Constructing and interpreting X-bar and R charts

Use of a control chart

A control chart is a plot of a sample statistic against time with a centre line and upper and lower control limits (UCL, LCL) at $\pm 3\sigma$. Points inside the limits show only chance variation; a point outside (or a trend) shows an assignable cause that has to be found and removed. $\bar X$ chart monitors the process average and $R$ chart monitors the spread.

Calculation (all ten samples)

$$\bar{\bar X} = \frac{500.21}{10} = 50.021\ \text{mm}, \qquad \bar R = \frac{0.65}{10} = 0.065\ \text{mm}$$

$\bar X$ chart:

$$UCL = \bar{\bar X} + A_2\bar R = 50.021 + 0.577(0.065) = 50.0585$$ $$LCL = 50.021 - 0.0375 = 49.9835$$

$R$ chart:

$$UCL_R = D_4\bar R = 2.114(0.065) = 0.1374, \qquad LCL_R = D_3\bar R = 0$$

Interpretation

All ranges (0.04 to 0.09) are below 0.1374, so the variation within samples is stable. In the $\bar X$ chart, sample 8 has $\bar X = 50.12 > 50.0585$ and is out of control; all others lie between 49.9835 and 50.0585.

Revised limits (without sample 8)

$$\bar{\bar X} = \frac{450.09}{9} = 50.010, \qquad \bar R = \frac{0.58}{9} = 0.0644$$ $$UCL = 50.010 + 0.577(0.0644) = 50.0472, \qquad LCL = 50.010 - 0.0372 = 49.9728$$ $$UCL_R = 2.114(0.0644) = 0.1362, \qquad LCL_R = 0$$

The remaining nine samples (49.99 to 50.04) lie within the revised limits, so the process is now in statistical control. The estimated process standard deviation is $\sigma = \bar R/d_2 = 0.0644/2.326 = 0.0277$ mm.

Answer: initial $\bar X$ limits 49.9835 to 50.0585 mm, $R$ limit 0 to 0.1374 mm (sample 8 out); revised $\bar X$ limits 49.9728 to 50.0472 mm, $R$ limit 0 to 0.1362 mm.