Group Xbar-s Chart Example

Group Xbar-s Charts

Group Xbar-s charts help you assess changes in averages and the standard deviation across measurement subgroups for a characteristic. Review the following example—an excerpt from Innovative Control Charting1—to get a sense of how a group Xbar-chart works.

Group Xbar s Chart Example
Figure 1. Three width measurements from a yoke.
 

Case Description

This yoke is machined from an aluminum casting. There have been complaints from the assembly department that some of the yokes have a taper on the inside width. To monitor the uniformity of the inside widths, a group chart is set up at the milling machine to track the width at locations a, b, and c.
 

Sampling Strategy

Because the production volume is very high, and the same characteristic is being measured at three different locations on the part, a group Xbar-s chart is selected. Ten yokes are measured every hour.
 

Data Collection Sheet

Table 1. Data collection sheet for the group Xbar-s chart. MAX and MIN plot points are shown in bold.

Group Xbar s Chart Data Collection Sheet 1
Group Xbar s Chart Data Collection Sheet 2
Group Xbar s Chart Data Collection Sheet 3
 

Group Xbar-s Chart


Chart Example group Xbar s

Figure 2. Group Xbar-s chart representing three different yoke width locations.

 

Chart Interpretation

Group s chart: Location a appears in the MAX position for all groups. This suggests that location a has the largest standard deviation. Locations b and c appear randomly in the MIN position. This indicates that locations b and c have similar standard deviations and they are less than location a’s.
Note: The centerline on the group s chart is the average of all the 5 values on the data collection sheet.

Group Xbar chart: The difference between the MAX and MIN for each group represents taper within the yokes. Locations a, b, and c appear randomly in the MAX position. However, location a appears five out of nine times in the MIN position. This might indicate that location a has a smaller diameter than either of the two other locations. However, this supposition is not as strong as it would be if location a represented the MIN position for all groups.
Note: The centerline on the group Xbar chart is the average of all the Xbar plot points found on the data collection sheet.
 

Recommendation

The repeated presence of location a in the MAX position in the group s chart may be the result of the inability of tooling to hold the work piece consistently during the manufacturing of the yokes. Notice that location a is found at the end of the yoke. This may signify the need for tooling changes that will hold the outer ends more rigidly during manufacturing.

 

Estimating the Process Average

Process average estimates should be performed separately for each characteristic or location on the group chart (see Calculation 1).

Group Xbar and s chart process average

Calculation 1. Estimate of the process average for yoke width at location a.
 

Estimating Sigma

Estimates of sigma are also calculated separately for each characteristic or location on the group chart. Continuing with yoke width location a, see Calculations 2 and 3.

average sample standard deviation

Calculation 2. Calculation of the average sample standard deviation for yoke width location a.


Estimated standard deviation

Calculation 3. Estimated standard deviation for yoke width location a.
Note: To ensure reliable estimates, the number of groups should be at least 20. In this example, the number of groups is only nine. Therefore, these estimates and those found in Table 2 are only for illustration purposes.
 

Calculating Process Capability and Performance Ratios

Calculations 4, 5, and 6 show the process capability and performance calculations for yoke width location a.

Cp calculation width
Calculation 4. Cp calculation for width location a.


Cpk upper calculation
Calculation 5. Cpk upper calculation for width location a.


Cpk lower calculation
Calculation 6. Cpk lower calculation for width location a.
 

Group Xbar-s Chart Advantages

  • Graphically illustrates the variation of multiple product or process characteristics simultaneously and relative to each other.
  • Pinpoints the characteristics that are in need of the most attention.
  • Separates variation due to changes in the average from variation due to changes in the standard deviation.
  • Multiple measurement locations can be tracked on one chart.
 

Group Xbar-s Chart Disadvantages

  • No visibility of the characteristics that fall between the MAX and MIN plot points.
  • Cannot detect certain out-of-control conditions because the group charts described here have no control limits.
  • Given the large amounts of data used in s charts, efficient analysis typically requires software.
 

An Additional Comment About the Case

The process capability and performance ratio calculations for yoke widths at locations b and c are shown in Table 2.
 
Table 2. Summary statistics and process capability and performance ratios for yoke widths at locations b and c.

Group Xbar and s chart
 
When you use SPC software from InfinityQS, consuming the information provided by group Xbar-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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