# Distribution results table for Automated Capability Analysis

Find definitions and interpretation guidance for the distribution results that are provided with Automated Capability Analysis.

## ML estimates of distribution parameters

The Maximum Likelihood (ML) method estimates the values of the distribution parameters that maximize the likelihood function for each distribution. The goal is to obtain the best agreement between the distribution model and the observed sample data.

Using the ML method, values are estimated for one or more parameters of each distribution. Each combination of parameters produces a specific distribution curve to fit the data.
Location
This parameter affects the location of a distribution. For example, with different location parameters, a logistic distribution can be shifted along the horizontal axis.
Shape
This parameter affects the shape of the distribution. For example, with different shape parameters, a Weibull distribution can appear more skewed or more symmetric.
Scale
This parameter affects the scale of the distribution. For example, with different scale parameters, a logistic distribution can appear more stretched out or more compressed.
Threshold
This parameter affects the minimum value of a random variable. For example, with different threshold parameters, an exponential distribution can be defined over a different range of values.
###### Note

Minitab calculates the parameter estimates using maximum likelihood method for all the distributions except normal and lognormal distributions, which instead use unbiased parameter estimates.

### Interpretation

Use the ML estimates of the distribution parameters to understand the specific distribution model that is used for your data. For example, suppose a quality engineer decides that, based on historical process knowledge and the Anderson-Darling and LRT p-values, the 3-parameter Weibull distribution provides the best fit for the process data. To understand the specific 3-parameter Weibull distribution that is used to model the data, the engineer examines the ML estimates for shape, scale, and threshold that are calculated for the distribution.

## Distribution

The analysis provides goodness-of-fit statistics and distribution parameters for several commonly used distributions. Many of these distributions are versatile and can model a variety of continuous data, including data with positive values, negative values, and 0.

However, the following distributions can only be used to model data with positive values:
• Lognormal
• Exponential
• Weibull
• Gamma
• Loglogistic

Therefore, if your data contain negative values or 0, Minitab does not report results for these specific distributions. In that case, use the results for the higher-parameter version of each distribution. For example, if your data contain negative values, Minitab does not report results for the lognormal distribution. Instead, use the results for the 3-parameter lognormal distribution.

For more information on the distributions, go to Why is Weibull the default distribution for nonnormal capability analysis?.

###### Note

For information on the formulas that are used to calculate the PDF and CDF for each distribution, go to Methods and formulas for distributions in Individual Distribution Identification.

## P

For each distribution, Minitab reports a p-value (P) for the Anderson-Darling (AD) test. The p-value is a probability that measures the evidence against the null hypothesis. For an AD test, the null hypothesis is that the data follow the distribution. Therefore, lower p-values provide stronger evidence that the data do not follow the distribution.
###### Note

No p-value for the AD test is available for the 3-parameter distributions, except for the Weibull distribution.

### Interpretation

Use the p-value to assess the fit of the distribution.

Compare the p-value for each distribution or transformation to the significance level. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that the data do not follow the distribution when they actually do follow the distribution.
P ≤ α: The data do not follow the distribution (Reject H0)
If the p-value is less than or equal to the significance level, the decision is to reject the null hypothesis and conclude that your data do not follow the distribution.
P > α: Cannot conclude the data do not follow the distribution (Fail to reject H0)
If the p-value is greater than the significance level, the decision is to fail to reject the null hypothesis. There is not enough evidence to conclude that the data do not follow the distribution. You can assume the data follow the distribution.
When selecting a distribution to model your data, also rely on your process knowledge. If several distributions provide a good fit, use the following strategies to choose a distribution:
• Choose the distribution that is most commonly used in your industry or application.
• Choose the distribution that provides the most conservative results. For example, if you are performing capability analysis, you can perform the analysis using different distributions and then choose the distribution that produces the most conservative capability indices. For more information, go to Distribution percentiles for Individual Distribution Identification and click "Percents and percentiles".
• Choose the simplest distribution that fits your data well. For example, if a 2-parameter and a 3-parameter distribution both provide a good fit, you might choose the simpler 2-parameter distribution.
###### Important

Use caution when you interpret results from a very small or a very large sample. If you have a very small sample, a goodness-of-fit test may not have enough power to detect significant deviations from the distribution. If you have a very large sample, the test may be so powerful that it detects even small deviations from the distribution that have no practical significance. Use the probability plots in addition to the p-values to evaluate the distribution fit.

###### Note

For several distributions, Minitab also displays results for the distribution with an additional parameter. For example, for the lognormal distribution, Minitab displays results for both the 2-parameter and 3-parameter versions of the distribution. For distributions that have additional parameters, consider whether the additional parameter is compatible with what you know about the process. For example, if the process has a physical boundary at a non-zero value, then a distribution with a threshold parameter is compatible with the process.

## Ppk

Ppk is a measure of the overall capability of the process and equals the minimum of PPU and PPL. Ppk is a ratio that compares two values:
• The distance from the process mean to the closest specification limit (USL or LSL)
• The one-sided spread of the process (the 3-σ variation) based on its overall variation
Ppk evaluates both the location and the overall variation of the process.

### Interpretation

Use Ppk to evaluate the overall capability of your process based on both the process location and the process spread. Overall capability indicates the actual performance of your process that your customer experiences over time.

Generally, higher Ppk values indicate a more capable process. Lower Ppk values indicate that your process may need improvement.

• Compare Ppk to a benchmark value that represents the minimum value that is acceptable for your process. Many industries use a benchmark value of 1.33. If Ppk is lower than your benchmark, consider ways to improve your process.

• Compare Pp and Ppk. If Pp and Ppk are approximately equal, then the process is centered between the specification limits. If Pp and Ppk differ, then the process is not centered.

• Compare Ppk and Cpk. When a process is in statistical control, Ppk and Cpk are approximately equal. The difference between Ppk and Cpk represents the improvement in process capability that you could expect if shifts and drifts in the process were eliminated.

### Caution

The Ppk index represents only one side of the process curve and does not measure how the process performs on the other side of the process curve.

For example, the following graphs display two processes that have identical Ppk values. However, one process violates both specification limits, and the other process violates only the upper specification limit.

If your process has nonconforming parts that fall on both sides of the specification limits, consider using other indices, such as Z.bench, to more fully assess process capability.

## Cpk

Cpk is a measure of the potential capability of the process and equals the minimum of CPU and CPL. Cpk is a ratio that compares two values:
• The distance from the process mean to the closest specification limit (USL or LSL)
• The one-sided spread of the process (the 3-σ variation) based on the within-subgroup standard deviation
Cpk considers both the location and the variation (within subgroups) of the process.

### Interpretation

Use Cpk to evaluate the potential capability of your process based on both the process location and the process spread. Potential capability indicates the capability that could be achieved if process shifts and drifts were eliminated.

Generally, higher Cpk values indicate a more capable process. Lower Cpk values indicate that your process may need improvement.

• Compare Cpk with a benchmark that represents the minimum value that is acceptable for your process. Many industries use a benchmark value of 1.33. If Cpk is lower than your benchmark, consider ways to improve your process, such as reducing its variation or shifting its location.

• Compare Cp and Cpk. If Cp and Cpk are approximately equal, then the process is centered between the specification limits. If Cp and Cpk differ, then the process is not centered.

• Compare Ppk and Cpk. When a process is in statistical control, Ppk and Cpk are approximately equal. The difference between Ppk and Cpk represents the improvement in process capability that you could expect if shifts and drifts in the process were eliminated.

### Caution

The Cpk index represents only one side of the process curve, and does not measure how the process performs on the other side of the process curve.

For example, the following graphs display two processes with identical Cpk values. However, one process violates both specification limits, and the other process only violates the upper specification limit.

If your process has nonconforming parts that fall on both sides of the specification limits, consider using other indices to more fully assess process capability.

## Cnpk

Cnpk is a measure of the overall capability of the process and equals the minimum of Cnpu and Cnpl.

Cnpu is a ratio that compares two values:
• The one-sided specification spread, from the process median to the upper specification limit
• One-half the process spread, from the process median to the estimate of the upper end of the process
Cnpl is a ratio that compares two values:
• The one-sided specification spread, from the process median to the lower specification limit
• One-half the process spread, from the process median to the estimate of the lower end of the process

### Interpretation

Use Cnpk to evaluate the overall capability of your process based on both the process location and the process spread. Overall capability indicates the actual performance of your process that your customer experiences over time.

Generally, higher Cnpk values indicate a more capable process. Lower Cnpk values indicate that your process may need improvement.

• If Cnpk < 1, then the specification spread is less than the process spread.

• Compare Cnpk to a benchmark value that represents the minimum value that is acceptable for your process. Many industries use a benchmark value of 1.33. If Cnpk is lower than your benchmark, consider ways to improve your process.

###### CAUTION

The Cnpk index represents the process capability for only the "worse" side of the process measurements, that is, the side that exhibits poorer process performance. If your process has nonconforming parts that fall on both sides of the specification limits, check the capability graphs and the probabilities of parts outside both specification limits to more fully assess process capability.