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7. Product and Process Comparisons


Comparisons based on data from one process

Questions answered in this section For a single process, the current state of the process can be compared with a nominal or hypothesized state. This section outlines techniques for answering the following questions from data gathered from a single process:
  1. Do the observations come from a particular distribution?
    1. Chi-Square Goodness-of-Fit test for a continuous or discrete distribution
    2. Kolmogorov- Smirnov test for a continuous distribution
    3. Anderson-Darling and Shapiro-Wilk tests for a continuous distribution
  2. Are the data consistent with the assumed process mean?
    1. Confidence interval approach
    2. Sample sizes required
  3. Are the data consistent with a nominal standard deviation?
    1. Confidence interval approach
    2. Sample sizes required
  4. Does the proportion of defectives meet requirements?
    1. Confidence intervals
    2. Sample sizes required
  5. Does the defect density meet requirements?
  6. What intervals contain a fixed percentage of the data?
    1. Approximate intervals that contain most of the population values
    2. Percentiles
    3. Tolerance intervals
    4. Tolerance intervals based on the smallest and largest observations
General forms of testing These questions are addressed either by an hypothesis test or by a confidence interval.
Parametric vs. non-parametric testing All hypothesis-testing procedures can be broadly described as either parametric or non-parametric/distribution-free. Parametric test procedures are those that:
  1. Involve hypothesis testing of specified parameters (such as "the population mean=50 grams"...).
  2. Require a stringent set of assumptions about the underlying sampling distributions.
When to use nonparametric methods? When do we require non-parametric or distribution-free methods? Here are a few circumstances that may be candidates:
  1. The measurements are only categorical; i.e., they are nominally scaled, or ordinally (in ranks) scaled.
  2. The assumptions underlying the use of parametric methods cannot be met.
  3. The situation at hand requires an investigation of such features as randomness, independence, symmetry, or goodness of fit rather than the testing of hypotheses about specific values of particular population parameters.
Difference between non-parametric and distribution-free Some authors distinguish between non-parametric and distribution-free procedures.

Distribution-free test procedures are broadly defined as:

  1. Those whose test statistic does not depend on the form of the underlying population distribution from which the sample data were drawn, or
  2. Those for which the data are nominally or ordinally scaled.
Nonparametric test procedures are defined as those that are not concerned with the parameters of a distribution.
Advantages of nonparametric methods. Distribution-free or nonparametric methods have several advantages, or benefits:
  1. They may be used on all types of data-categorical data, which are nominally scaled or are in rank form, called ordinally scaled, as well as interval or ratio-scaled data.
  2. For small sample sizes they are easy to apply.
  3. They make fewer and less stringent assumptions than their parametric counterparts.
  4. Depending on the particular procedure they may be almost as powerful as the corresponding parametric procedure when the assumptions of the latter are met, and when this is not the case, they are generally more powerful.
Disadvantages of nonparametric methods Of course there are also disadvantages:
  1. If the assumptions of the parametric methods can be met, it is generally more efficient to use them.
  2. For large sample sizes, data manipulations tend to become more laborious, unless computer software is available.
  3. Often special tables of critical values are needed for the test statistic, and these values cannot always be generated by computer software. On the other hand, the critical values for the parametric tests are readily available and generally easy to incorporate in computer programs.
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