TECHNOLOGY OVERVIEW

Exponentially Weighted Moving Average (EWMA) and Range/Sd Control Charts

The BIS.Net Team BIS.Net Team

The EWMA Chart

The EWMA chart has the same application that the moving average chart has. It is 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.

Whereas the Moving average has equal weights the EWMA weights decrease r so that the most recent samples are weighted most while the most distant samples contribute very little.

The points on the EWMA chart are obtained with the following formula.

E= alpha* subgroup average + (1- alpha) * Previous E

Where alpha is the smoothing constant which can take values between 0 and 1. If 0 the value will be equal to the first value for all values. If 1 then there is no averaging. A value of .2 to .25 is a common value used.

For Individuals the sub group average is replaced with the individual measurement.

Control limits are placed around an estimate of the process mean or target according to the following formula.

LCL= Process Mean/Target – 3 * Sigma* Sqrt((1 - (1 - Alpha) ^ (2 * i)) * Alpha / (2 - Alpha))

UCL= Process Mean/Target + 3 * 3 * Sigma* Sqrt((1 - (1 - Alpha) ^ (2 * i)) * Alpha / (2 - Alpha))

I is the sample number in the chronological order , i.e. 1, 2, 3, …n

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/Target – 2.66*(Average Moving Range/sqrt(period)) * Sqrt((1 - (1 - Alpha) ^ (2 * i)) * Alpha / (2 - Alpha))

UCL= Process Mean/Target + 2.66*(Average Moving Range/sqrt(period))* Sqrt((1 - (1 - Alpha) ^ (2 * i)) * Alpha / (2 - Alpha))

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/Target – 3*(Average Sub-group Standard deviation/(C4*sqrt(period))) * Sqrt((1 - (1 - Alpha) ^ (2 * i)) * Alpha / (2 - Alpha))

UCL= Process Mean/Target + 3*(Average Sub-group Standard deviation/(C4*sqrt(period))) * Sqrt((1 - (1 - Alpha) ^ (2 * i)) * Alpha / (2 - Alpha))

Where C4 is an unbiasing factor

Please note that EWMA Chart is autocorrelated and hence run and trend tests cannot be applied

Setting alpha is usually arbitrary. 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

Both the Moving Average and EWMA chart lag behind when a change in the process mean occurs.

Both the EWMA and Moving Average Chart cannot be used to detect assignable causes of variation other than those that cause a shift in the process mean

There are alternative charts that can be used, which are often preferred. One is the Cusum Chart

The R/ Sd Chart

As with the Xbar and I chart a Range or Sd chart are usually plotted to control variability.

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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