Short Run Transformations

This topic discusses how the data is transformed for short run charts (see also Montgomery, 1996; Bhote, 1988). The transformations are described for the different types of  variables (X-bar and R, MR, and S, and MA and EWMA charts) charts and attribute (C, U, Np, and p) charts.

Note: the Statistica S chart differs from the Montgomery formulas because in Statistica, C4 is calculated more precisely than by Montgomery, who used lookup tables.

Transformations for Variable Control Charts. The transformations of the data for short run variable control charts use the general equation:

x'i,j,k = ( Xi,j,k - tk)/vk          

For each case i in sample j, for part k, the raw measurement xi,j,k is transformed to x'i,j,k by first subtracting a part-specific constant tk, and then dividing by the part-specific constant vk. For the default target X-bar short run chart, tk is the part mean, and vk is equal to1 for all parts. For short run X-bar and R (standardized) charts, vk is equal to the average of the sample ranges for the respective part. For short run X-bar and S (standardized) charts, vk is equal to the average of the sample standard deviations for the respective part. For short run X-bar and S2 charts, vk is equal to the square root of the pooled within-sample mean square (of deviations from the sample means; this is an estimate of k). Values of both tk and vk can be replaced by user-defined values. The sample means (for X-bar charts), and the sample ranges or standard deviations (for R and S charts, respectively) that are plotted in the requested control charts are computed from the transformed data (for aggregated data the corresponding computations are performed directly on the sample means, ranges, standard deviations, etc.).   

Short run moving range charts
When short run charts for variables are requested with n = 1 for each sample j in each part k, a short run moving range chart is produced. For this chart, the individual observations are first transformed within parts, according to the general equation described above. Then the moving ranges for each sample j (except for the first sample) are computed from the transformed data.  

The moving ranges are computed using the formula:

rj = abs(mj - mj-1 )

where rj  is the moving range for each sample j (j>1), and mj is the transformed mean for each sample j  

Transformations for Attribute Control Charts.

Short run C charts
The transformations of the number of defects found in each sample for short run C charts use the equation:

c' j,k  = (cj,k - tk )/tk1/2

For each sample j and part k, the plot points c' j,k for the short run C chart are computed by standardizing the deviations of the observed sample Poisson frequencies from a target frequency tk. By default, tk is equal to the average Poisson frequency for each respective part k (see also the Parts tab on the Results dialog).

Short run U charts
The transformations of the rate of defects found in each sample for short run U charts use the equation:

u' j,k = u j,k - tk /(tk /nj)1/2

For each sample j and part k, the plot points u' j,k for the short run U chart are computed by standardizing the deviations of the observed sample Poisson rates from a target rate tk. By default, tk is equal to the Poisson rate for each respective part k (see also the Parts tab on the Results dialog).

Short run Np charts
The transformations of the binomial frequencies found in each sample for short run Np charts use the equation:

np'j,k = npj,k - nj * tk /(nj * tk (1 - tk )) 1/2

For each sample j and part k, the plot points np'j,k for the short run Np chart are computed by standardizing the deviations of the observed binomial frequencies (np'j,k) from a target binomial frequency njtk; By default, the binomial proportion tk is equal to the binomial proportion for each respective part k (see also the Parts tab on the Results dialog).

Short run P charts
The transformations of the binomial proportions found in each sample for short run P charts use the equation:

p'j,k = pj,k - tk /(tk * (1 - tk )/nj) 1/2

For each sample j and part k, the plot points p'j,k for the short run P chart are computed by standardizing the deviations of the observed sample binomial proportions from a target proportion tk.  By default, tk is equal to the binomial proportion for each respective part k (see also the Parts tab on the Results dialog).

For a description of the computations involved in standard quality control charts, refer to the Computational Details for Quality Control Charts topic.