A manufacturer of a very expensive shampoo brand has lately experienced problems with what store managers called “the labels dancing around on the bottles.” A detailed study has shown that the height of labels on the bottles, in fact, varies quite randomly, giving the bottles, when lined up on the store shelf, a very unappealing and low-quality appearance that was judged to have drastically reduced the sales of this otherwise very popular high-margin product. A Six Sigma GB team was, therefore, formed around investigating this issue. To get a baseline, they collected data on 60 bottles coming off the production line, measured the label height, and immediately made an X-bar R control with subgroups of 6. The collected data is in the file Shampoo-Bottles-Data.JMP. Note that the data is measured as the height of the labels from the base of the bottle minus the nominal height (all we need here is consistency of the label heights – the actual height is less important). 1. Construct an X-bar R chart using subgroups of size 6.
The Correct Answer and Explanation is:
To construct an X-bar R chart using subgroups of size 6 for the given shampoo bottle label height data, we’ll go through the steps of calculating the necessary statistics, such as the average of each subgroup (X-bar), the range within each subgroup (R), and then use these to plot the control charts.
Step-by-step Guide:
1. Data Collection and Organization
First, we organize the data into subgroups of size 6, ensuring that each subgroup represents a set of measurements taken from the production line.
2. Calculate Subgroup Averages (X-bar)
For each subgroup, calculate the average of the measurements. This will represent the center of the subgroup. The formula for the average of each subgroup is:Xˉ=1n∑i=1nXi\bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_iXˉ=n1i=1∑nXi
Where XiX_iXi represents the label height measurements in the subgroup and nnn is the subgroup size (6 in this case).
3. Calculate Subgroup Range (R)
For each subgroup, calculate the range, which is the difference between the maximum and minimum label heights in that subgroup:R=Xmax−XminR = X_{\text{max}} – X_{\text{min}}R=Xmax−Xmin
Where XmaxX_{\text{max}}Xmax and XminX_{\text{min}}Xmin are the maximum and minimum values, respectively, in each subgroup.
4. Calculate Overall Average and Range
After calculating the subgroup averages (X-bar) and ranges (R), calculate the overall average of the X-bar values and the average of the R values:Xˉoverall=1m∑i=1mXˉi\bar{X}_{\text{overall}} = \frac{1}{m} \sum_{i=1}^{m} \bar{X}_iXˉoverall=m1i=1∑mXˉiRoverall=1m∑i=1mRiR_{\text{overall}} = \frac{1}{m} \sum_{i=1}^{m} R_iRoverall=m1i=1∑mRi
Where mmm is the number of subgroups.
5. Calculate Control Limits
Using the overall average and range, calculate the control limits for the X-bar and R charts. These control limits are based on standard constants for subgroup sizes (n = 6), which can be obtained from statistical tables:
- For the X-bar chart, the control limits are: UCLX=Xˉoverall+A2RoverallUCL_X = \bar{X}_{\text{overall}} + A_2 R_{\text{overall}}UCLX=Xˉoverall+A2Roverall LCLX=Xˉoverall−A2RoverallLCL_X = \bar{X}_{\text{overall}} – A_2 R_{\text{overall}}LCLX=Xˉoverall−A2Roverall Where A2A_2A2 is a constant based on the subgroup size (for n=6, A2=0.483A_2 = 0.483A2=0.483).
- For the R chart, the control limits are: UCLR=D4RoverallUCL_R = D_4 R_{\text{overall}}UCLR=D4Roverall LCLR=D3RoverallLCL_R = D_3 R_{\text{overall}}LCLR=D3Roverall Where D4D_4D4 and D3D_3D3 are constants based on the subgroup size (for n=6, D4=2.114D_4 = 2.114D4=2.114 and D3=0D_3 = 0D3=0).
6. Plot the Control Charts
- X-bar chart: Plot the subgroup averages against the control limits. Any points outside the control limits suggest an issue with the process.
- R chart: Plot the subgroup ranges against the control limits. Again, any points outside the limits suggest variability in the process that requires attention.
Conclusion:
By following these steps, you’ll be able to construct the X-bar R control chart, which helps identify whether the label height variability is within acceptable limits or if corrective actions need to be taken. This analysis will provide insights into whether the production process is stable or if adjustments are needed to improve product quality.
