Table 6.1
Factors for Calculating Three-Sigma Limits for the $\bar { X }$ -chart and R-chart
$$\begin{array} { | c | c | c | c | }
\hline \begin{array} { c }
\text { Size of } \
\text { Sample } \
( n )
\end{array} & \begin{array} { c }
\text { Factor for UCL } \
\text { and LCL for } X \text { - } \
\text { Charts } \
\left( A _ { 2 } \right)
\end{array} & \begin{array} { c }
\text { Factor for } \
\text { LCL for } R \text { - } \
\text { Charts } \
\left( D _ { 2 } \right)
\end{array} & \begin{array} { c }
\text { Factor for } \
\text { UCL for } R \text { - } \
\text { Charts } \
\left( D _ { 1 } \right)
\end{array} \
\hline 2 & 1.880 & 0 & 3.267 \
3 & 1.023 & 0 & 2.575 \
4 & 0.729 & 0 & 2.282 \
5 & 0.577 & 0 & 2.115 \
6 & 0.483 & 0 & 2.004 \
7 & 0.419 & 0.076 & 1.924 \
8 & 0.373 & 0.136 & 1.864 \
9 & 0.397 & 0.184 & 1.816 \
10 & 0.308 & 0.233 & 1.777 \
\hline
\end{array}$$
Source: 1950 Manual on Quality Control of Materials, copyright
American Society for Testing Materials. Reprinted with permission.
-A milling operation has historically produced an average thickness of 0.005 inch with an average range of 0.015 inch.Currently,the first three items from each batch of 20 are inspected.Use Table 6.1.What is the value of the LCL $\bar { X }$ ?
A) less than or equal to 0.00100
B) greater than 0.00100 but less than or equal to 0.00299
C) greater than 0.00299 but less than or equal to 0.00499
D) greater than 0.00499

Question

Table 6.1
Factors for Calculating Three-Sigma Limits for the $\bar { X }$ -chart and R-chart
 $$\begin{array} { | c | c | c | c | } 
\hline \begin{array} { c } 
\text { Size of } \
\text { Sample } \
( n )
\end{array} & \begin{array} { c } 
\text { Factor for UCL } \
\text { and LCL for } X \text { - } \
\text { Charts } \
\left( A _ { 2 } \right)
\end{array} & \begin{array} { c } 
\text { Factor for } \
\text { LCL for } R \text { - } \
\text { Charts } \
\left( D _ { 2 } \right)
\end{array} & \begin{array} { c } 
\text { Factor for } \
\text { UCL for } R \text { - } \
\text { Charts } \
\left( D _ { 1 } \right)
\end{array} \
\hline 2 & 1.880 & 0 & 3.267 \
3 & 1.023 & 0 & 2.575 \
4 & 0.729 & 0 & 2.282 \
5 & 0.577 & 0 & 2.115 \
6 & 0.483 & 0 & 2.004 \
7 & 0.419 & 0.076 & 1.924 \
8 & 0.373 & 0.136 & 1.864 \
9 & 0.397 & 0.184 & 1.816 \
10 & 0.308 & 0.233 & 1.777 \
\hline
\end{array}$$ 
Source: 1950 Manual on Quality Control of Materials, copyright
American Society for Testing Materials. Reprinted with permission.
-A milling operation has historically produced an average thickness of 0.005 inch with an average range of 0.015 inch.Currently,the first three items from each batch of 20 are inspected.Use Table 6.1.What is the value of the LCL $\bar { X }$ ?
A) less than or equal to 0.00100
B) greater than 0.00100 but less than or equal to 0.00299
C) greater than 0.00299 but less than or equal to 0.00499
D) greater than 0.00499

Quizplus · Accepted Answer

Using Table 6.1 with n=3, we have A2=1.023 and D2=0. Therefore, the LCL for $\bar { X }$ is given by:
LCL $\bar { X }$= $\bar { X }$ - (A2 $\times$ $\bar{R}$/d2) = 0.005 - (1.023 $\times$ 0.015/0) = 0.005.

Note: The calculation gives an answer of 0.005, which is also the process mean. This can happen when the range is 0 or close to 0. However, the response choices do not account for this possibility, so we select the closest answer which is C.

Table 61 Factors for Calculating Three-Sigma Limits for The The \bar

Table 61
Factors for Calculating Three-Sigma Limits for The The \bar