2. Statistical Process Control (SPC)

2.1 Control Charts

• Basic plots of statistical variation to show trends. Uses basic values like,


standards deviation



• the uses for control charts,

these give a measure of performance, and therefore we can estimate the benefits of process parameter adjustment.

process capability can be determined

process specifications can be made greater than process capability

can indicate when a process is out of control, and be used to reject a batch of product.

2.1.1 Sampling

• values used for control charts should be numerical, and express some desired quality.

• selecting groups of parts for samples are commonly done 2 ways.

INSTANT-TIME METHOD: at predictable times pick consecutive samples from a machine. This tends to reduce sample variance, and is best used when looking for process setting problems.

PERIOD-OF-TIME METHOD: Samples are selected from parts so that they have not been presented consecutively. This is best used when looking at overall quality when the process has a great deal of variability.

• Samples should be homogeneous, from same machine, operator, etc. to avoid multi-modal distributions.

• Suggest sample group size can be based on the size of the production batch


2.1.2 Creating the Charts

• The central line is an average of ‘g’ historical values.


• The control limits are +/-3 σ of the historical values


• At start-up these values are not valid, but over time it is easy to develop a tight set of values.

• For the non-technical operators there are a couple of techniques used.

- to simplify calculation of the control limits we can approximate σ with

2.1.3 Maintaining the Charts

• Over time σR and σX should decrease.

• If some known problems occurred that created out of control points, we can often eliminate them from the data and recreate the chart for more accurate control limits.


• If we recalculate values from the beginning there is no problem, but if we are using the numerical approximation (using constants from table B)


2.1.4 The s-Chart

• Instead of an R-chart we can use an s-chart to measure variance

• This chart will reduce the effect of extreme values that will occur with R charts. And, as the number of samples grows, so does the chance of extreme values to throw off the R-chart.


• the approximate technique is,


2.1.5 Interpreting the Control Charts

• We consider a point that lies outside of the 3σ control limits to be very unlikely, therefore the process is ‘out of control’

• In some cases a process may have points within the control limits, but in highly unlikely trends that indicate a process is out of control


2.1.6 Using the Charts for Process Control

1. change or jump in level (X)


indicates a discontinuity in the process, could be caused by material, new operator, etc.

e.g. measurement gauge has slipped.

2. Change or jump in level (R)

indicates a change in process accuracy, caused by failure of small parts, material, etc.

e.g. measurement gauge has stretched

3. trend, or steady change in level (X and R)


indicates a gradual change in the process caused by wear, aging, etc.

e.g. a tool is aging

4. Cycles


indicates time variance in process

e.g. shift change, day-night temp changes, etc.

5. Mixed Data


indicates multimodal distributions caused by mixed material batches, alternate operators, etc.

e.g. steel from scrap dealer, and from steelco is used.

6. Error


readings are incorrect because of misreadings, transcription error, etc.

e.g. a new operator measured the wrong dimension

2.1.7 Problems

Problem 2.1 Draw the detailed X, R, and s charts for the data below.

Problem 2.2 What problems can be seen in this control chart?

Problem 2.3 Draw the Pareto diagram for the data below. The data indicates the number of reported errors made when taking fast food orders by telephone

2.1.8 Control Charts for Attributes

• ATTRIBUTE: a nonconformity that cannot, or will not be measured

• NONCONFORMITY- A flaw in a product that will not always make it unsellable.

• p-chart (for nonconforming units)


• p-chart control limits can vary if the sample size varies.


• np-chart (nonconforming units): almost identical to p-chart, except not normalized to 0 to 1 scale, so it is more sensitive to sample size changes, but it is easier to maintain.


• c-chart (for count of nonconformities): Best used when there is a small number of errors in a huge sample space, and the nonconforming events are independent.


• u-chart: similar to c-chart, except it can handle different sample group sizes (e.g. a rivet department dealing with different sizes of riveted parts).


• choosing an attribute chart


2.2 Inspection for Quality

2.2.1 Acceptance of Lots

• What? screening of arriving product lots to ensure adequate quality for use

• Why? Because putting parts into production when they are out of spec. will probably result in out of spec. product. This will cost much more.

• Advantages,

focuses quality problems on source

reduces inspection required

can use destructive testing

rejection of entire lots increases supplier quality incentives

uses known risks and probabilities

• Disadvantages,

good/bad lots may be rejected/accepted without reason

planning and documentation required

only describes part of a lot

2.2.2 Screening

• When every part in an incoming lot is checked, this is called screening.

• The non-conforming parts can be removed, and replaced with good parts

• Screened lots can be mixed with good lots to improve the AOQ


2.2.3 The Cost of Sampling

• A simplified economic comparison of inspection versus screening can be made with the following method,



2.3 Single/Double/Multiple Sampled Plans

• SINGLE: just take a sample from the lot, examine, and reject. e.g. a lot of 5000 has 200 samples removed, if more than 2 non-conforming samples found, the lot is sent back.

• DOUBLE: One sample is taken. If it conforms, keep the lot. If there are too many non-conforming send back the lot, otherwise take a second sample. If the combined first and second samples have too many rejects, then send the lot back.

• MULTIPLE: like the double sampled plan, but extended for a larger number of tests.

• the accept/retest/reject limits are set by statistical methods.

• Lot Sample Selection, must be as random as possible,

use a random number generator,

use a paper based method (e.g. pg. 288 in text)

• What to do when a lot is rejected


2.4 Operating Characteristic (OC) Curves

• Used to estimate the probability of lot rejection, and design sampling plans.


• Drawing the single sampling curve (assuming Poisson distribution)


• Double sampling curves



• Factors that vary OC curves


• Producers/Consumers risk


• The basic trade-off to be considered when designing sampling plans.

The producer does not want to have lots with higher rejects than the AQL to be rejected. Typically lots have acceptance levels at 95% when at AQL. This gives a producers risk of α = 100%: 95% = 5%. In real terms this means if products are near the AQL, they have a 5% chance of being rejected even though they are acceptable.
The consumer/customer does not want to accept clearly unacceptable parts. If the quality is beyond a second unacceptable limit, the LQL (Lower Quality Level) they will typically be accepted 10% of the time, giving a consumers risk of β = 10%. This limit is also known as the LTPD (Lot Tolerance Percent Defective) or RQL (Rejectable Quality Level).

• AOQ (Average Outgoing Quality)


• AOQ (Average Outgoing Quality): a simple relationship between quality shipped and quality accepted.

• ASN (Average Sample Number): the number of samples the receiver has to do




• On the other hand, given consumers risk (β) and Lower Quality Level (LQL), we can follow a similar approach, still using the table on pg. 314

• Given α and AQL, and β and LQL we can also find a best fit plan through trial and error.


2.4.1 Problems

Problem 2.4 Show the effect of lot screening if the sample size is n=100 and the reject limit is c=1.

Problem 2.5 a) Develop a double sampling plan OC curve given that,

N = 1000

n1 = 50

c1 = 2

r1 = 4

n2 = 100

c2 = 3

b) What is the AOQL?

Problem 2.6 a) Develop Operating Characteristic (OC) curves for the three cases below,

i) N = 1000, n = 20, c = 4

ii) N = 1000, n = 40, c = 8

iii) N = 1000, n = 80, c = 16

b) For each curve indicate the AQL and RQL for a producer risk of 5% and a consumer risk of 10%

Answer 2.6 for n=20,c=4, AQL=10%, RQL=40%, for n=40,c=8, AQL=11.8%,RQL=33%, for n=80,c=16, AQL=13.8%,RQL=N.A.
2.5 MIL-STD-105D AND ANSI/ASQC Z1.4-1981


• Began development 1942

• A set of tables for single, double and multiple inspections.

• These tables are designed to favor producers risk.

• Three types of inspection levels,

(III) Tightened: for new suppliers, or suppliers with poor quality history

(II) Normal: for poor quality history

(I) Reduced: for suppliers with a good quality history

• AQL (Acceptable Quality Level): chosen by experience and requirement


1. Sample size code can be found on table (pg. 329) using lot size and inspection level.


2. Using AQL, and sample size code, find the appropriate sample plan table, and look up the values. (plan varies by single, double, multiple sampling)


3. Start at normal inspection level, then tighten or loosen as required.


• An example of use for these tables is an aerospace manufacturer is receiving batches of parts in batches of 5000. These parts will be used in the landing gear, and are considered critical. Find sampling plans (single, double, and multiple) for this batch. Find a single sampling plan if the two last batches were rejected.

2.5.1 Problems

Problem 2.7 Given a lot of 3000 and an AQL of 1%, compare level I, II, and III plans using OC curves on the same graph.

Problem 2.8 Draw the X, R, and s charts for the data below, using exact calculations: Then calculate the Control Limits using approximate techniques.

Answer 2.8 1. Based on exact calculations

Approximate control limits

UCLX = 13.671

LCLX = 11.157

UCLR = 5.772

LCLR = 0.228

UCLs = 1.954

LCLs = 0.122

Problem 2.9 The data below was collected for a factory that manufactures telephones. Each day they make 10,000 phones, inspect 500, and expect to find 4 nonconforming (e.g. sometimes a button is put in upside down). The chart below lists nonconformity data for the previous week. Select the appropriate type of control chart, and plot the data.

Answer 2.9 We have nonconforming units with a constant sample size, therefore select an np chart.

Problem 2.10 Use a Frequency Division Analysis Sheet and 2 other methods to determine if the data listed below is normal. Date: 5, 7, 4, 3, 5, 9, 6, 4, 6, 7, 2, 3

Answer 2.10

Problem 2.11 xVery Short answers

a) what is the difference between the two histograms below?

b) Give a practical example of what a bi-modal distribution would indicate in quality control.

c) Describe the difference between Cp and Cpk.

d) How can Design of Experiments help an engineer improve a process?

e) Why would a single sampled plan be preferred to a double sampled Lot Acceptance Plan. Use an example in your answer.

f) Give a single sampling plan for a sample of 188 with an Acceptance Quality Level (AQL) of 0.4.

Answer 2.11 a) The histogram on the left indicates grouped data, the histogram on the right indicates ungrouped data.

b) Bimodal distributions indicate a mixture of two sample populations, as if a box of parts has been filled by two different machines.

c) Cp indicates process variance to tolerance only, while Cpk incorporates the shift of the process center also.

d) DOE allows the most sensitive process parameters to be controlled first.

e) Double sampled plans are good when we have something like a good supplier, and destructive testing. Single sampled plans are better when quality must be very high.

f) Look at charts in book. sample = 188, 2 or less must fail for acceptance.

Problem 2.12 Draw the Operating Characteristic (OC) curve for a batch of 10,000 with a sample size of 30, and a non-conforming reject level of 2. Identify the producers risk when a quality of 2% nonconforming is produced.

Answer 2.12

Problem 2.13 a) Develop an Ishikawa diagram to identify problems when painting a house.

b) make a tally sheet based on your Ishikawa diagram, and suggest some data.

c) Draw a Pareto chart of the data.

Answer 2.13 a)



Problem 2.14 Short Answer Questions

a) What is meant by grouped and ungrouped data?

b) What is a mode? Give an example.

c) Describe the Instant Time Method

d) What problems can be seen in this control chart?

Answer 2.14 Short answer questions

a) Ungrouped data is continuous, and can be grouped into “slots”

b) It is a value in a set of grouped data that reoccurs the most. It tends to indicate where a distribution is centered, or some unnatural patterns. e.g., 10 out of 30 students get a mark of 85% on the test, indicating copying.

c) A set of samples are taken from a process in one instant in time. The samples are all from a very short period of production time.


Problem 2.15 a) Is this data normal?

b) For the data draw an X and R chart

Answer 2.15 a) May as well use a3 and a4, but other methods could be used. X = 148.3125, s = 9.0459, a3 = .3900 (skewed to right, positive direction), a4 = 1.958 indicates mesokurtic.


Problem 2.16 Write the values displayed on the vernier scales below.

Problem 2.17 Draw the appropriate diagram for the data below. The data indicates sources of photocopy problems.

Problem 2.18 a) The data below has been plotted from QC samples, add the lines required to complete the X chart.

b) We are designing a new product and want to produce it on the process described in #3 a). What tolerance is required to obtain 6 sigma quality?(6%)

Problem 2.19 Given the attached “Frequency Distribution Analysis Sheet”

a) What is the average?

b) What value is at +2 sigma?

c) What is the mode?

d) What is the median?

Problem 2.20 Are the ISO9000 quality standards part of SPC? Justify your answer.(5%)

Problem 2.21 a) Given the results from a Designed Experiment, as listed below, what are the main effects of A and B?

b) Draw a graph from the data in a), and explain the significance of the effects.(5%)

Problem 2.22 At Joe’s Barbatorium a once a day inspection is done of one customer at random. The QC inspector checks 1000 hairs for length, any over 1” are considered defective. For the last week the counts have been 10, 2, 25, 0, 7.

a) Select the correct type of attribute chart to track quality.(4%)

b) Draw the chart selected in a).(11%)

Problem 2.23 Develop a double sampling plan OC curve given that, (20%)

N = 1000

n1 = 50

c1 = 2

r1 = 4

n2 = 100

c2 = 3

Problem 2.24 Generally, why should production try to meet specifications, not exceed them?

Problem 2.25 a) Draw a fishbone diagram for the production of cookie dough. The quality to be measured is the ratio of chocolate chips to dough per cubic meter. Note: the components are weighed separately, and then mixed together in a large tub.

b) Select the most reasonable causes from a), make up a tally sheet, fill it with some data, and draw a Pareto chart. You must consider that there are three different operators that may do the weighing and measuring.

Problem 2.26 a) We have found a box of gum for practical jokes. Most of the gum is normal, but some pieces will result in purple tongues when chewed. Would inductive, or deductive statistics be used to determine how many of the sticks are for jokes without chewing them all?

b) Assume we are counting the number of fish in a pet store aquarium. Give an example of a grouped, and ungrouped count.

Problem 2.27 Four samples have been taken at the start of a new process run. But, one of the values, X1, was accidentally erased after the calculations were done. Using the data below, find the missing value.

Problem 2.28 Is the data below normal? Justify your answer.

Problem 2.29 The data below was measured over a two week period for a 1.000” shaft with a tolerance of +/- 0.010” .

a) Draw accurate X, R and s control charts.

b) Determine if all the data values are normal (56 in total) using the normal distribution graph paper.

c) Determine Cp and Cpk.

d) What would the tolerance have to be if we required 6 sigma quality?

Problem 2.30 A manufacturer ships 10,000 balloons a day to McDonald’s. A daily sample of 50 are removed and tested. The table below lists the number that burst when inflated.

a) Select the correct type of control chart, and draw it accurately.

b) If McDonalds sets a maximum of 3 rejects in 50 samples, draw the OC curve.

c) For the OC curve drawn in b), identify the consumers risk when 3% of the balloons are non-conforming.

Problem 2.31 List some material and process variables that can affect quality.

Answer 2.31 Quality can be affected by speeds, feeds, hardness, surface contamination, purity, temperature, etc.

Problem 2.32 Why is standard deviation important for process control?

Answer 2.32 Standard deviation is a measure of distribution in a process that varies randomly with a Gaussian distribution.

Problem 2.33 Describe SPC (Statistical Process Control).

Answer 2.33 Statistical process control uses statistical methods to track process performance, and then probability to estimate when it is undergoing some non-random or systematic change. When this occurs the process is no longer under control.

Problem 2.34 What is the purpose of control limits in process monitoring?

Answer 2.34 Control limits are hard boundaries that the process should not work outside. If a value is outside these limits the process should be stopped.

Problem 2.35 What is process capability and how is it used?

Answer 2.35 Process capability is used to measure the precision of a machine.

Problem 2.36 What would happen if the SPC control limits were placed less than +/-3 standard devistions.

Answer 2.36 There will be more of the distribution outside the limits, and hence more rejects.

Problem 2.37 What factors can make a process out of control.

Answer 2.37 Factors can put a process out of control. A sudden jump will be caused by a change in tools/operators/components/materials. A slow change in the mean will result from slipping gages/tools, or tool wear. A single anomaly can lead to a single control point outside the range (e.g. somebody drops their gum in the machine).

Problem 2.38 What is acceptance sampling and when should it be used?

Answer 2.38 Acceptance sampling is used for parts that are being made by another manufacturer who does not provide process quality information. This technique uses random inspection of parts as they arrive to ensure conformance to quality limits.
2.6 Quality Control Projects

2.6.1 Measuring for Quality Control


• You will be expected to gain practice using the Metrology equipment. Beyond this the students will be expected to make sense of the numbers collected, as well as how they relate to variations caused by processes. The student should also gain an appreciation for software support tools.

Suggested Procedure:

1. Students will be assigned a set of similar parts.

2. Each part will be measured for specified dimensions/features during normal laboratory time. EACH part is to be measured by EACH student 5 times. Each measured dimension will be measured using at least two different Metrology methods.

3. The results may be analyzed using methods, such as,

distribution for individuals, parts, overall, etc.
calculation of parameters; σ, mean, etc.
process capability.

4. TurboSPC is used on the measured data, and is available on a number of computers.

5. A report and presentation (10 minutes) will be prepared for the lab session

2.6.2 Evaluation of Metrology Equipment


• To understand the limitations in the equipment used for Metrology.

Suggested Procedure:

1. Select a measuring machine used in Project 1. This should not have been used by any other groups in your section.

2. Measurement standards will be used to make repeated measurements, as in Project 1.

3. The capabilities of the process will be determined using known methods.

4. Comparison to results in Project 1 can be made.

5. A report and presentation will be completed.