Control Charts

Run Chart (runs tests)

Median plus four runs tests (clustering, mixtures, trends, oscillation) with p-values from a normal approximation.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Quality Tools > Run Chart

Description

Run-chart engine for pre-assessment of a data series before setting up a control chart. Computes the median, the count of runs about the median (points equal to the median are excluded) and the count of runs up/down. From the Wald-Wolfowitz expectation and standard deviation it produces four z-statistics and p-values: clustering (too few median runs → level shift), mixtures (too many → mixed sources), trends (too few up/down runs → drift) and oscillation (too many → zigzag/overshoot).

Formulas

Runs about median

E [R] = \frac{2 m _{1} m _{2}}{N} + 1, Var [R] = \frac{2 m _{1} m _{2} ( 2 m _{1} m _{2} - N )}{N ^{2} ( N - 1 )}, z = \frac{R _{o b s} - E [ R ]}{Var [ R ]}

m₁/m₂ = points above/below median; N = m₁+m₂

Runs up or down

E [R] = \frac{2 n - 1}{3}, Var [R] = \frac{16 n - 29}{90}

n = number of points (after removing ties)

p-Werte

p_{cluster/trend} = Φ (z), p_{mix/oszi} = 1 - Φ (z)

Φ is the standard normal CDF (Abramowitz/Stegun 7.1.26)

Assumptions

Data order matches collection order — sorted data invalidates the tests
Values equal to the median are excluded from the tests (Minitab convention)
Normal approximation needs ~10 points or more for reliable p-values

Limitations

Four simultaneous tests inflate the false-alarm rate; single p < 0.05 is weak evidence
No control limits — run chart does not replace a control chart for ongoing monitoring

References

Bradley, J. V. (1968), Distribution-Free Statistical Tests, Prentice-Hall
Wald, A. & Wolfowitz, J. (1940), On a Test Whether Two Samples Are From the Same Population, Annals of Math. Stat.
Minitab — Run chart documentation