The XmR Control Chart

Purpose: A control chart is a run chart which includes statistically determined limits, calculated from the data of the process. Control limits allow you to analyze the data to determine if variation is due to common causes or to a special cause.

Note: Control limits ARE NOT specification limits or desired goals or standards. They are calculated based on the data from the process.

For our purposes, we will use the method given by Brian Joiner in his book Fourth Generation Management.  The following steps are used to contruct the control limits based on the moving range of successive data points, hence the name XmR (X is the individual values, mR is the moving range).  This is also called an "Individuals and Moving Range Chart". There are other types of control charts but these will not be covered in this course.

Steps:

  1. List the data in its time series order.
  2. Calculate the average. This becomes the center line of the control chart.
  3. Calculate the absolute value differences (ranges) between each set of points.   There will be one less range than there are number of data points.
  4. Determine the median range. List the ranges from highest to lowest and find the middle of the list.
  5. Multiply range by 3.14. This determines the distance of the control limits from the center line.
  6. Calculate the control limits: Add the results of Step 2 to Step 5 to get the Upper Control Limit (UCL). Subtract Step 5 from Step 2 to get the Lower Control Limit (LCL).
  7. Plot the data in time series order and draw a solid center line at X, the average.
  8. Draw dashed lines to indicate the control limits.
    9.

Example:

  Data

Range
1 6  
2 9 3
3 15 6
4 8 7
5 8 0
6 7 1
7 4 3
8 9 5
9 17 8
10 22 5
11 6 16
12 4 2
13 4 0
14 10 6
15 13 3
16 12 1
17 15 3
18 17 2
19 3 14
20 4 1
21 12 8
22 7 5
23 6 1
24 6 0
25 8 2
Total: 233  
  1. The data have been ordered and the ranges calculated, as given in the table to the left.
  2. Calculate the average, X, which will be the center line on the control chart.  X = 233 / 25 = 9.32
  3. Determine the median range.  This step requires ordering the ranges from smallest to largest and finding the value(s) in the middle.  Ordered ranges: 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 6, 6, 7, 8, 8, 14, 16.   R = 3
  4. Multiply the range by 3.14.  3 x 3.14 = 9.42
  5. Calculate the upper control limit (UCL) by adding the distance obtained in step 4 to the average calculated in step 1. UCL = 9.32 + 9.42 = 18.74
  6. Calculate the lower control limit (LCL) by subtracting the distance obtained in step 4 from the average calculated in step 1.  LCL = 9.32 - 9.42 = -0.10   If the data collected cannot take on values less than zero, the lower control limit is adjusted to this minimum value, as in this case.  LCL = 0
  7. The XmR control chart can now be plotted.
    8.

Analysis:

Given that the upper control limit is 18.74, any point larger than 18 would be an outlier and, therefore, signal a possible special cause of variation.  Data point #10 has a value of 22 and should be investigated to discover why the system was out of statistical control at that point.

 

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