SPC: R-bar Charts

R-bar charts utilize chart factors that are typically found in statistical references. Table 2 shows a portion of such a chart for 3-sigma control:

Table 2: 3-Sigma Control Chart Factors

Sub-Groups	D3	D4
2	0	3.27
3	0	2.57
4	0	2.28
5	0	2.11
6	0	2.00
7	0.08	1.92
8	0.14	1.86

The UCL (calculated) are LCL (calculated) are defined by:

Eq. (6) Eq. (7) Eq. (8)

where MR (moving range) is the absolute value of the difference between the current data point and the preceding data point. The number of sub-groups is an area that most people do not agree upon. For example, if you group results by week and you're analyzing data for a month you could use 4 sub-groups. As a general rule, if your continuous process has been operating under the same specifications over the time of your analysis, you may assume 2 sub-groups. This is the approach we'll use for our system.

Let's assume that our plant also produces glycol which has an average specific gravity of 1.11. An R-bar chart provides an effective means of monitoring the transition from n-hexane (SG=0.65) to glycol (SG=1.11). Monitoring the individual results (X-bar chart) in conjunction with the R-bar chart will paint a very clear picture of the transition. The data in Table 3 shows the 4-day transition. The feeds were changed just prior to this data being recorded. What we're seeing is the n-hexane leaving the system and the glycol showing up gradually.

Table 3: n-hexane to Glycol Transition Data

Date	Time	SG results	Moving Range results
3/15/99	0300	0.65	---
Â	0900	0.63	0.02
	1500	0.67	0.04
	2100	0.74	0.07
3/16/99	0300	0.77	0.03
Â	0900	0.82	0.05
	1500	0.95	0.13
	2100	0.99	0.04
3/17/99	0300	1.02	0.03
Â	0900	1.10	0.08
	1500	1.08	0.02
	2100	1.13	0.05
3/18/99	0300	1.10	0.03
Â	0900	1.08	0.02
	1500	1.07	0.01
	2100	1.10	0.03

Now, to form the R-bar chart, we graph the data/time versus the moving range. We can assume 2 sub-groups (n-hexane and glycol over a short period of time). The moving range for 3/15/99 at 0900 was calculated by |(0.63-0.65)|=0.02. Typically, there will be no process or quality control limits for R-bar charts. For this transition:

UCL(calculated) = (3.27)(0.043) = 0.14
LCL(calculated) = (0)(0.043) = 0.00

Figure 3 shows the R-bar chart for the transition from n-hexane to glycol. While using an X-bar chart alone during times of change in a system is feasible, it is sometimes difficult to graph the data due to the potentially large differences in results. Another consideration is that in data compiling, a R-bar chart appearing between X-bar charts is a nice way to show a transition phase has occurred. R-bar charts are also useful when plotting data over a large time span. To show the contrast in the two types of charts, the x-bar chart during the transition is shown in Figure 4.

The arrow in Figure 4 shows the point at which the SG of the glycol has entered the quality control range. This is a very important point because the system output must be directed to a glycol storage tank at this point (rather than the waste container used during the transition). This is why the charts should be used in conjunction with one another, and an R-bar chart should not be used alone during a transition.

Uses of R-bar charts include:
1. Keep a record of when process changes or feed changes occurred
2. Record of how long the process took to stabilize
3. Show long history of a process or piece of equipment

If or whenever you use R-bar charts, remember that they tell you nothing about the actual value of the results, only deviations from one result to the next.