You are operating a process that makes approximately $150$ gallons of dibutyl ether. Due to the changes in the viscosity of this product as it is drummed, you must sample each drum as it is filled. Because each drum within a batch is logically linked to the other drums, you decide to implement an XR chart for this activity. Given the initial data below, calculate the x and range for each batch, the mean and R for the entire data set, and the control limits you would expect to set for this activity. Using the template at the end of this chapter, plot your data with the control limits$.$ What can you conclude from your initial chart?

Observation

Drum $1$

Drum $2$

Drum $3$

$1$

$55\mathrm{.}8$

$56\mathrm{.}8$

$55\mathrm{.}8$

$2$

$54\mathrm{.}3$

$53\mathrm{.}3$

$53\mathrm{.}4$

$3$

$52\mathrm{.}7$

$52\mathrm{.}4$

$54\mathrm{.}8$

$4$

$53\mathrm{.}7$

$54\mathrm{.}9$

$59\mathrm{.}3$

$5$

$55\mathrm{.}8$

$54\mathrm{.}7$

$55\mathrm{.}0$

$6$

$57\mathrm{.}8$

$57\mathrm{.}2$

$55\mathrm{.}5$

$7$

$54\mathrm{.}2$

$53\mathrm{.}5$

$54\mathrm{.}8$

$8$

$57\mathrm{.}0$

$57\mathrm{.}6$

$57\mathrm{.}8$

$9$

$57\mathrm{.}4$

$57\mathrm{.}3$

$59\mathrm{.}5$

$10$

$54\mathrm{.}1$

$55\mathrm{.}6$

$56\mathrm{.}6$

$11$

$56\mathrm{.}1$

$55\mathrm{.}4$

$57\mathrm{.}6$

$12$

$57\mathrm{.}6$

$59\mathrm{.}1$

$59\mathrm{.}9$

$13$

$48\mathrm{.}3$

$62\mathrm{.}0$

$58\mathrm{.}1$

$14$

$56\mathrm{.}1$

$56\mathrm{.}6$

$57\mathrm{.}2$

$15$

$54\mathrm{.}0$

$55\mathrm{.}6$

$56\mathrm{.}5$

$16$

$57\mathrm{.}5$

$56\mathrm{.}8$

$54\mathrm{.}1$

$17$

$54\mathrm{.}1$

$54\mathrm{.}6$

$58\mathrm{.}2$

$18$

$57\mathrm{.}4$

$56\mathrm{.}4$

$53\mathrm{.}9$

$19$

$57\mathrm{.}3$

$56\mathrm{.}5$

$55\mathrm{.}4$

$20$

$53\mathrm{.}2$

$54\mathrm{.}2$

$55\mathrm{.}3$

$21$

$55\mathrm{.}2$

$53\mathrm{.}3$

$52\mathrm{.}3$

$22$

$55\mathrm{.}4$

$56\mathrm{.}9$

$58\mathrm{.}1$

$23$

$55\mathrm{.}8$

$56\mathrm{.}6$

$58\mathrm{.}2$

$24$

$53\mathrm{.}7$

$55\mathrm{.}4$

$52\mathrm{.}1$

$25$

$55\mathrm{.}5$

$55\mathrm{.}3$

$57\mathrm{.}3$