A$2/$D$3/$D$4$ Table

Factors for Calculating Three$-$Sigma Limits for the x$-$bar Chart and R$-$Chart

Size of

Sample

$($n$)$

Factor for UCL and LCL for

x$-$bar$-$Charts

A$2$

Factor for

LCL for

R$-$Charts

D$3$

Factor for

UCL for

R$-$Charts

D$4$

$2$

$1\mathrm{.}880$

$0$

$3\mathrm{.}267$

$3$

$1\mathrm{.}023$

$0$

$2\mathrm{.}575$

$4$

$0\mathrm{.}729$

$0$

$2\mathrm{.}282$

$5$

$0\mathrm{.}577$

$0$

$2\mathrm{.}115$

$6$

$0\mathrm{.}483$

$0$

$2\mathrm{.}004$

$7$

$0\mathrm{.}419$

$0\mathrm{.}076$

$1\mathrm{.}924$

$8$

$0\mathrm{.}373$

$0\mathrm{.}136$

$1\mathrm{.}864$

$9$

$0\mathrm{.}337$

$0\mathrm{.}184$

$1\mathrm{.}816$

$10$

$0\mathrm{.}308$

$0\mathrm{.}223$

$1\mathrm{.}777$

Source: $1950$ ASTM Manual on Quality Control of Materials, copyright American Society for Testing Materials. Reprinted with permission.

$($LO$2$DM$1)$ Jus de Fruit Co$.$ has set up for automated production of its new bottled Triple Berry Colada. Five samples were taken during the first week of production. The OM team wants to check if the dispersion of the process is in control. What will be the upper and lower control limits$,$ respectively?

Sample

Bottle $1$

Bottle $2$

Bottle $3$

$1$

$20\mathrm{.}1$

$20\mathrm{.}1$

$19\mathrm{.}8$

$2$

$20\mathrm{.}0$

$19\mathrm{.}1$

$20\mathrm{.}0$

$3$

$19\mathrm{.}9$

$20\mathrm{.}1$

$19\mathrm{.}4$

$4$

$18\mathrm{.}0$

$21\mathrm{.}8$

$20\mathrm{.}5$

$5$

$20\mathrm{.}0$

$19\mathrm{.}7$

$20\mathrm{.}0$

$21\mathrm{.}13$ and $18\mathrm{.}67$

$3\mathrm{.}09$ and $0$

$2\mathrm{.}54$ and $0$

$21\mathrm{.}1$ and $18\mathrm{.}7$

$20\mathrm{.}59$ and $19\mathrm{.}21$