A Control Chart for Monitoring Variation and Center Can Be $\overline {s}$

question 77

Essay

A control chart for monitoring variation and center can be based on standard deviations instead of on ranges.
An s chart for monitoring variation is made by plotting sample standard deviations with a centerline at $\overline {s}$ (the
mean of the sample standard deviations) and control limits at $\mathrm { B } _ { 4 } \overline { \mathrm { s } } \text { and } \mathrm { B } _ { 3 } \overline { \mathrm { s } }$ , where $\mathrm { B } _ { 4 } \text { and } \mathrm { B } _ { 3 }$ are found in the
Control Chart Constants table. Construct an s chart for the data below.
A manufacturer of lightbulbs picks 10 lightbulbs at random each day for quality control. The mean, standard
deviation, and range (in hours) of the lifetimes of the 10 lightbulbs are calculated. The results for 12 consecutive
days are shown in the table below. $$\begin{array} { c | c | c | c } \text { Day } & \bar { x } & \text { Range } & \mathrm { s } \\\hline 1 & 201.1 & 3.4 & 1.2 \\2 & 203.4 & 4.7 & 1.7 \\3 & 200.9 & 3.1 & 0.9 \\4 & 205.6 & 6.2 & 2.3 \\5 & 201.3 & 4.7 & 1.3 \\6 & 200.5 & 3.7 & 1.7 \\7 & 205.8 & 5.8 & 2.1 \\8 & 203.7 & 6.2 & 2.9 \\9 & 199.8 & 2.1 & 0.8 \\10 & 202.7 & 4.4 & 1.6 \\11 & 200.0 & 2.7 & 0.7 \\12 & 204.6 & 7.8 & 3.0\end{array}$$

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Definitions:

Asset-Specific Risks

Asset-Specific Risks are those risks that are unique to an individual asset or investment, such as company performance or sector volatility, as opposed to systemic risks affecting the entire market.

Standard Deviation

A statistical measure of the dispersion or spread in a set of data, indicating how much variation there is from the average (mean).

Stock

A category of investment that evidences a holder's part ownership in a corporation, entitling them to a share of the entity's assets and income.

Expected Return

The anticipated profit or loss from an investment, reflecting the potential rewards and risks.