ARTICLES Process performance analysis is used to see how good or bad your current process is performing in relation to non-conformances. Such knowledge can lead to dramatic quality improvements.

Process performance analysis has traditionally been applied in manufacturing but can be equally applied to other areas, such as health and wellbeing. For example, the human body is a process and has performance variables such as blood pressure and blood glucose levels which must fall within medically set specification limits.

Process performance analysis requires that a distribution curve is fitted so that process performance and capability indexes can more accurately be calculated. The usual assumption is that the distribution follows a Normal (bell-shaped) curve. Unfortunately, this assumption is a bad practice because many processes do not follow a normal distribution, which results in erroneous indexes and wrong decisions.

There are many theoretical distribution curves that can be fitted to the data. If you know that a distribution from the list below applies to your measurements, then you should select the distribution from the distribution list. Otherwise the automatic option is recommended.

Table I

Description
Log Normal The log-normal distribution is one where the logarithms of the values follows a Normal distribution. The distribution often applies in the description of natural phenomena, for example body weight.
Rayleigh A Rayleigh distribution can arise when the total magnitude of a vector is related to its directional components and each component is independent and normally distributed. One application is when analysing wind speed in its two orthogonal vector components
Exponential The exponential distribution is the probability distribution that describes the time between events in a Poisson process. Examples include the time between telephone calls, time between accidents, etc.
Gamma Applications for the gamma distribution include modelling insurance claims, rainfall and multi-path feeding of signal power.
Logistic The logistics distribution is similar to the Normal distribution, but with heavier tails. A common application is in logistics regression.
Box-Cox The Box-Cox distribution is a family of functions that are applied to create a power transformation to make data more normal, allowing performance and capability statistics to be determined.
Johnson SU The Johnson SU distribution is a family of a wide range of unbounded distributions. Its parameters are used in a transformation function to make data more normal, allowing performance and capability statistics to be determined.
Johnson SU The Johnson SB distribution is a family of a wide range bounded distributions. Its parameters are used in a transformation function to make data more normal, allowing performance and capability statistics to be determined.
Truncated Normal The truncated normal distribution is a Normal distribution which has been truncated at either the lower tail end, upper tail end, or both. Truncation means that data above, or below the truncation point has been removed, such as through a check weighing process.
Folded Normal The folded normal distribution is a Normal distribution where negative values have been changed to positive values. A half normal distribution is a folded normal distribution where the mean of the unfolded distribution is zero.
Maximum Extreme Value The folded normal distribution is a Normal distribution where negative values have been changed to positive values. A half normal distribution is a folded normal distribution where the mean of the unfolded distribution is zero.
Minimum Extreme Value Normal The folded normal distribution is a Normal distribution where negative values have been changed to positive values. A half normal distribution is a folded normal distribution where the mean of the unfolded distribution is zero.
Distributionless The folded normal distribution is a Normal distribution where negative values have been changed to positive values. A half normal distribution is a folded normal distribution where the mean of the unfolded distribution is zero. FIGURE 1: Histogram used in a process peformance analysis FIGURE 2: Probability plot FIGURE 3: Process Performance Analysis - Summary Report

Your report will show a histogram to show how your measurements are distributed. The probability plot can be used to see how well the fitted curve superimposed on top of the histogram fits the data. However, a visual inspection of the curve is equally effective.

The fitted curve is affected by sample size. If the sample size is relatively small different samplings may result in different curves. It is thus important that sample sizes are not too small. At least 100 is recommended.

The adjusted (for sample size) Anderson Darling Statistic quantifies the fit. The smaller the better. As a rule of thumb a value of less than .6 should be sought. However, if the fit appears reasonable a higher value may be acceptable. The value of 1-p is the probability of the data not fitting the distribution. Thus, if p=.05 then there is a .95, or 95% chance that the data does not fit the chosen distribution. The higher the value above .05 the better.

The automatic fitting option fits the distribution with the lowest Anderson Darling Statistic for that data. A different sample may result in a different chosen curve.

You will see several sample statistics displayed. Most are self-explanatory. The lower and upper limits are 95% confidence intervals within which the true population parameters are expected to lie in. For proportions confidence intervals are based on the score method instead of the Wald method, as used by other software. The score method has been shown to provide a more reliable interval than the Wald based interval, especially for smaller sample sizes.

Observed % non-conformance is the actual percent of non-conformance obtained by counting the non-conforming measurements. The expected value is the theoretical percentage assuming the fitted distribution. There are advantages and disadvantages with both estimates. The expected values are based on ‘all’ data, but if the fitted distribution is wrong then actual estimates will be more accurate for the snap shot.

Pp and Ppk are capability indexes based on the fitted distribution and hence are only as good as the fit. The Pp value assumes your process is centred such that 50% of the non-conformance measurements fall above the upper specification and 50% fall below the lower specification limit. For a normal distribution, this is where the average is centred between the specification limits. An upper and lower specification limit is needed for this index

The Pp value should at least be equal to 1. Equal to 1 means that the process is producing .27% of defectives. Less than one means the process is producing more non-conforming product. Greater than 1 means the process is producing less than .27% non-conformances.

The two images below show a non-normal and normal process, respectively, with a Pp of 1. These images demonstrate that only symmetrical distributions, such as a normal distribution, are centred on the mean for Pp calculations. FIGURE 4: Non-Normal Distributions FIGURE 5: Normal Distributions

The Pp index is calculated by (US-LS)/(Upper Quantile – Lower Quantile). The lower quantile is equal to the value where for a 3-sigma process approx. 0.135% of defectives fall below that value. For a normal distribution, this is equal to the average-3*Sd. The upper quantile is equal to the value where for a 3-sigma process approx. 0.135% of defectives fall above that value. For a normal distribution, this is equal to average+3*Sd.

In the example below the upper specification limit will be used in the calculations. FIGURE 6: Histogram with upper limit used in the calculations

If the Lower Specification limit is used then BIS.Net Analyst will obtain the Ppk using this formula: Ppk=(Median-Lower Specification)/(Median-Lower Quantile) where for a 3-sigma process the lower quantile is the value that has approx. 0.135% of non-conformance below the value. For a normal distribution process, this is equal to (Average-Lower Specification)/(3*Sd).

If the Upper Specification limit is used then BIS.Net Analyst will obtain the Ppk using this formula: Ppk=(Upper Specification Limit-Median)/(Upper Quantile -Median) where for a 3-sigma process the upper quantile is the value that has approx. 0.135% of non-conformance above the value. For a normal distribution process, this is equal to (Upper Specification-Average)/(3*Sd).

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