When performing process capability analysis in JMP, there are platforms for "Process Screening" and "Process Capability," but the most straightforward way is to use the "Process Capability" option in the "Distribution" platform.
If you run a capability analysis on the JMP sample data set "Pickles.jmp" for the continuous column "Acidity" with specification limits (lower = 8, upper = 17) using "Distribution", the following report is displayed:
The report includes sections called "Histogram," "Process Summary," "Within Sigma Capability," "Overall Sigma Capability," and "Defective Rate." Understanding these graphs and statistics is important when conducting a process capability analysis.
You may think that the goal is simply to calculate process capability indices such as Cpk and Ppk and that it is enough to look at the relevant sections of the report, but it is also important to understand how these indices are calculated.
Therefore, in this blog, we will explain how to read the report of the process capability analysis of "univariate distribution" shown above.We will provide a detailed explanation of how the displayed statistics are calculated. You can also download a script (Calculate_Capability.jsl) that shows the calculation formulas.
Notice
- The example presented does not set a target value, but specifies a lower specification limit (LSL) and an upper specification limit (USL).
- Calculating confidence intervals for process capability indices is complex and will not be discussed here.
- You can perform capability analysis by specifying subgroup columns, but an example of this is not provided here.
- The reports described here assume a normal distribution for the process. JMP can perform capability analysis for distributions other than normal, but we will not discuss them here.
Process Summary: What is Within-Group Sigma and Overall Sigma?
A histogram for the column "Acidity" specified as Y is displayed along with the specified specification limits (red solid lines). This graph allows you to visually see how much data is below the LSL and how much is above the USL.
Within-subgroup Sigma : An estimate of the variability calculated in the short term (using σ from the control chart)
Overall Sigma : An estimate of the variability calculated over time (using either the data or the overall standard deviation)
It is recommended that you use process capability analysis after checking whether the target process is stable using a control chart, etc. If the process is stable, the within-group sigma and overall sigma values will be approximately equal, but if the process is not stable, these values will diverge.
The histogram shows a normal distribution curve (blue line) with parameters of the sample mean and within-group sigma, and a normal distribution curve (black dotted line) with parameters of the sample mean and overall sigma. Comparing the two curves, the within-group curve appears to be a better fit for the distribution of this process.
If the process is considered stable, it is a good idea to consider the overall sigma value. If the process is not stable, the overall sigma value becomes less reliable. In that case, it is a good idea to consider the within-group sigma value.
Conversely, by comparing the overall sigma value with the within-subset sigma value, we can determine whether the process is stable or not. The index for this is the stability index, which is defined as follows:
Stability index : Overall Sigma ÷ Within-group Sigma
If the process is stable, the stability index will be close to 1 because, as mentioned above, the within-group sigma and overall sigma values are nearly equal. A higher value indicates that the process is less stable. In this example, the stability index is 1.446, which is high, so we can say that the process is not stable.
*The Process Performance Graph in the Process Screening platform, which allows you to compare multiple processes, uses a stability index threshold of 1.25. A process is considered unstable if its stability index is greater than 1.25.
Calculation method
Within-group sigma is calculated based on the average of the moving range. The moving range is the absolute value of subtracting the previous value from the current value. To calculate the sigma equivalent to the R control chart, |value in row 2 - value in row 1|, |value in row 3 - value in row 2| ... |Nth value - (N-1)th value| are calculated and summed. This sum is then divided by a constant called d2(2). d2(2) is the value of d2 when the number of subgroups is 2 (in the case of the moving range, the subgroup is the value from the current value to the previous value). This is a constant that is also used to calculate sigma on the control chart, and its value is 1.128. This value is then divided by the number of subgroups (the number of data in this example is 24, so the number of subgroups is 23, which is 1 less than the number of data).
Within-group sigma = (sum of moving ranges / d2(2)) / 23 = 1.160
Overall Sigma is calculated using the standard deviation calculated from the data. You can confirm that it matches the standard deviation value in "Summary Statistics."
Engineering Capability Index
As stated in the JMP manual, the within-group sigma process capability index is calculated using the following formula.
In this formula, μ is estimated by the sample mean, and σ is estimated by the short-term sigma. Using the values shown in the "Process Summary" report above, the estimates of the process capability indices are calculated as follows:
Cp = (17 - 8) 8) / (6 * 1.1598) = 1.293
Cpl = (10.5667 - 8) 8) / (3 * 1.1598) = 0.738
Cpu = (17 - 10.5667) /(3 * 1.1598) = 1.849
Cpk = min(0.738, 1.849) = 0.738
In the overall sigma process capability index, σ in the above equation is estimated by overall sigma.
Pp = (17 - 8)8)/ (6 * 1.6766) = 0.895
Ppl = (10.5667 - 8) 8) / (3 * 1.6766) = 0.510
Ppu = (17 - 10.5667) /(3 * 1.6766) = 1.279
Ppk = min(0510, 1.279) = 0.510
In this example, Cpk is 0.738 and Ppk is 0.510, both low values.
unfit rate
Observed % : The percentage of data that falls outside the specification limits.
In this example, there is one piece of data below LSL and no data above USL, so we divide these values by the number of pieces of data (24).
Less Than LSL = (1 / 24)*100 = 4.1667
Above USL = (0/ 24 )*100 = 0
The within-group σ % and overall σ % indicate the probability of being below the LSL or above the USL for the normal distribution curve shown in the histogram.
Within % : Area (probability) outside the specification limits for the normal distribution density curve using the within-group sigma
The cumulative distribution function of the normal distribution can be calculated using the Normal Distribution(q, mu, sigma) function using the following formula:
Less than LSL = Normal Distribution(LSL, sample mean, within-group sigma) * 100 = 1.3448
Above USL = (1- Normal Distribution(USL, Sample Mean, Within Sigma)) * 100 = 0.0000
Overall %: Area (probability) outside the specification limits for the normal distribution density curve with overall sigma
Less Than LSL = Normal Distribution(LSL,Sample Mean,Overall Sigma) * 100 = 6.2901
Above USL = (1- Normal Distribution(USL, Sample Mean, Overall Sigma)) * 100 = 0.0062
To improve process capability
As mentioned above, the Cpk and Ppk for this example process are 0.738 and 0.510, respectively, so the process capability is low. So how can we improve the process to improve its capability? JMP provides an option to give you numerical hints to answer this question.
Select the "Interactive Capability Plot" option from the red triangle button to the left of the report "Capability Analysis." This will add the following report. The graph shows the (sample) mean for the current data, the normal distribution with overall sigma as a parameter, LSL, and USL lines. This shows the current process capability.
So, if you could improve the process, shift the average to a larger value, and reduce the variability, how much would the process capability improve? You can check this by moving the "New Process Capability" slider. For example, here is a report with the average set to 12.5 and the overall sigma (variability) set to 1.1.
This improves the Ppk to 1.364, exceeding the commonly used guideline of 1.33.
Although this is merely a simulation of a hypothetical case, I think it will be useful to know the reference value for future process improvements.
The following video introduces process capability analysis using the "Process Capability" and "Process Screening" platforms, including the "Univariate Distribution" process capability analysis introduced in this article.
Process Capability Analysis with JMP (Japanese)
by Naohiro Masukawa (JMP Japan)
Naohiro Masukawa - JMP User Community
