TECHNOLOGY OVERVIEW

# X-bar and S (Standard Deviation) Control Charts

#### The X-bar Chart

Because it is so easy to calculate and understand the range, over the standard deviation, the X-bar and Range chart is often preferred over the X-Bar and S control chart. However, with modern computing power and better trained quality practitioners one can argue that these reasons no longer apply and that the X-bar and S chart should be the preferred option. Practically in most instances the differences are negligible, and the chosen chart is one of personal preference. However, if the sub group size is variable then the X-bar S chart is the only one applicable.

To set up control limits for the X-bar chart an estimate of the process standard deviation sigma is required if sigma is not known (as is usually the case). There are two ways of estimating sigma. One is averaging the subgroup standard deviations and the other is using weighted standard deviations if the subgroup size varies. BISNET Analyst uses both as required. The standard deviation is then divided by an unbiasing factor of c4 which is dependent on subgroup size only, to compensate for bias.

The X-bar chart control limits themselves are based on the standard error of the data which is equal to sigma/sqrt(sub group size n[i] for sample i), where sigma if unknown is replaced with the estimate above.

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. To ensure there is a small probability a control chart factor of 3 is used.

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

Hence control limits are

LCL= Estimate of population average (or target) - 3 * estimate of sigma /sqrt(n[i]) where the estimate of sigma was obtained as described above and n is the subgroup size of the ith sample

UCL= Estimate of population average (or target) - 3 * estimate of sigma /sqrt(n[i]) where the estimate of sigma was obtained as described above.

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

#### The S Chart

Control limits are placed around an estimate of sigma, the process standard deviation, as explained above.

Control limits are set to

LCL= estimate of sigma*(1-3*sqrt(1-c4^2)/c4)

UCL= estimate of sigma*(1_3*sqrt(1-c4^2)/c4)

Where c4 is an unbiasing factor depended on the sub group size

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