Methods and formulas for Z-MR Chart

Z chart

Each data point, z_i, in a Z chart

MR chart

Each data point, R_i, the moving range of the z values in each group. R_i is not plotted for i < w because it is undefined.

Notation

Minitab provides four methods for estimating σ, the process standard deviations. You should choose an estimation method based on the properties of your particular process/product. You can also choose to enter a historical value. You need to make assumptions about the process variation.

Use the following information to help you choose a method:

Constant (pool all data)

This option pools all the data across runs and parts to obtain a common estimate of σ.

Use this option when all the output from your process has the same variance—regardless of the size of the measurement.

Relative to size (pool all data, use log (data))

This option takes the natural log of the data, pools the transformed data across all runs and all parts, and obtains a common estimate of σ for the transformed data. The natural log transformation stabilizes the variation in cases where variation increases as the size of the measurement increases.

Use this option when the variance increases in a fairly constant manner as the size of the measurement increases.

By Parts (pool all runs of same part/batch)

This option combines all runs of the same part to estimate σ for that part.

Use this option when all runs of a particular part or product have the same variance.

By Runs (no pooling)

This option estimates σ for each run independently.

Use this option when you cannot assume all runs of a particular part or product have the same variance.

Z-MR Chart estimates the mean for each different part or product separately. Z-MR Chart pools all the data for a common part, and obtains the average of the pooled data. The result is the estimate of μ for that part. The part name data define the groupings for estimating the process means. When you use the Relative to size (combine all observations, use ln) option for estimating σ, the means are also taken on the natural log of the data.

You can also center data using historical values. Using historical means allows you to compare your process with past performance. When you use known means to center the data, the chart reflects whether the process is functioning as it has in the past with respect to location. That is, the chart shows whether each part/product mean is the same as previously established. When the process is functioning the same in respect to location, the values are distributed (equally) around the center line.

You can also use nominal specifications—target values—for each part/product to center the data. Using nominal specifications to center the data allows you to compare your process to desired performance. The nominal specifications are established target values for the dimension of interest for each part/product. When you use nominal specifications to center the data, the chart reflects whether the process is producing parts/products that are on target, or whether the process is biased.

Z chart

Center line

The center line represents the process average. For the Z chart, the center line is always located at 0 because the data are standardized.

Lower control limit (LCL)

The lower control limit is always −3 because the data are standardized.

Upper control limit (UCL)

The upper control limit is always 3 because the data are standardized.

MR chart

Center line

The center line represents the moving range average. For the default average moving range estimation method, the center line is always 1.128 because the data are standardized. For the median moving range estimation method, the center line is always 0.954.

Lower control limit (LCL)

The lower control limit is always 0 because the data are standardized.

Upper control limit (UCL)

For the default average moving range estimation method, the upper control limit is always 3.686 because the data are standardized. For the median moving range estimation method, the upper control limit is always the median moving range 0.954 multiplied by D4 (3.26729), which equals 3.12.