DSQR Training Attribute Control Charts HEATING, COOLING & WATER HEATING PRODUCTS DSQR Training Attribute Control Charts Ted Fisher/Fred Nunez Corporate Quality

Attribute Control Charts * - Constant Sample Size Required

To decide which to use, you need to ask: ATTRIBUTE CONTROL CHARTS Two Major Types p charts and np charts C charts and U charts To decide which to use, you need to ask: What is being counted? Two ways to count: Count how many sample units have (or do not have) the attribute (use a p chart or np chart) Count how many occurrences of the attribute are in the sample (use a C chart or U chart)

The Objective of Attribute Control Charts is Similar to Control Charts for Measurement Data Sample data is collected in an ordered sequence of subgroups Control limits are placed at + 3 Sigma above and below a center line Objective is to differentiate between Common and Special causes by detecting non-random patterns of variation Attribute Control Charts are used to evaluate performance in terms of: p - proportion of units having one or more nonconformances np - number of units having one or more nonconformances C - number of nonconformances in a sample U - number of nonconformances per unit

Selecting a Control Chart Monitoring Selecting a Control Chart Start Type of data Counting items with an attribute or counting occurrences? Equal Sample sizes? Opportunity? p chart np chart Individuals Individuals chart EWMA EWMA chart Continuous Yes No Rational Subgroups Discrete u chart c chart Do limits look right? Try individuals chart Need to detect small shifts quickly? Individual measurements or subgroups? Try transformation to make data normal Either/Or Individual measurements Occurrences X, R chart Items with attribute Start Type of data ? Discrete Continuous Counting items with an attribute or counting occurrences? Need to detect small shifts quickly? Yes Items with attribute Occurrences No Equal sample sizes ? Equal opportunity ? Individual measurements No Yes No Yes Individual measurements or subgroups ? Either/Or Rational Subgroups Individuals chart 5 EWMA chart p chart np chart u chart c chart X, R chart Do limits look right? Do limits look right? Yes Yes No No Try individuals chart Try transformation to make data normal

Which chart to use? Defects Defectives Variable Sample Size u chart p chart Constant Sample Size c chart np chart

Computation of p, np, C, U # of units having one or more occurrences # of units in the subgroup p = np = Average # of units in the subgroup having that attribute C = Average # of occurrences in the subgroup having that attribute # of occurrences of the attribute # of units in the subgroup U =

Constructing Control Charts for Discrete Data Monitoring Constructing Control Charts for Discrete Data Chart Control Limit Calculations p chart n ) - (1 3 ± np c u a p (or np) chart assumptions are based on the Binomial distribution: Two attributes only (e.g., defective vs. non-defective) The expected proportion of items with the attribute is constant (the same) for each sample Occurrence of the attribute is independent from item to item c (or u) chart assumptions are based on the Poisson distribution: Can count occurrences, but not non-occurrences Probability of an occurrence is relatively rare (less than 10% of the time) Occurrences are independent (one does not influence the occurrence of another)

Considerations Notes: Data for p charts may be expressed either as a fraction or as a percentage. For p charts: sample size (n) should be >50 such that the average number of occurrences is >4. To simplify calculations, equal size samples are recommended. However, if n for all subgroups is within 25% of the average sample size, you may use the average sample size in control limit calculations. For np charts: n should be >50, such that the average number of occurrences is >4. Subgroups must be of equal size. For C charts: n should be large enough, such that the average number of occurrences is >1. Subgroups must be of equal size. For U charts: n should be large enough, such that the average number of occurrences is >1. To simplify calculations, equal size samples are recommended. However, if n for all subgroups is within 25% of the average sample size, you may use the average sample size in control limit calculations.

p Chart The calculation of control limits for this example could have been simplified by the use of average sample sizes for some, though not all subgroups.

np Chart Because calculated LCL < 0, LCL = 0.

c Chart Sample Defects 1 7 2 5 3 4 6 8 9 10 11 12 13 14 15 16 17 18 19 20   Total 82 Average 4.1 UCL 10.2 LCL -2.0 Because calculated LCL < 0, LCL = 0 or n/a.

u Chart Because calculated LCL < 0, LCL = 0 or n/a. Total 25.4 75.0 Sample Size (sq ft) Num Holes Holes per sq ft UCL Calc LCL Actual LCL 1 1.0 4 4.0 8.11 -2.20 n/a 2 5 5.0 3 3.0 6 1.3 1.5 7.47 -1.57 7 3.8 8 2.3 9 10 0.8 11 12 13 3.1 14 15 1.2 7.66 -1.75 16 3.3 17 0.0 18 1.7 4.7 6.91 -1.00 19 1.8 20 Because calculated LCL < 0, LCL = 0 or n/a. Total 25.4 75.0 Average   2.95

(Pass/Fail; Count Data) Comparison of Control Charts Variables Data (Measurements) Attribute Data (Pass/Fail; Count Data) Uses Average, Range, Standard Deviation Uses, p, np, C, or U Efficient - can use small sample sizes Not efficient - must use large sample sizes Costs per test may be expensive Costs per test generally small Generally used to monitor a single quality characteristic Can monitor single or multiple quality characteristics

Questions? Comments?

DSQR Training Process Capability & Performance HEATING, COOLING & WATER HEATING PRODUCTS DSQR Training Process Capability & Performance Fred Nunez Corporate Quality

Goals At the end of this section you’ll be able to Process Capability At the end of this section you’ll be able to apply the basic methods for assessing process capability (Cp, Cpk, Pp, Ppk). use the properties of a Normal Curve that are important to process capability calculations. explain the Minitab “6pack” and Minitab capability report for the Normal distribution case.

Process Capability Indices Process capability and performance indices are ways for measuring how the process distribution is aligned with the specification. USL LSL Voice of the Customer Voice of the Process -3s +3s

Process Capability What shall we do? 15 % above the Upper Spec Limit 3 % below the Lower Spec Limit Normal process variation This is an example of a situation where the distribution of values for this in-control process does not fall within the allowable tolerances. What shall we do? Since this process is in-control but not capable, one of three actions must occur: Change the specifications “Suffer and Sort” Make a fundamental change to the process Don’t tell anyone

Process Capability Process Capability Here is a situation in which the process average is on-target, but the spread of the values are barely within allowable tolerances. Target But if we only control process variability but don’t control the process average, we might get this. Target And if we control process average only, and fail to control process variability, we might get this. 20

The Normal Curve and Process Capability When continuous data are normally distributed, calculating a process capability index is really equivalent to finding the area under the normal (or bell-shaped) curve that is outside the spec limits, as depicted in the diagram below. LSL USL The normal curve can provide a good model for process observations, but the model does not apply to every situation. The usual measures and interpretation of process capability depend on the properties of the normal curve.

Normal Distribution –3S +3S +2S +1S –2S –1S 34.13% 13.60% 2.14% 0.13% Process Capability Normal Distribution –3S +3S +2S +1S –2S –1S 34.13% 13.60% 2.14% 0.13% As you can see, the curve is divided into a series of equal increments, each representing one standard deviation from the mean. The area under the curve represented by the first standard deviation out from the center in either direction represents approximately 34% of the total area. Together, the area represented within one standard deviation of the center is about 68% of the total area. In other words, given a data set that is normally distributed, approximately 68% of the data values should fall within one standard deviation of the center. Going out plus or minus two standard deviations represents approximately 95% of the total area; go out 3 standard deviations in either direction and you’ve accounted for more than 99% of the area. Tables have been developed that contain these listed areas, plus many more. Thus by knowing a normal curve’s mean and standard deviation, we can figure out the yields within specified “zones.” 68.26% 95.46% 99.73%

Common Process Capability Indices: The C and P families Pp - Process Performance This is the performance index which is defined as the tolerance width divided by the performance, irrespective of process centering. Remember that this discussion applies equally to both Pp and Cp. Since is a “P” family, it used the standard deviation of all the data to estimate variation. If it was the “C” family, it would have used the average range from a control chart to estimate variation. Where: USL = upper specification limit LSL = lower specification limit 6s = 6 times the sample standard deviation

Common Process Capability Indices: The C and P families Pp - Process Performance You can think of Pp as a measure of how many times the process spread will go into the tolerance. The high the quotient, the better. This implies that the spread is small compared to the tolerance. Pictorially, the process performance Pp is the tolerance width divided by the process spread.

Common Process Capability Indices: The C and P families Pp - Process Performance The Pp is determined by the tolerance and spread of the process, location is not considered. The red (left) and blue (right) distributions have the same Pp. Virtually all of the parts produced on the red (left) process will be in specification, while virtually all of the parts from the blue (right) process will be out of specification. Important to realize that the location of the distribution has no impact on the Pp (Cp) calculation.

Common Process Capability Indices: The C and P families Ppk - Process Performance The process performance index, Ppk,, which accounts for process centering and is defined as: Pp (Cp) did not consider location in the calculations. Ppk (Cpk) is influenced by the location of the distribution. To estimate the Ppk perform both calculation above and report the smaller value. A quicker way is to determine which specification limit (USL or LSL) is closest to the average and only do that calculation, it will be the smallest.

Common Process Capability Indices: The C and P families Ppk - Process Performance The Ppk is determined by the tolerance, spread and distance from specification.

Common Process Capability Indices: The C and P families Ppk - Process Performance Here we can see the impact of the specification in the definition of Ppk. Both processes above will have the same Pp, same spread and tolerance. The Ppk for the blue (left) process will be lower because (Xbar-LSL) is smaller.

Interpreting Ppk 29 LSL USL LSL USL Process Capability Interpreting Ppk LSL USL LSL USL Ppk ~ 2.0 Ppk ~ 0.4 Ppk ~ 1.3 Ppk ~ 0.0 Ppk ~ 1.0 Ppk ~ -1.0 Notice that although the variability is the same for each distribution and Pp=2.0, Ppk changes dramatically depending on where the process is centered. 29

Common Process Capability Indices: The C and P families Cp, Cpk, Pp, Ppk The only difference between the C and P capability indices is the method used to estimate the standard deviation. When you see an index with a “C” the standard deviation was estimated using the average range from a control chart. When you see an index with a “P” the standard deviation was estimated using the standard deviation of all the data. Both calculations assume that the data is normally distributed.

Ppu or Ppl vs. Amount Out-of-Spec Process Capability Ppu or Ppl vs. Amount Out-of-Spec Ppu or Ppl % Out-of-Spec p.p.m. Out-of-Spec These values apply only to the tail of the distribution with the higher % out-of-spec (smaller index). Actual % out-of-spec may be as much as double these values when both tails (upper & lower) are considered. 31

Common Process Capability Indices: The C and P families Index Symbol Index Name Default Formula in Minitab Notes (Normal Distribution Case) C Capability Index ( USL-LSL)/6 swithin The index is not defined p unless both the upper and lower specification limits are used. P Performance Index p ( USL-LSL)/ 6 soverall The index is not defined unless both the upper and lower specification limits are used. CPU Upper Capability ( USL- X )/3 swithin Index PPU Upper Performance ( USL- X )/3 soverall Index Different software programs will use different algorithms to estimate sigma. You may get a slightly different estimate using the same data in Excel, Statgraphics or Minitab. CPL Lower Capability ( X - LSL)/3 swithin Index PPL Lower Performance ( X - LSL)/3 soverall Index C Capability Index Minimum of { CPU,CPL} Cpk takes into account the pk process center while Cp does not. P Performance Index Minimum of { PPU,PPL} Ppk takes into account the pk process center while Pp does 32 not.

Process Capability Statistics Data: C:\SixSigma\Data\pHexample.mtw Use the following Minitab command to obtain descriptive statistics: Stat>Basic Statistics> Graphical Summary Situation: 100 hourly batch values after one day ramp up, following major replacement of process equipment. The product is pH buffer solution which is supposed to measure 4.0 ± 0.02 at 25° C. 33

Process Capability Statistics 34 The TrMean is calculated after removing the highest & lowest 5%. The SE Mean is the standard error of the mean; the standard deviation of the distribution of the means – Central Limit Theorem.

Process Capability Statistics Data: C:\SixSigma\Data\pHexample.mtw Use the following Minitab command to compute the process capability statistics: Stat>Quality Tools>Capability Analysis (Normal) Enter these values 35

Process Capability Statistics Mean Standard Deviation Specification Cp & Cpk Pp & Ppk Actual Observed PPM Estimated PPM 36

Process Capability Capability Sixpack Data: C:\SixSigma\Data\pHexample.mtw Use the following Minitab command to compute the create the Sixpack analysis: Stat>Quality Tools>Capability Sixpack (Normal) Data: C:\SixSigma\Data\pHexample.mtw Use the following Minitab command to compute the process capability statistics: Stat>Quality Tools>Capability Sixpack (Normal) 37 Situation: 100 hourly batch values after one day ramp up, following major replacement of process equipment. The product is pH buffer solution which is supposed to measure 4.0 ± 0.02 at 25° C.

Capability Sixpack 38 See next slide Enter these values Process Capability Capability Sixpack See next slide Enter these values 38

Capability Sixpack, cont. Process Capability Capability Sixpack, cont. You may select all the test or only one. For subgroup size >1, default is pooled standard deviation to obtain denominator in C family 39

Process Capability Sixpack for Production pH 2 1 4 3 1. The control charts show no signals of special causes, so the C and P family indices should give about the same values. 2. The individual observations match the reference line, so the normal distribution will provide a useful model. 3. The capability plot shows how well the process is centered as well as shows the amount of tolerance used by the process. As Cp is about 1.0, the short-term process tolerance length is about the same length as the distance between upper and lower specifications. It is worth noting that Minitab uses different estimates for sigma than AIAG.

Process Capability Sixpack for Production pH The control charts show no signals of special causes, so the C and P family indices should give about the same values. The individual observations match the reference line, so the normal distribution will provide a useful model. The capability plot shows how well the process is centered as well as shows the amount of tolerance used by the process. As Cp is about 1.0, the short-term process tolerance length is about the same length as the distance between upper and lower specifications.

Application Exercise – Process Capability Which two processes are the most variable? __________ and _________ 2. Which two processes are not potentially capable? __________ and _________ a 3. Which two processes have the highest % out-of-spec? __________ and _________. b 4. Which process has the highest Cp? __________ c 5. For which three processes are Cpk and Cp equal? __________, ________ and ____________ 6. Which process is potentially capable but not meeting specs? __________ d 7. Which process is supplying the most material at target value? __________ e 8. If these materials were stored in the warehouses of six different suppliers (all other things being equal) which one would you prefer? __________ f 9. If the process average could be adjusted for future production, which supplier would you prefer? __________ 10. Calculate Cp and Cpk for c) & d).

Application Exercise – Process Capability Which two processes are the most variable? __________ and _________ e f 2. Which two processes are not potentially capable? __________ and _________ e f a 3. Which two processes have the highest % out-of-spec? __________ and _________. c f b 4. Which process has the highest Cp? __________ d Cp=1.00 Cpk=0.53 5. For which three processes are Cpk and Cp equal? __________, ________ and ____________ c a b e 6. Which process is potentially capable but not meeting specs? __________ c d Cp=2.00 Cpk=1.33 7. Which process is supplying the most material at target value? __________ a 8. If these materials were stored in the warehouses of six different suppliers (all other things being equal) which one would you prefer? __________ e a 9. If the process average could be adjusted for future production, which supplier would you prefer? __________ f d 10. Calculate Cp and Cpk for c) & d). 43

Process Capability and Normality The calculated capability indices and the corresponding % out-of-spec values are only valid when the individual data points are normally distributed. Special causes tend to distort the Normal curve. 44