 # X-Bar and R Chart

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Definition

A statistical process control chart for use with continuous data collected in subgroups at set time intervals - usually between 3 to 5 pieces per subgroup. The Mean (X-Bar) of each subgroup is charted on the top graph and the Range (R) of the subgroup is charted on the bottom graph. Out of Control points or patterns can occur on either the X-bar or R chart.

Examples A cereal manufacturer was concerned that the packaging line was overfilling the 8.0 oz. boxes and decided to investigate. Data were collected over twenty hours of operation consisting of a subgroup of 5 observations per hour and an X-bar and R chart was created.

The chart shows that the process is stable, but the process mean is well above the target of 8 oz., the reasons for which need further investigation.

Application

The procedure to construct the chart is as follows :

1. Calculate the average (X-bar) and range (R) for each subgroup.

2. Calculate the control limits for the R chart: the average range, R-bar, is the centerline. The upper and lower limits for the R chart are calculated as:
UCL(R) = R-bar x D4
LCL(R) = R-bar x D3
Where D3 and D4 are SPC constants to get the 3 sigma control limits corresponding to the given subgroup size.

3. Calculate the control limits for the X-bar chart: the average of all the subgroup averages, or X-bar-bar, is the centerline. The upper and lower limits for the X-bar chart are calculated as:
UCL (X-bar) = X-bar-bar + (A2 x R-bar)
LCL(X-bar) = X-bar-bar - (A2 x R-bar)
Again, A2 is the SPC constant to get the 3 sigma control limits for the X-bar chart corresponding to the given subgroup size.

4. Plot the X-bar and R values for each subgroup on their respective charts. Evaluate the stability of the Range Chart before drawing any conclusions about the Averages (X-bar) Chart - if the Range Chart is out of control, the control limits on the Averages Chart will be unreliable.

Note: If the subgroup size exceeds 10, the range chart is replaced by a chart of the subgroup standard deviation, or S chart.