An X Chart Based on Standard Deviations (Instead of Ranges)

question 19

Essay

An x chart based on standard deviations (instead of ranges) is made by plotting sample means with a centerline
at x and control limits at x + A3 s and x - A3 s where A3 is found in the Control Chart Constants table and s is
the mean of the sample standard deviations. Construct an x chart based on standard deviations for the data
below.
A manufacturer of high intensity video projector lightbulbs picks 6 lightbulbs at random each day for quality
control. The mean, standard deviation, and range (in hours) of the lifetimes of the 6 lightbulbs are calculated.
The results for 12 consecutive days are shown in the table below. $$\begin{array} { c | c | c | c } \text { Day } & \bar { x } & \text { Range } & \mathrm { s } \\\hline 1 & 21.1 & 3.4 & 1.2 \\2 & 23.4 & 4.7 & 1.7 \\3 & 20.9 & 3.1 & 0.9 \\4 & 25.6 & 6.2 & 2.3 \\5 & 21.3 & 4.7 & 1.3 \\6 & 20.5 & 3.7 & 1.7 \\7 & 25.8 & 5.8 & 2.1 \\8 & 23.7 & 6.2 & 2.9 \\9 & 29.8 & 2.1 & 0.8 \\10 & 22.7 & 4.4 & 1.6 \\11 & 20.0 & 2.7 & 0.7 \\12 & 24.6 & 7.8 & 3.0\end{array}$$

---

Definitions:

Manufacturing Overhead

All indirect costs associated with the manufacturing process, excluding direct materials and direct labor.

Controllable Variance

The difference between actual and expected amounts that can be directly managed or controlled by responsible individuals.

Fixed Overhead Costs

Indirect expenses of running a business that are not affected by changes in the volume of goods or services produced, such as rent, salaries, and insurance.

Overhead Costs

Ongoing expenses related to the operation of a business, such as rent, utilities, and insurance, that are not directly tied to production.