Process Capability with Variable Data

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About this lesson

Process Capability is often correlated with process Sigma. The calculation of process capability is quite different depending upon whether the data is variable or attribute data. This lesson will present the technique and practice using the variable data process capability ratios.

Quick reference

Process Capability with Variable Data

The calculation of process capability is quite different depending upon whether the data is variable or attribute data. This lesson will explain the formulas for calculating process capability with variable data.

When to use

Process data is either variable data or attribute data.  Variable data is measured on a scale and a value is determined.  Attribute data consists of placing the data value in a category.  If the process generates variable data, the variable data Process Capability ratios must be used.

Instructions

Variable data process capability is established using ratios of the customer expected process performance divided by a value representing normal process spread.  There are four different indices.  They represent the difference between using near-term recent process data versus long-term historical data and the difference between considering the best-case process performance as compared to the current actual process performance.  The differences will be in what process performance measures are used in the calculations.

Cp and Pp

The Cp and Pp indices provide a best-case view of the process capability.  These indices are often used during the product and process design process to determine whether a particular type of process is capable of delivering satisfactory performance.  These indices have the difference between the upper spec limit and the lower spec limit in the numerator.  That means the numerator is the spread of allowable variation that will meet the customer requirements.  The denominator in the ratio is the value from minus three standard deviations to plus three standard deviation.  Which is a spread of six standard deviations.  This is where the difference between Cp and Pp come in. 

Cp uses recent, near-term data for calculating the standard deviation.  Pp uses long-term historic data.  The reason that a spread of minus three standard deviations to plus three standard deviations is used is because that is the way in which Walter Shewhart designed the ratios in the early 1900s.  At that time, he felt that process that could deliver all the process output within the limits of minus three standard deviations to plus three standard deviations was a n excellent process  With this philosophy, Shewhart set as a target for a process to be considered capable it must have a ration greater than or equal to one.  Meaning the process normal spread or variation was less than the allowed variation in the customer specifications.

The Cpk and Ppk process capability indices are derived from the Cp and Pp indices.  These indices modify the Cp and Pp indices by taking into consideration the current mean or average point of the process performance.  The values for the numerator and denominator are both split into two parts.  The numerator is split at the value of the mean.  So one numerator will be the difference between the mean and the lower spec limit and the other numerator will be the difference between the upper spec limit and the mean.  The denominator is split exactly in half.  The value for three standard deviations is used in each ratio.  This splitting creates two ratios for Cpk and Ppk instead of just one.  The process capability function selects the smaller of the two to report as the process capability.  When the process is exactly centered (the mean is half way between the upper and lower spec limits); the numerator for both ratios will be the same. However, if the mean is not centered, one ratio will be larger than the other.  A process then can have a Cp or Pp value that is greater than one – showing that the basic design of the process is capable of delivering good results.  But if the process management has not centered the process, the actual results can be much less than one, indicating that process has drifted outside the bounds of acceptable performance.  And like with the other indices, Cpk uses short-term recent data to calculate the mean and standard deviation while the Ppk relies on long-term historic data.

Hints & tips

  • Cpk and Ppk can never be greater than the Cp or Pp value.  Once the process is exactly in the center of the allowable limits, the two Cpk/Ppk ratios will be identical and they will also equal the corresponding Cp or Pp ratio.
  • When looking at process results, you only need to calculate the Cpk/Ppk ratio for the condition where the mean is closest to the spec limit.  That will always be the smallest value.’
  • Cp and Pp normally do not change unless there is a major change to the process.  But Cpk and Ppk can often change if machines wear out or different operators setup the process slightly differently.
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  • 00:04 Hi, I'm Ray Sheen.
  • 00:05 Let's now take a few minutes and look at the process capability ratio calculations,
  • 00:10 when we have process performance measurments that are with variable data.
  • 00:16 I'll start with a description of the process capability indices.
  • 00:21 For starters there're four indices that give insight into four slightly different
  • 00:25 perspectives on the process.
  • 00:27 All four indices follow the same pattern which is a ratio.
  • 00:32 The ratio is the voice of the customer in terms of the required or
  • 00:35 desired process performance, divided by the voice of the process or
  • 00:40 actual performance expressed with values from the process descriptive statistics.
  • 00:45 Now I said that there were four different perspectives.
  • 00:48 These four perspectives are the combinations of best case versus
  • 00:52 actual performance, and the near term versus long term performance.
  • 00:57 Cp is short term best case performance, Cpk is short term actual performance.
  • 01:03 Pp is long-term best performance and
  • 01:06 Ppk is the long-term actual process capability.
  • 01:11 The Cpk and Ppk are the most closely associated with SPC control charting.
  • 01:17 That's because SPC control charts are actual performance and the Cpk and
  • 01:21 Ppk are tracking actual performance.
  • 01:24 The Cp and Pp ratios are more commonly used in the design processes to determine
  • 01:29 the theoretical possibilities.
  • 01:32 Cpk and Ppk are grounded in the reality of today.
  • 01:37 However let's start with Cp and Pp.
  • 01:39 They're the simple ratios that provide a good foundation for us.
  • 01:44 Now the definition of Cp and
  • 01:45 Pp are the ratio of the allowable range of processed performance
  • 01:49 divided by the normal level of processed variation due to common cause variation.
  • 01:54 And keep in mind these ratios where first created by Walter Shewhart in the early
  • 01:58 1900s.
  • 01:59 At that time, we thought that the limits of plus or minus 3 sigma were quite good
  • 02:03 for process performance, so that is what he used in these calculations.
  • 02:08 So if a Cp or Pp of one was equivalent of three sigma, a Cp or
  • 02:13 Pp of two will be the equivalent of Six Sigma.
  • 02:16 As you can see from the formulas for Cp and Pp, the only difference is that,
  • 02:21 Cp is using their term process standard deviation, and
  • 02:24 Pp is using the long term standard deviation.
  • 02:28 Let's look at a picture of what this looks like.
  • 02:30 The numerator and the ratio is the desire or customer performance requirement.
  • 02:34 This is expressed using the minimum acceptable level, the lower spec limit and
  • 02:38 the maximum acceptable level, known as the upper spec limit.
  • 02:42 So the numerator is the upper spec limit minus lower spec limit and
  • 02:46 this represents the range of what is allowable.
  • 02:49 The denominator is the ratio of the range from minus three standard deviations to
  • 02:54 plus three standard deviations.
  • 02:56 By going from minus three to plus three the spread is a total of six times
  • 03:00 the standard deviation.
  • 03:02 In this example the range of acceptable customer performance is greater than
  • 03:07 the range of six standard deviations, therefore, the ratio is greater than one.
  • 03:12 This ratio is used by product and process designers to design processes which should
  • 03:17 be able to consistently deliver acceptable quality.
  • 03:21 Let's contrast that with this case, the spec limits are the same.
  • 03:25 The spread from the upper spec limit to the lower spec limit didn't change.
  • 03:29 However, the process has much higher common cause variation.
  • 03:33 Therefore, if our standard deviation is larger then that means that
  • 03:36 the denominator of the ratio is now much larger.
  • 03:39 So in this case the Cp or Pp would be less than one.
  • 03:43 A process that is designed with this level of Cp will consistently produce process
  • 03:48 results that are outside the customer requirement for performance.
  • 03:53 Now let's look at Cpk and Ppk.
  • 03:56 The ratios are structured in a similar manner as
  • 03:59 we've already seen with Cp and Pp.
  • 04:01 But it's been adjusted for the reality of current process performance.
  • 04:06 In particular,
  • 04:07 it's adjusted to the fact that the actual process performance normally is
  • 04:10 not precisely centered in the middle of the allowable range from the customer.
  • 04:15 So this ratio brings the mean value into the numerator.
  • 04:18 This then resets the numerator range based upon the distance from the mean
  • 04:22 to the closest spec limit.
  • 04:24 But now, since there is one spec limit involved,
  • 04:27 only half of the denominator needs to be used.
  • 04:31 That means that both the numerator and denominator are split but
  • 04:34 we actually have two ratios.
  • 04:36 Since we are splitting the numerator at the mean value,
  • 04:39 there's normally one numerator value than the smaller than the other.
  • 04:43 If the mean is exactly in the centre the two values would be the same, but
  • 04:48 that is uncommon.
  • 04:49 The denominator though is split exactly in half which means that instead
  • 04:54 of using the spread of six sigma, we only use three sigma.
  • 04:58 Finally, the ratio that we use is the smaller of those two ratios.
  • 05:02 That is why the calculation will be expressed using the minimum function.
  • 05:07 So let's look at the formulas.
  • 05:09 The difference between the Cpk and Ppk is whether the mean value and
  • 05:12 standard deviation are from recent process performance or
  • 05:15 if they represent long term historical performance.
  • 05:18 And while we can calculate the two different ratios,
  • 05:22 the smaller ratio is the one that matters.
  • 05:25 So let's look at a graph of the situation where both the Cp AND
  • 05:28 Cpk are greater than one.
  • 05:30 As you can see, both the ratio of the mean minus the lower spec limit divided by
  • 05:35 3 sigma, and the ratio of the upper spec limit minus the mean divided by 3 sigma,
  • 05:41 is greater than one.
  • 05:42 The ratio on the lower spec limit will be the smaller of those two,
  • 05:46 since the performance is shifted to the low side.
  • 05:49 But it is still greater than one, and
  • 05:51 the process should be providing acceptable performance.
  • 05:54 Now let's shift things even farther to the left.
  • 05:57 We're at a point where the mean minus the lower spec limit is now less than three
  • 06:01 standard deviations.
  • 06:03 So the Cpk is less than 1 for the ratio on the low side.
  • 06:07 But in this case the Cp would still have been greater than one,
  • 06:10 because if the process performance had been centered within the spec limits,
  • 06:14 the ratios would be greater than one on both sides.
  • 06:16 So, as you can see, the process capability indices of Cp, Cpk,
  • 06:22 Pp AND Ppk are excellent measures of actual process performance,
  • 06:27 with respect to the customer's desired performance.

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