Control Charts: The X-bar Chart and the R Chart

A control chart is a graph with a Central Line (CL), an Upper Control Limit (UCL), and a Lower Control Limit (LCL), on which sample statistics are plotted in the order they were taken. As long as the plotted points stay inside the two control limits, with no unusual pattern, the process is judged to be in statistical control (chance causes only); a point falling outside a control limit is the chart's signal that an assignable cause has crept in and needs to be investigated immediately.

For a variable (a measurement, like weight or length, rather than a simple pass/fail count), two charts are used together, from small samples of a fixed size $n$ drawn periodically from the process:

• $\bar{X}$ chart (mean chart) — tracks whether the process average stays on target.
• $R$ chart (range chart) — tracks whether the spread (consistency) of the process stays on target.

The control limits are computed from two grand statistics: $\bar{\bar{X}}$, the mean of all the sample means (the average of averages), and $\bar{R}$, the mean of all the sample ranges — together with standard control-chart factors ($A_2$, $D_3$, $D_4$) that depend only on the sample size $n$ and are looked up from a standard table. For a commonly used sample size $n=5$, the standard factors are $A_2 = 0.577$, $D_3 = 0$, and $D_4 = 2.115$.

Worked Example. Five samples, each of size $n=5$, were drawn from a production process; the mean and range of each sample were:

Using $n=5$ factors ($A_2=0.577$, $D_3=0$, $D_4=2.115$):

$\bar{X}$ chart:

$R$ chart:

$UCL = (2.115)(5) = 10.58 \qquad LCL = (0)(5) = 0$ …