X̄–R control charts

Paired charts of subgroup means and subgroup ranges with 3-sigma limits from the within-subgroup variation, used to tell routine process variation from special causes.

\text{X-bar chart: } \bar{\bar x} \pm A_2\bar R, \qquad \text{R chart: } [D_3\bar R, \; D_4\bar R], \qquad \hat\sigma = \frac{\bar R}{d_2}

Centre lines and control limits

$\bar{\bar x}$: grand mean of the subgroup means
$\bar R$: average subgroup range (max − min within each subgroup)
$d_2, d_3$: mean and sd of the range of $n$ standard normal values (computed by the engine for subgroup size $n$ )
$A_2 = \frac{3}{d_2\sqrt n}$: factor giving 3-sigma limits for the means
$D_3, D_4$: $max(0, 1 - 3d_3/d_2)$ and $1 + 3d_3/d_2$

When to use

Monitoring a measured characteristic (diameter, fill weight, time) sampled in small rational subgroups (typically 2–10 items made close together). Equal subgroup sizes of 2–25.
Phase I: establish limits from historical data and remove assignable causes; Phase II: monitor new data against those limits.

Assumptions

Subgroups are rational: variation within a subgroup reflects only common-cause noise.
Measurements are independent over time and roughly normal (the limits are fairly robust for means).
Read the R chart first: the X̄ limits are only meaningful if the spread is in control.

How to interpret

A point beyond a limit, or a non-random pattern (see Western Electric rules), signals a likely special cause to investigate.
Control limits describe what the process does; specification limits describe what the customer needs. A process can be in control and still incapable (process capability).
A negative lower limit for the range is set to 0.

Common mistakes

Drawing specification limits on the X̄ chart as if they were control limits.
Computing limits from the overall standard deviation of all data (inflated by shifts between subgroups).
Adjusting the process after every point inside the limits (tampering increases variation).

Example

Five subgroups of 4: (10.1, 9.8, 10.0, 10.3), (9.9, 10.2, 10.1, 9.7), (10.4, 10.0, 9.9, 10.2), (10.0, 9.6, 10.1, 10.0), (10.2, 10.3, 9.8, 10.1). Grand mean 10.035, $\bar R$ = 0.5; for $n = 4$ : $d_2 = 2.059$ , $A_2 = 0.729$ , $D_4 = 2.282$ , $D_3 = 0$ . X̄ limits 9.671 to 10.399; R limits 0 to 1.141; $\hat\sigma = 0.243$ . No point is out of control.