91Ó°ÊÓ

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

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

The ith subgroup mean is calculated using;

${}_{\text{i}}\text{=}\frac{{\displaystyle \underset{\text{j=1}}{\overset{\text{n}}{∑}}\text{χ¾±Âá}}}{\text{n}}$

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

$\begin{array}{l}{}_{\text{i}}\text{=}\frac{{\displaystyle \underset{\text{i=1}}{\overset{\text{n}}{∑}}\underset{\text{j=1}}{\overset{{\text{n}}_{\text{i}}}{∑}}\text{χ¾±Âá}}}{{\displaystyle \underset{\text{j=1}}{\overset{\text{k}}{∑}}\text{ni}}}\\ \text{=0.499}\end{array}$

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

${}_{\text{i}}\text{=}{\underset{\text{i=1}}{\overset{\text{k}}{∑}}}_{\text{i}}=\frac{{}_{\text{i}}{+}_2+\mathrm{...}{+}_k}{k}$

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

$\begin{array}{l}\frac{{\displaystyle \underset{\text{i=1}}{\overset{\text{k}}{∑}}{R}_{\text{i}}}}{k}\\ \text{=0.037}\end{array}$

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

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

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

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

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.