Question

The following table contains the measurements of the key length dimension from a fuel injector. These samples of size five were taken at one-hour intervals.

Construct a three-sigma __ -chart and R-chart (use Exhibit 13.7) for the length of the fuel injector. What can you say about this process?

Short answer

The X-bar graph demonstrates how the mean, as well as average, varies over time, whereas the R chart depicts how the spectrum of subgroups evolves.

Step-by-step solution

Step-by-Step Solution Step 1: Consider the data

Given data is as follows:

The sample mean and range are calculated as follows:

The Sample mean for set 1 = Sum of all the samples for set 1/Number of samples in set 1 =2.47/5 = 0.49

The Range for day 1 = Maximum value of all the samples in set 1 - Minimum value of all samples in set 1 = 0.511 -0.481 = 0.03

The details for all the sets are as follows

Calculations for UCL, LCL, and X bar chart are as follows

The calculations for UCL, LCL, and X bar chart are as follows:

The ith subgroup mean is calculated using;

${}_{\text{i}}\text{=}\frac{{\displaystyle \sum _{\text{j=1}}^{\text{n}}\text{χij}}}{\text{n}}$

The grand mean (x-bar-bar) is calculated as follows:

$\begin{array}{l}{}_{\text{i}}\text{=}\frac{{\displaystyle \sum _{\text{i=1}}^{\text{n}}\sum _{\text{j=1}}^{{\text{n}}_{\text{i}}}\text{χij}}}{{\displaystyle \sum _{\text{j=1}}^{\text{k}}\text{ni}}}\\ \text{=0.499}\end{array}$

Also, the following is another formula for the calculation of the grand mean (x-bar-bar):

${}_{\text{i}}\text{=}{\sum _{\text{i=1}}^{\text{k}}}_{\text{i}}=\frac{{}_{\text{i}}{+}_2+\mathrm{...}{+}_k}{k}$

The range of the above data is calculated using the formula below:

$\begin{array}{l}\frac{{\displaystyle \sum _{\text{i=1}}^{\text{k}}{R}_{\text{i}}}}{k}\\ \text{=0.037}\end{array}$

We also have the following A2, D3, and D4 values for a sample size of n = 5. For the chosen sample size (as determined by the table of constants):

$\begin{array}{l}\text{A2}=0.\text{577}\\ \text{D3}=0\\ \text{D4}=\text{2}.\text{115}\end{array}$

The following are the calculations for UCL, LCL, as well as centerline for the X-bar chart

$\begin{array}{l}\text{UCL}(\text{X}-\text{bar})=\text{X}-\text{bar}-\text{bar}+(\text{A2xR}-\text{bar})=0.\text{4985}+0.\text{577}\times 0.0\text{373}=0.\text{52}\\ \text{LCL}(\text{X}-\text{bar})=\text{X}-\text{bar}-\text{bar}-(\text{A2xR}-\text{bar})=0.\text{4985}-0.\text{577}\times 0.0\text{373}=0.\text{477}\\ \text{CenterlineofX}-\text{barchartisX}-\text{bar}-\text{bar}\left(\text{grandmean}\right)=0.\text{4985}\end{array}$

The R chart's UCL, LCL, and centerline calculations are as follows.

$\begin{array}{l}\text{UCL}\left(\text{R}\right)=\text{R}-\text{bar}\times \text{D4}=0.0\text{373}\times \text{2}.\text{115}=0.0\text{789}\\ \text{LCL}\left(\text{R}\right)=\text{R}-\text{bar}\times \text{D3}=0.0\text{373}\times 0=0\\ \text{CenterlineofRchartisR}-\text{bar}\left(\text{Rmean}\right)=0.0\text{373}\end{array}$

The x-bar and range computed values are within the UCL as well as LCL limits determined previously for the x-bar and r-chart, indicating that the process is statistically controlled.