X AND MOVING RANGE (MR) CHART

Sometimes, a process may not work continuously or does not generate many items. When that occurs, a single measurement is used as the sample size ( n = 1). At that point, we say that we are in the process of using the X and moving R chart. This is a special-purpose control chart for variables data and should be used when more sensitive charting techniques are impossible . The X and moving R chart is used in the following instances:

• The process offers no piece-to-piece variation (e.g., fluid viscosity , fuel octane, solder temperature).

• Measurements are extremely difficult or expensive in terms of money, time, or safety (e.g., auto crash testing, space exploration).

• The process does not produce measurements that can be clustered into subgroups (e.g., speed of an assembly line, measurements of prototype parts ).

Because this control chart is extremely insensitive to changes in the process, the X and moving R chart should not be used as a substitute for the Xbar and R chart.

The steps for constructing and interpreting an X and moving R chart are similar to those for the Xbar and R chart. There are three differences between the Xbar and R chart and the X and moving R control charts :

1. Because the sample size is a single measurement ( n = 1), it is impossible to determine a true range or measure of piece-to-piece variation. A moving range indicates the amount of movement from one sample reading to the next reading and is calculated with the following formula:

   Rm = X _i - X _i - 1

   where

   Rm	=	moving range.
   i	=	a measurement in a sequence.

   (e.g., when i = 5, X ₅ represents the 5th measurement).

   X i	=	the ith measurement.
   X i	-	1 = the measurement previous to the ith measurement.

   It is very important to realize that no moving range can be calculated for the first measurement. On the other hand, depending on the amount of data, the moving range may be based on two, three, four, or more actual samples. However, the most common one is two.

2. Individual readings (Xs) are plotted on the control chart for location. This control chart should not be used if the distribution for the readings does not follow the normal distribution.

3. Control limits are based on the center line of the Rm chart (Rbar m), and constants are based on the number of readings included in the moving range ( n ).

   

   The constants are listed in Table 8.2.

   Table 8.2: Table of Constants and Formulas for Individuals and MR

   		Chart for Ranges
   Subgroup Size	Chart for Individuals (X) Factors for Control Limits	Divisors Estimate of Standard Deviation	Factors for Control Limits
   n	E 3	d 2	D 3	D 4
   2	2.660	1.128	-	3.267
   3	1.772	1.693	-	2.574
   4	1.457	2.059	-	2.282
   5	1.290	2.326	-	2.114
   6	1.184	2.534	-	2.004
   7	1.109	2.704	0.076	1.924
   8	1.054	2.847	0.136	1.864
   9	1.010	2.970	0.184	1.816
   10	0.975	3.078	0.223	1.777

Because there is only one measurement per sample, control limits should not be calculated until 100 samples have been collected.

Figure 8.16 shows a sample X and moving R control chart.

Figure 8.16: A sample X and moving R control chart.