# How To Calculate NDC

If you are attempting a statistical analysis of data, you need more than just the assortment of numbers generated by whatever collection process you used. You also need to be sure of the trustworthiness of the collection process itself. In other words, if someone told you that a neighborhood bakery's cakes varied in quality by 15 percent from one batch to the next, you would have to know if the measurements used to determine this quality were themselves of sufficient quality. What if the cakes are all more or less the same across batches and it is actually the quality-assessment system that shows real variation from one data set to the next?

Such concerns lie at the heart of measurement system analysis, or MSA. The concept of **number of distinct categories**, or NDC, in MSA is an important way to keep track of the means by which you evaluate the quality of your data acquisition, and it is derived from Gage R&R. These statistical tools are very useful in situations where a large number of items are being produced and they are, in theory, identical (e.g., a kind of automotive part that goes into one type of vehicle but is manufactured at the level of thousands per year).

## MSA Explained

An MSA calculation explores how much variation in a measurement results from the measurement tools, measuring process, work environment, the people doing the measuring and other factors outside the item actually being studied. Returning to the example about cakes, you would want to know how much of the reported variation in their quality was the result of variation in the perception of their quality. Were they in fact "too sweet" last week compared to six months ago, or could this be the result of how people taste things in the winter versus the summer?

The idea behind invoking MSA is to use the results to refine a production process and eliminate errors. It is a relatively sophisticated aspect of quality control. Most, including the Gage R&R and the NDC information it produces, are done not by hand but by using statistics software packages.

## The Gage R&R

The "R&R" part of "Gage R&R" stands for "reliability and reproducibility." Reliability refers to the ability of a single operator (often a person) to get the same result over and over; reproducibility refers to the measurements of multiple operators falling within as tight a numerical cluster as possible.

This type of MSA involves up to three **operators** (that is, measurement tools), five to 10 **parts** or **items**, and up to three **repeat measurements**. These analyses are structured so that each distinct part is handled individually by every operator, and that measurements from each part-operator pairing are repeated at least once.

The Gage R&R measures only the variability in measurements. Note that this says nothing about the accuracy of measurements, which can only be assured through calibration. A favorable reproducibility calculation is useless if the data itself is suspect.

## The NDC Calculation

When you run a Gage R&R on your software program, the results will include an NDC. It is useful, however, to understand where this number comes from.

The formula is:

\(NDC=\sqrt{2}\frac{\sigma_{part}}{\sigma_{gage}}=1.41\frac{\sigma_{part}}{\sigma_{gage}}\)

Here, σ_{part} represents the square root of variance of the part component of the Gage R&R, while σ_{gage} represents the square root of variance of the entire Gage R&R analysis. An NDC value of 5 or greater is considered desirable. Less than 2 is too few because there is nothing to make comparisons between; values of 2 and 3 can be used to create "more/less" and "low/middle/high" categories but are suboptimal.

### References

- Purdue University: Gage R&R
- The Minitab Blog: Understanding "Number of Distinct Categories" in Your Gage R&R Output
- Lean Manufacturing and Six Sigma Definitions: Gage R&R
- Western Kentucky University: The Application of Gage R&R Analysis in a Six Sigma Case of Improving and Optimizing an Automotive Die Casting Product's Measurement System

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