TECHNOLOGY OVERVIEW
# Moving Average and Range/Sd Control Charts

#### The Moving Average Chart

#### The R/ Sd Chart

The moving average chart is used as an alternative to the Shewhart X-Bar or Individuals chart when it is important to detect relatively small changes in the process mean. The first two are more appropriate for detecting large changes, or the presence of assignable or special cause variation.

The points on a moving average chart are determined by calculating successive averages for a specified period, as shown below

**Table - Period 3**

Measurement | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Value | 2 | 3 | 1 | 4 | 6 | 0 | 3 | 0 | 1 | 3 |

Moving Average | 2/1 | 5/2 | 6/3 | 8/3 | 11/3 | 10/3 | 9/3 | 3/3 | 4/3 | 4/3 |

Example: Point 4 (3+1+4)=8/3

Control limits are placed around an estimate of the process mean or target.

i.e.

LCL= Process Mean or Target – 3 * Sigma/sqrt(period)

LCL= Process Mean or Target + 3 * Sigma/sqrt(period)

When the measurement number is less than period, period is replaced by the measurement number.

Sigma is the process standard deviation and unless available is replaced with an estimate.

BIS.Net Analyst uses the moving range estimate when the moving average is based on individual measurements i.e. the sample size for each measurement equals 1.

Hence

LCL= Process Mean or Target – 2.66*Average Moving Range/sqrt(period)

UCL= Process Mean or Target + 2.66*Average Moving Range/sqrt(period)

The value of 2.66 is derived from the control chart factor (3), an anti-biasing factor and the moving period of 2

When the sample size of each measurement is more than 1 BISNET Analysis uses the standard deviation method, based on the average or weighted average of the subgroup standard deviations, to estimate the process standard deviation

Hence

LCL= Process Mean or Target – 3*Average Sub-group Standard Deviation/(C4*sqrt(period))

UCL= Process Mean or Target + 3*Average Sub-group Standard Deviation/(C4*sqrt(period))

Where C4 is an unbiasing factor

Please note that the Moving Average Chart is autocorrelated and hence run and trend tests cannot be applied

Setting the period is usually arbitrary, with a value of 3 or 6 commonly used. It is also possible to specify an amount of change that needs to be detected and the average number of points required to detect this.

The underlying data is assumed to be reasonably normal

There are alternative charts that can be used, which are often preferred. These are the Cusum and Exponentially Weighted Moving Average Charts

For Individuals data a moving range chart of 2 is plotted with the following Control Limits

UCL=3.267*Average Moving Range

LCL=0

The factor is derived from the control chart factor 3 and a constant obtained from readily available tables.

When the sub-group size is greater than 1 control limits for the Sd chart 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 and estimate of sigma is based on the average of the sub group standard deviations or a weighted estimate if the sub-group size varies

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