9.4 Short-Run SPC and Moving Average Charts

The short-run problem

A conventional X-bar and R chart needs roughly 20 to 25 subgroups to establish trustworthy limits. A job shop that runs 30 pieces of a part number and then changes over never accumulates them, and it may never run that part again. Yet the process -- the machine, the operator, the method -- is continuous even though the parts are not.

Short-run SPC solves this by charting the process rather than the part, transforming each measurement onto a common scale so that different part numbers can be plotted on one chart.

The DNOM chart

Deviation from nominal subtracts each part's own target from its measurement:

$x_{DNOM} = x_{measured} - \text{Target}_{part}$

All part numbers then plot around a common centre line of zero, and standard X-bar and R limits apply to the transformed values.

Worked example. Three part numbers on one lathe:

All five points plot on one chart around zero, and the chart now monitors the lathe's ability to hold nominal regardless of what it is making.

Requirement: the parts must share approximately the same process standard deviation. If part C-315 is turned to a much tighter tolerance with a different feed and speed, its variation will differ from A-100's and pooling them on one DNOM chart produces limits that are wrong for both.

The standardized Z chart

When the parts do not share a common standard deviation, standardize as well as centre:

$Z = \frac{x - \text{Target}_{part}}{\hat{\sigma}_{part}}$

Now every point is expressed in the part's own standard deviations, so the chart has a centre line of 0 and limits at $\pm 3$ regardless of the part or even the unit of measurement. A diameter in millimetres and a hardness in HRC can share one chart.

The Z chart's cost is that it needs a prior estimate of sigma for every part number. For a brand-new part with no history, an engineering estimate or a pooled estimate from similar parts is used, and the chart is provisional until data accumulates.

Related short-run tools include the target X-bar and R chart (target lines drawn per part on a common chart) and short-run attribute charts that standardize proportions the same way.

Why Shewhart charts are slow to small shifts

A Shewhart chart makes its decision from the current subgroup only. Its average run length to detect a shift of size $k\sigma$ in the subgroup mean is roughly:

A sustained half-sigma shift takes on average 155 subgroups to trigger a point beyond a control limit -- weeks or months of production. Run rules (such as eight points on one side of the centre line) improve this considerably, but the fundamental limitation remains: a chart that looks only at the newest point throws away the information in the previous ones.

Moving average and EWMA charts

Simple moving average

Plots the average of the last $w$ observations:

$MA_t = \frac{x_t + x_{t-1} + \cdots + x_{t-w+1}}{w}, \qquad \sigma_{MA} = \frac{\sigma}{\sqrt{w}}$

Averaging suppresses noise, so small sustained shifts become visible. The costs are that consecutive points are correlated (they share observations), which makes run rules invalid, and that the chart lags a real change by up to $w$ periods.

Exponentially weighted moving average

EWMA weights all past observations, with weights decaying geometrically:

$z_t = \lambda x_t + (1 - \lambda) z_{t-1}, \qquad z_0 = \mu_0$

with control limits

$\mu_0 \pm L\sigma\sqrt{\frac{\lambda}{2 - \lambda}\left[1 - (1-\lambda)^{2t}\right]}$

which widen from a narrow start and converge to a steady-state value.

Typical settings are $\lambda = 0.2$ with $L = 3$, or $\lambda = 0.1$ with $L = 2.7$. The trade-off is explicit: small $\lambda$ means more memory, so small shifts are found faster and large shifts more slowly.

CUSUM

The cumulative sum chart accumulates deviations from target:

$C_i^{+} = \max\left[0,\; x_i - (\mu_0 + K) + C_{i-1}^{+}\right], \qquad C_i^{-} = \max\left[0,\; (\mu_0 - K) - x_i + C_{i-1}^{-}\right]$

with a signal when either statistic exceeds the decision interval $H$. Conventional settings are $K = 0.5\sigma$ (half the shift to be detected) and $H = 5\sigma$.

CUSUM and EWMA have similar performance and are chosen largely by preference; EWMA is easier to plot and explain, CUSUM has a slightly more direct interpretation of the shift magnitude.

Choosing a chart

A practical combination is a Shewhart chart plus an EWMA on the same data: the Shewhart chart catches large sudden shifts and supports run rules, and the EWMA catches the slow drift the Shewhart chart would take months to find.