u-chart

u-chart
Originally proposed by	Walter A. Shewhart
Process observations
Rational subgroup size	n > 1
Measurement type	Number of nonconformances per unit
Quality characteristic type	Attributes data
Underlying distribution	Poisson distribution
Performance
Size of shift to detect	≥ 1.5σ
Process variation chart
Not applicable
Process mean chart
Center line	${\displaystyle {\bar {u}}={\frac {\sum _{i=1}^{m}\sum _{j=1}^{n}{\mbox{no. of defects for }}x_{ij}}{mn}}}$
Control limits	${\displaystyle {\bar {u}}\pm 3{\sqrt {\frac {\bar {u}}{n_{i}}}}}$
Plotted statistic	${\displaystyle {\bar {u}}_{i}={\frac {\sum _{j=1}^{n}{\mbox{no. of defects for }}x_{ij}}{n}}}$

In statistical quality control, the u-chart is a type of control chart used to monitor "count"-type data where the sample size is greater than one, typically the average number of nonconformities per unit.

The u-chart differs from the c-chart in that it accounts for the possibility that the number or size of inspection units for which nonconformities are to be counted may vary. Larger samples may be an economic necessity or may be necessary to increase the area of opportunity in order to track very low nonconformity levels.^[1]

Examples of processes suitable for monitoring with a u-chart include:

• Monitoring the number of nonconformities per lot of raw material received where the lot size varies
• Monitoring the number of new infections in a hospital per day
• Monitoring the number of accidents for delivery trucks per day

As with the c-chart, the Poisson distribution is the basis for the chart and requires the same assumptions.

The control limits for this chart type are ${\displaystyle {\bar {u}}\pm 3{\sqrt {\frac {\bar {u}}{n_{i}}}}}$ where ${\displaystyle {\bar {u}}}$ is the estimate of the long-term process mean established during control-chart setup. The observations ${\displaystyle u_{i}={\frac {x_{i}}{n_{i}}}}$ are plotted against these control limits, where x_i is the number of nonconformities for the ith subgroup and n_i is the number of inspection units in the ith subgroup.

• c-chart