Rbar by the appropriate A2 factor (based on subgroup size) and subtracting that value from the average (Xbar bar).
Plot the Lower Control Limit on the chart.
10. After constructing the control chart, follow the same rules to assess stability that are used on XmR charts. Make sure to evaluate the stability of the Range
Chart before drawing any conclusions about the Averages () Chart if the Range Chart is out of control, the control limits on the Averages Chart will be unreliable.
Control Chart Constants
and R chart  
Subgroup Size (n) 




2  1.880  0  3.267  1.128 
3  1.023  0  2.574  1.693 
4  0.759  0  2.282  2.059 
5  0.577  0  2.114  2.326 
6  0.483  0  2.004  2.534 
7  0.419  0.076  1.924  2.704 
8  0373  0.136  1.864  2.847 
9  0.337  0.184  1.816  2.970 
10  0.308  0.223  1.777  3.078 
and S chart  
Subgroup Size (n) 



11  0.927  0.322  1.678 
12  0.886  0.354  1.646 
13  0.850  0.382  1.619 
14  0.817  0.407  1.593 
15  0.789  0.428  1.572 
Exercise: Following is a table of data sampled twelve times from a process; the process mean is supposed to be 74.000 inches. Plot these data in an Xbar R chart to determine if the process is in statistical control
(note: n = 6).
Sample #  Average  Range  
1  74.030  74.002  74.019  73.992  74.006  73.999  74.008  0.038  
2  73.989  74.000  74.001  73.995  73.998  74.000  73.997  0.012  
3  73.998  74.002  74.050  73.989  74.005  74.030  74.012  0.061  
4  74.005  74.030  73.995  74.000  73.998  73.990  74.003  0.040  
5  74.006  73.990  74.004  73.998  74.000  74.002  74.000  0.016  
6  73.982  73.984  73.998  73.999  73.998  73.995  73.993  0.017  
7  74.003  74.002  74.008  73.992  74.006  73.999  74.002  0.016  
8  73.989  74.001  74.001  73.998  73.998  74.000  73.998  0.012  
9  73.998  74.002  74.001  73.989  74.006  74.030  74.004  0.061  
10  74.005  74.021  73.995  73.990  73.998  73.993  74.000  0.012  
11  74.004  73.995  73.998  73.998  74.006  74.002  74.001  0.061  
12  73.982  73.984  73.998  73.999  73.998  73.995  73.993  0.012  
 74.001 
 0.026 