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Are Cpk & Ppk acronyms? If so, what do they actually mean or represent?
Since their introduction, there has been a lot of speculation as to meaning. Here is my two cents worth.
Cp has always been know as capability of the process since I became aware of it and it has been around for some time. My connection with Cpk came through the Ford “Continuous Process Control and Process Capability Improvement Manual” probably more than 20 years ago. In the Ford manual, a k value was used to represent the number of standard deviations between the Target. I would assume that the Cpk came literally from Cp with a k factor adjustment. In reference to the Pp and Ppk, the reference from the beginning has been to Process Performance as opposed to Process Capability.
What is the difference between Cp and Pp?
The technical difference is that the 6 sigma used for the Cp calculation (or the 3 sigma used for the Cpk calculation) comes from the estimate of sigma based on the average range, and the 6 sigma used for Pp calculation (or 3 sigma used for the Ppk calculation) comes from the estimate of sigma based on using all the data and the classical formula for the standard deviation. View the formulas for Cp and Cpk; view the formulas for Pp and Ppk.
In general, if the process is in control and normally distributed (standard assumptions when doing capability analysis), both values should be close. However, since most processes wander around a little bit (and are in control), an intuitive interpretation is that the Cpk is what you could be doing and Ppk is what you are doing.
What is the difference in the formulas for Cpk and Ppk? The only difference I see is the i and r after the sigma symbol. What are these referring to?
The six sigma used for the Cpk calculation comes from the estimate of sigma based on the average range (r). The six sigma used for the Ppk calculation comes from the estimate of sigma based on using all the Individual data (i) and the classical formula.
In general, if the process is in control and normally distributed (standard assumptions when doing capability analysis), both values should be close. However, since most processes wander around a little bit (and are still in control), an intuitive interpretation is that the Cpk is what you could be doing and Ppk is what you are doing.
Are there maximum values for Cp, Cpk, Pp and Ppk?
No. As long as the spec range does not change and you continually reduce the variation, you will increase these indices. I have seen as high as 36 and have heard of higher.
How can I improve Cpk value, when it is less than 1.0?
First, compare Cpk to Cp. If Cpk is less than Cp and Cp is greater than one, center the process in the specification. This should make Cpk comparable to Cp. If Cp and Cpk are less than one, there are two actions you can take. The first (an unadvisable one) is to widen the specification particularly on the side that has the spec limit closest to the center of the process. The second and more advisable answer is to improve the process by reducing variation in the process. If the process is off-center, it would be advisable to try to center it as you try to improve it.
Is it possible to have a Ppk value of 10 and a Pp number of 5?
This should not occur. You might have a negative number for the Ppk that is larger in absolute value then the Pp number. This implies that the process mean lies outside the specification limits.
What do the letters in Cp and Cpk stand for?
There is no authoritative answer. Cp has been around for a long time and many believe it stands for Capability of the Process. Others say Process Capability, but that would reverse the letters.
As for Cpk, in the literature that I first saw about Cpk, k was the amount of the difference in the target value and in standard deviations (the number of standard deviations that the process is off target). Before you ask, Pp generally is said to be Process Performance.
Why do capability indices formulas divide by 3?
When calculating Cp you divide the specification range by six sigma. This is plus and minus three sigma on each side of the mean of the process which would include about 99.7% of the distribution of output if the process is normal. Cp considers only the spread and not the centering of the process. Consequently, you can have a capable process (Cp > 1) and not be making any good product. Cpk considers the mean of the process and calculates two values. Since the specification has been split into two pieces, the process spread is split into two as well.
What is an ideal Cpm value?
Generally there is no “ideal.” Bigger is always better. The difference in Cpm as defined in SQCpack is in the calculation of the stand deviation or variance term. The standard deviation for Cpm is based on using the target value rather than the mean which will make sigma(pm) larger and Cpm smaller when the process is not centered on the target value. You could say “ideally” the process should be centered in the specification making Cpm = Cp. However, Cp might only be 0.80 which clearly is not “ideal.”
Can I compare two processes based on only the Cpk values of each of them? Is there any other tool by which I can say that one process is better than other?
It depends on what you mean by better. If the processes are producing the same product dimension, then you can compare them more or less directly.
Cpk includes a centering factor as well as the variation factor. Unless you want to compare centering as part of the two processes, use Cp.
Can the process performance index Ppk be applied on the ongoing process? If yes, how?
The capability indices are designed to be applied to ongoing processes. They are an indication of what a customer can expect in terms of quality from a particular process.
If you have a control chart on a characteristic for a process, SQCpack will calculate these values for you if you enter the specifications. If you do not have the software, the capability analysis article series provides information on calculating capability.
Why would I have Cp and Cpk indices well over 1 when some readings are outside the specification limits?
My first guess would be that if you look at a control chart of the data, it is out of control. Before you can do capability analysis, the process should be predictable and that requires that it be stable (in-control). For a more detailed discussion, see How can Cpk be good with data outside the specification?
The key to chart interpretation is to initially ascertain the type of variation in the system—that is, whether the variation is coming from special or common causes. When the system has only common causes of variation, it is referred to as stable or in control. If, however, the system has special causes of variation, it is referred to as unstable, or out of control.
Look any of the conditions listed below, which indicate that the process is statistically unstable:
>> Any point lying outside the control limits >> 7 or more points in a row above or below the centerline >> 7 or more points in one direction >> Any nonrandom pattern >> Too close to the average >> Too far from the average >> Cycles >> Trends >> Clusters >> Sawtooth >> 2 of 3 points beyond 2 sigma >> 4 of 5 points beyond 1 sigma
When you have determined whether or not there is special cause variation, declare the system stable or unstable.
Capability analysis is a set of calculations used to assess whether a system is statistically able to meet a set of specifications or requirements. To complete the calculations, a set of data is required, usually generated by a control chart; however, data can be collected specifically for this purpose.
Specifications or requirements are the numerical values within which the system is expected to operate, that is, the minimum and maximum acceptable values. Occasionally there is only one limit, a maximum or minimum. Customers, engineers, or managers usually set specifications. Specifications are numerical requirements, goals, aims, or standards. It is important to remember that specifications are not the same as control limits. Control limits come from control charts and are based on the data. Specifications are the numerical requirements of the system.
All methods of capability analysis require that the data is statistically stable, with no special causes of variation present. To assess whether the data is statistically stable, a control chart should be completed. If special causes exist, data from the system will be changing. If capability analysis is performed, it will show approximately what happened in the past, but cannot be used to predict capability in the future. It will provide only a snapshot of the process at best. If, however, a system is stable, capability analysis shows not only the ability of the system in the past, but also, if the system remains stable, predicts the future performance of the system.
Capability analysis is summarized in indices; these indices show a system’s ability to meet its numerical requirements. They can be monitored and reported over time to show how a system is changing. Various capability indices are presented in this section; however, the main indices used are Cp and Cpk. The indices are easy to interpret; for example, a Cpk of more than one indicates that the system is producing within the specifications or requirements. If the Cpk is less than one, the system is producing data outside the specifications or requirements. This section contains detailed explanations of various capability indices and their interpretation.
Capability analysis is an excellent tool to demonstrate the extent of an improvement made to a process. It can summarize a great deal of information simply, showing the capability of a process, the extent of improvement needed, and later the extent of the improvement achieved.
Capability indices help to change the focus from only meeting requirements to continuous improvement of the process. Traditionally, the focus has been to reduce the proportion of product or service that does not meet specifications, using measures such as percentage of nonconforming product. Capability indices help to reduce the variation relative to the specifications or requirements, achieving increasingly higher Cp and Cpk values.
Before capability analysis is completed, a histogram and control chart need to be completed. Easily create these charts and perform capability analysis using software like SQCpack.
Use the standard method for calculating capability analysis when you can answer “yes” to all of the following questions:
Note that if you cannot answer these questions with a “yes,” Practical Tools for Continuous Improvement includes capability analysis for one-sided specifications, trial runs, attributes data, and nonnormal data.
A courier company has set up a team to look at the actual arrival time at customers’ locations to pick up packages, in comparison to the scheduled arrival time. The company guarantees pick up of packages within 14 minutes of the scheduled time. It is unacceptable to customers for the courier to arrive early. Therefore, there are two requirements: on time and up to 14 minutes late. The result of the capability analysis for this example follows.
A capability analysis like this can be accomplished using software like SQCpack.
These steps assume that variables data has been collected over time, and that a control chart and histogram have been completed. The control chart should show no special causes, and the histogram should reveal that the data is normally distributed.
In the example, the team examined the arrival time of couriers in comparison to the scheduled arrival time over a month. Since time is being measured, the data is variables. The team completed a histogram and found the data to be normal. An X- R control chart was also completed, showing no special causes of variation. Since a control chart must be completed before performing capability analysis, the calculations from the control chart can be utilized. The information taken from the control chart follows:
The numerical specifications or requirements should also be known. Information for the example is shown below:
n = 5 = 10.00 = 4.653 USL = upper specification limit = 14 minutes LSL = lower specification limit = 0 minutes (on time)
n = 5
= 10.00
= 4.653
USL = upper specification limit = 14 minutes
LSL = lower specification limit = 0 minutes (on time)
Steps:
The above article is an excerpt from the “Capability Analysis” section of Practical Tools for Continuous Improvement.
>> Cpk or Ppk: Which should you use? >> How can Cpk be good with data outside the specification? >> How do we determine process capability if the process isn’t normal? >> Is Cpk the best capability index? >> Should you calculate Cpk when your process is not in control? >> The capability index dilemma: Cpk, Ppk, or Cpm >> Calculating capability indices with one specification
A histogram can be created using software such as SQCpack. How would you describe the shape of the histogram?
Bell-shaped: A bell-shaped picture, shown below, usuallypresents a normal distribution.
Bimodal: A bimodal shape, shown below, has two peaks. This shape may show that the data has come from two different systems. If this shape occurs, the two sources should be separated and analyzed separately.
Skewed right: Some histograms will show a skewed distribution to the right, as shown below. A distribution skewed to the right is said to be positively skewed. This kind of distribution has a large number of occurrences in the lower value cells (left side) and few in the upper value cells (right side). A skewed distribution can result when data is gathered from a system with has a boundary such as zero. In other words, all the collected data has values greater than zero.
Skewed left: Some histograms will show a skewed distribution to the left, as shown below. A distribution skewed to the left is said to be negatively skewed. This kind of distribution has a large number of occurrences in the upper value cells (right side) and few in the lower value cells (left side). A skewed distribution can result when data is gathered from a system with a boundary such as 100. In other words, all the collected data has values less than 100.
Uniform: A uniform distribution, as shown below, provides little information about the system. An example would be a state lottery, in which each class has about the same number of elements. It may describe a distribution which has several modes (peaks). If your histogram has this shape, check to see if several sources of variation have been combined. If so, analyze them separately. If multiple sources of variation do not seem to be the cause of this pattern, different groupings can be tried to see if a more useful pattern results. This could be as simple as changing the starting and ending points of the cells, or changing the number of cells. A uniform distribution often means that the number of classes is too small.
Random: A random distribution, as shown below, has no apparent pattern. Like the uniform distribution, it may describe a distribution that has several modes (peaks). If your histogram has this shape, check to see if several sources of variation have been combined. If so, analyze them separately. If multiple sources of variation do not seem to be the cause of this pattern, different groupings can be tried to see if a more useful pattern results. This could be as simple as changing the starting and ending points of the cells, or changing the number of cells. A random distribution often means there are too many classes.
Follow these steps to interpret histograms.
A run chart is a line graph of data plotted over time. By collecting and charting data over time, you can find trends or patterns in the process. Because they do not use control limits, run charts cannot tell you if a process is stable. However, they can show you how the process is running. The run chart can be a valuable tool at the beginning of a project, as it reveals important information about a process before you have collected enough data to create reliable control limits.
Run charts show individual data points in chronological order.
An individuals and moving range (X-MR) chart is a pair of control charts for processes with a subgroup size of one. Used to determine if a process is stable and predictable, it creates a picture of how the system changes over time. The individual (X) chart displays individual measurements. The moving range (MR) chart shows variability between one data point and the next. Individuals and moving range charts are also used to monitor the effects of process improvement theories.
The individuals chart, on top, shows each reading. It is used to analyze central location. The moving range chart, on the bottom, shows the difference between consecutive readings. It is used to study system variability.
A median chart is a special purpose variation of the X-bar chart. This chart uses the median instead of the subgroup average to show the system’s central location. The median is the middle point when data points are arranged from high to low. The chart shows all the individual readings. Use charts to determine if the system is stable and predictable or to monitor the effects of process improvement theories.
Although median charts show both central location and spread, they are often paired with range charts.
The median chart shows each reading in a subgroup. Subgroup medians are connected by the data line and are used to analyze the central location.
Use the median chart when you want to plot all the measured values, not just subgroup statistics. This may be the case when subgroup ranges vary a great deal, as showing all the points will emphasize the spread. It shows users that individual data points can fall outside the control limits, while the central location is within the limits.
Use the median chart when you can answer yes to these questions:
Collect as many subgroups as possible before calculating control limits. With smaller amounts of data, the median chart may not represent variability of the entire system. The more subgroups you use in control limit calculations, the more reliable the analysis. Typically, twenty to twenty-five subgroups will be used in control limit calculations.
Median charts have several applications. When you begin to improve a system, use them to assess the system’s stability.
After the stability has been assessed, determine if you need to stratify the data. You may find entirely different results between shifts, among workers, among different machines, among lots of materials, etc. To see if variability on the median chart is caused by these factors, you should collect and enter data in a way that lets you stratify by time, location, symptom, operator, and lots.
You can also use median charts to analyze the results of process improvements. Here you would consider how the process is running and compare it to how it ran in the past. Do process changes produce the desired improvement?
Finally, use median charts for standardization. This means you should continue collecting and analyzing data throughout the process operation. If you made changes to the system and stopped collecting data, you would have only perception and opinion to tell you whether the changes actually improved the system. Without a control chart, there is no way to know if the process has changed or to identify sources of process variability.
Variables data is normally analyzed in pairs of charts which present data in terms of location or central location and spread. Location, usually the top chart, shows data in relation to the process average. It is presented in X-bar, individuals, or median charts. Spread, usually the bottom chart, looks at piece-by-piece variation. Range, sigma, and moving range charts are used to illustrate process spread. Another aspect of these variables control charts is that the sample size is generally constant.
Use the following types of charts and analysis to study variables data:
These charts, and more, can be created easily using software packages such as SQCpack.
An X-bar and s (sigma) chart is a special purpose variation of the X-bar and R chart. Used with processes that have a subgroup size of 11 or more, X-bar and s charts show if the system is stable and predictable. They are also used to monitor the effects of process improvement theories. Instead of using subgroup range to chart variability, these charts use subgroup standard deviation. Because standard deviation uses each individual reading to calculate variability, it provides a more effective measure of the process spread. X-bar and sigma charts To create an X-bar and sigma chart using software, download a copy of SQCpack.
The X-bar chart, on top, shows the mean or average of each subgroup. It is used to analyze central location. The sigma chart, on the bottom, shows how the data is spread and used to study system variability.
An X-bar and R (range) chart is a pair of control charts used with processes that have a subgroup size of two or more. The standard chart for variables data, X-bar and R charts help determine if a process is stable and predictable. The X-bar chart shows how the mean or average changes over time and the R chart shows how the range of the subgroups changes over time. It is also used to monitor the effects of process improvement theories. As the standard, the X-bar and R chart will work in place of the X-bar and s or median and R chart. To create an X-bar and R chart using software, download a copy of SQCpack.
The X-bar chart, on top, shows the mean or average of each subgroup. It is used to analyze central location. The range chart, on the bottom, shows how the data is spread. It is used to study system variability.
You can use X-bar and R charts for any process with a subgroup size greater than one. Typically, it is used when the subgroup size falls between two and ten, and X-bar and s charts are used with subgroups of eleven or more.
Use X-bar and R charts when you can answer yes to these questions:
Collect as many subgroups as possible before calculating control limits. With smaller amounts of data, the X-bar and R chart may not represent variability of the entire system. The more subgroups you use in control limit calculations, the more reliable the analysis. Typically, twenty to twenty-five subgroups will be used in control limit calculations.
X-bar and R charts have several applications. When you begin improving a system, use them to assess the system’s stability.
After the stability has been assessed, determine if you need to stratify the data. You may find entirely different results between shifts, among workers, among different machines, among lots of materials, etc. To see if variability on the X-bar and R chart is caused by these factors, collect and enter data in a way that lets you stratify by time, location, symptom, operator, and lots.
You can also use X-bar and R charts to analyze the results of process improvements. Here you would consider how the process is running and compare it to how it ran in the past. Do process changes produce the desired improvement?
Finally, use X-bar and R charts for standardization. This means you should continue collecting and analyzing data throughout the process operation. If you made changes to the system and stopped collecting data, you would have only perception and opinion to tell you whether the changes actually improved the system. Without a control chart, there is no way to know if the process has changed or to identify sources of process variability.