# Analyzing Data bounded at zero

Tools

Intervals & Tests

Hypothesis Test Of Sample Mean Example

Hypothesis Test Of Two Sample Variances Example

Hypothesis Test Of A Standard Deviation Compared To A Standard Value Example

Distributions

Area Under the Standard Normal Curve

Non-Normal Distributions in the Real World

Rayleigh Distribution for True Position

03/13/2008:

I would like to know how to analyze surface finish in your SPC IV Excel software. I am attaching a sample of data. The Upper Specification Limit for the finish is 63.

Cheryl D.

It would appear your data was collected with a rational subgroup size of one. Given that, we will use one of the charts useful for individuals (n=1) data, such as the Individual-X chart or the Moving Average Range chart.

When using an Individual-X / Moving Range chart, the choice of distribution (used to define the Individual-X control limits) is based on the characteristics of the process. In your case, the process is bounded at zero. The Normal distribution, by definition, will place the Individual-X control limits at plus and minus 3 times the process sigma value from the process mean, without regard to whether your process can logically operate there. (see calculations) Of course, the calculations have no advance knowledge of your process, and in this case assumes your process approximately follows the conditions of a Normal distribution. For some bounded processes this can be a poor assumption, as the process may be skewed. In the case of Surface Finish (or Customer Wait Time, Process Cycle Time, TIR, etc.), where the process is physically bounded at 0, process improvement efforts tend to move the mode and median closer to zero. As they move closer to zero, and further from the average, the Normal distribution will incorrectly predict a significant amount of the process below zero, a practical impossibility. This induces errors in your capability analysis, as well as your estimates of process control.

Since we need to understand the underlying distribution to define meaningful control limits for the Individual-X chart, and we cannot fit a meaningful distribution without verifying that the process is in control, the Individual-X chart is not the best tool at this point in the analysis.

Instead, you could use the Moving Average chart, with subgroup size of one and cell width of 3, to verify the process control. If in control, a distribution may be fit to the data for capability analysis or for use in the Individual-X chart. In your case, there is insufficient data for a complete conclusion, although mathematically we are able to generate this Moving Average control chart for an initial estimate of process control:

As shown in the chart, this initial estimate (based on limited data) suggests the process is in control. Viewing the same data on an Individual-X chart shows that the Normal distribution provides a reasonable estimate for the process distribution (note the K-S value of 0.72 and the reasonable fit of the curve to the histogram), and a Cpk of 1.105 (a marginal value):

Again, this is preliminary information, based on a small set of data (23 data values). For process control purposes, with a moving cell width of 3, we should have 50 or more subgroups to define meaningful control limits. For curve fitting purposes (such as needed for sound Capability estimates), two hundred or so data are recommended.

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