# Target Xbar-R Charts

Target Xbar and range (Xbar-R) charts can help you identify changes in the average and range of averages of a characteristic. Review the following example—an excerpt from Innovative Control Charting1—to get a sense of how a target Xbar-R chart works.

Figure 1. Relief valve with adjustable cracking pressure capabilities.

## Case Description

Cracking pressure, the pressure at which the relief valve opens, is a key characteristic. The valve can be adjusted during assembly to crack at different pressures. Each customer has his or her own crack pressure requirements.

## Sampling Strategy

Cracking pressure is the only characteristic, but the requirements change with each order (see Table 1). Because the production volume is steady and the standard deviation is expected to be consistent across all cracking pressure settings, a target Xbar-R chart is used to monitor the process. Valves are 100 percent tested, but for charting purposes, the test results from three out of every 30 valves are used for analysis on control charts.

Table 1. Crack pressure requirements for three valve customers.

## Data Collection Sheet

Table 2. Data collection sheet for relief valves.

## Target Xbar-R Chart

Figure 2. Crack pressure target Xbar-R control chart.

## Control Limit Calculations

Calculation 1. Calculations for the crack pressure target Xbar chart.

Calculation 2. Calculations for the crack pressure range chart.

## Chart Interpretation

Range chart: No out-of-control plot points. There are no shifts, trends, or runs. It appears that the ranges are stable. This normal pattern supports the assumption that the process standard deviation is not affected when the valves are adjusted to different cracking pressures.

Target Xbar chart: Plot point comparisons to both the coded Xbar and the zero line must be made. Relative to the coded Xbar ( –0.94) none of the jobs is centered; this is caused mainly by customer C’s job being run well below its target of 180 psi. These plot points are pulling down the entire average, thus causing there to appear significantly long runs of plot points above the coded Xbar.

Relative to the zero line, the valve for customer A is centered on target, valves for customer B are a little on the high side of the target, and customer C’s valves are running consistently low.

## Recommendations

If a characteristic is not centered on its target, either the process needs to be adjusted or the target needs to be changed.

Assuming the targets are desired values,

• Customer A valves are centered on target; no adjustment needs to be made.
• Customer B valves are a little on the high side. The benefit of centering the crack pressure on its target may not be worth the effort required if the Cp and Cpk values are high (greater than 1.3).
• Customer C valves need to be adjusted about 5 psi higher. However, before changing the process, people attending to the process should verify the off-target values are not caused by a faulty measurement system.

## Estimating the Process Average

The average difference from target is not the same for all three valve adjustments. So calculations for X need to be done separately for each of the three customer requirements. The following example focuses on customer A valves.

Calculation 3. Calculation for customer A's average cracking pressure.
Note: To ensure reliable estimates, k should be about 20. In this example k is only nine. Therefore, the calculations on these pages and in the additional comments section are used only for illustration purposes.

## Estimating Sigma

Because the range chart is in control across all three customer requirements, the estimate of sigma for all valves may be based upon the range chart’s centerline (see Calculation 4). If the range chart were not in control, separate, reliable R values would need to be calculated for each of the customer requirements.

Calculation 4. Estimating sigma using R.

## Calculating Process Capability and Performance Ratios

Because the R chart is in control, the same sigma may be used for separately calculating all process capability and performance ratios for the cracking pressures. Following are the Cp and Cpk calculations for customer A valves.

Calculation 5. Cp calculation for customer A valves.

Calculation 6. Cpk upper calculation for customer A valves.

Calculation 7. Cpk lower calculation for customer A valves.

• Multiple parts, specifications, or characteristics can be plotted on the same chart (provided they all exhibit similar variability).
• Data from gauges that are zeroed out on their target values can be plotted directly on the target Xbar without further data coding or transformation.
• Statistical control can be assessed for both the process and each unique part and/or characteristic being made.

• Control limits are valid only when the Rs from each part on the chart are similar. When they are not similar, the suspect part(s) must be monitored on a separate chart, or the data must be collectively evaluated on a short run chart.
• When interpreting the target Xbar chart, both the zero line and the coded Xbar must be taken into account. This accounts for some added complexity when interpreting the chart.

• The process capability and performance ratio calculations for the cracking pressure are shown in Table 3.
• When valves A, B, or C are run again, the new data can be combined with prior data.

Table 3. Cp and Cpk calculations for valves B and C.

When you use SPC software from InfinityQS, consuming the information provided by target Xbar-R charts becomes faster and easier than ever. See how this type of analysis is surfaced in InfinityQS solutions.

FOOTNOTE:
1 Wise, Stephen A. and Douglas C. Fair. Innovative Control Charting: Practical SPC Solutions for Today’s Manufacturing Environment. Milwaukee, WI: ASQ Quality Press.

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