TECHNOLOGY OVERVIEW
# X-bar and Range Control Charts

#### The Xbar Chart

#### The R Chart

Because it is so easy to calculate the range, the X-bar and Range chart is often preferred over the XBar and S control chart, However, it is recommended that the subgroup size is less than 10.

To set up control limits for the X-bar chart an estimate of the process standard deviation is required. The estimate of the process standard deviation is equal to the average range divided by a factor called d2. The factor of d2 depends on the sample size and is obtained from tables.

The X-bar chart is based on the standard error of the data which is equal to estimate of sigma/sqrt(sub group size n). Control limits are calculated such that there is a small probability of points falling outside these limits. If a point falls outside the limits it is assumed that the event is of such small probability that we can reasonably assume it is not a false alarm, but due to an assignable cause.

Control limits are then placed around an estimate of the population mean or target. The estimate is usually obtained from the sample mean.

Hence control limits are

LCL= Estimate of population average (or target) - 3 *R-bar*d2/sqrt(n)

UCL= Estimate of population average (or target) +3 *R-bar*d2/sqrt(n)

Warning limits: Shewhart charts are often supplemented with warning limits. Warning limits are calculated as above, after replacing 3 with 2. A warning is signalled when one point falls outside warning limits but inside control limits. An assignable cause is signalled if two consecutive points fall outside the same warning limit.

Control limits are placed around an estimate of the average range.

The standard deviation of the range is equal to 3*average range *d3/d2

i.e. 3*R-bar*d3/d2

Control limits are therefore set to

LCL= R-bar- 3 *R-bar*d3/d2 =R-bar*(1- 3 *d3/d2)

UCL= R-bar+3 *R-bar*d3/d2) = R-bar*(1+3*d3/d2)

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