7. Process Capability & SQC

Process Process Capability & Capability & Statistical Statistical

Quality ControlQuality Control

A B RajuA B Raju

Statistical Quality Control Statistical Quality Control (SQC)(SQC)

Define Statistics


Statistics is a science concerned with the collection, organisation, analysis, interpretation and presentation of data.


Statistics is a science concerned with the collection, organisation, analysis, interpretation and presentation of data.

SQC is defined asThe technique of applying statistical methods based upon the theory of probability to quality control problems with the purpose of establishing quality standard and maintaining adherence to those standards in most economical manner.

SQCSQCStatistical Processes & Methods:

SQCSQCStatistical Processes & Methods:• Data Collection

SQCSQCStatistical Processes & Methods:• Data Collection• Organisation & presentation

- Frequency distributions - Histograms



• Descriptive Measures- Central tendency (mean, median, mode)- Dispersion (range, standard deviation, variance)



• Descriptive Measures- Central tendency (mean, median, proportions)- Dispersion (range, standard deviation, variance)

• Statistical Inference - Hypothesis testing

- Experimental design



• Descriptive Measures- Central tendency (mean, median, proportions)- Dispersion (range, standard deviation, variance)

• Statistical Inference - Hypothesis testing

- Experimental design• Predictive Statistics

- Correlation analysis- Regression analysis

SQCSQCDescriptive statistics:

SQCSQCDescriptive statistics:- Tabular summarisation : tally marks,

frequency


frequency- Graphical summarisation: Histogram


frequency- Graphical summarisation: Histogram- Box and whisker plots


frequency- Graphical summarisation: Histogram- Box and whisker plots- Quantitative methods

- Mean / average- Standard deviation

SQCSQCDescriptive statistics:- Tabular summarisation : tally marks, frequency- Graphical summarisation: Histogram- Box and whisker plots- Quantitative methods

- Mean / average- Standard deviation

- Probability Distributions- Normal- Exponential- Weibull- Poisson- Bionomial

SQCSQC• Pareto Principle/Analysis

Analysis of data to where we see a few contributors are responsible for bulk of costs, damages etc.,

This principle can be applied to “cost of quality” , “suppliers” , “type of defects” etc.,

7 Quality Control Tools7 Quality Control Tools• Pareto Analysis –diagram for tallying the

percentage of defects resulting from different causes to identify major quality problems

• Flow Chart – diagram of the steps in a process• Check sheet – a fact finding tool for tallying the

number of defects for a list of previously identified problems causes

• Histogram – diagram showing the frequency of data related to a quality problem

• Scatter diagram – graph showing the relationship between 2 variables in a process

• Control chart – chart with upper and lower limits• Cause and effect diagram – also called fish bone

diagram

Statistical Process Control Statistical Process Control (SPC)(SPC)

SPC can be defined as the application of statistical methods to the measurement and analysis of variation in any process.

Quality Measurement in Manufacturing

- Quality of out put: First qualityAmount of scrap/reworkNo. of defects found after inspection

- Quality of input: Mfg.down-timeInput meeting specs.Instrument calibration

Process variationProcess variation• Processes that provide goods and services

usually exhibit some variation in their output.

• Variation that is caused by factors that can be clearly identified and possibly even managed is called “Assignable variation”

• Variation that is inherent in the process itself is called “Common variation”

SPCSPC• Statistical Control Charts

It is a graphic comparison of process performance data to compute control limits.

Process variations are because of two causes1. Common (random or chance)2. Special (Assignable)

The main objective of control charts is detecting assignable (special) causes of variation in a process.

A Process that is operating without special causes of variation is said to be in “in a state of statistical control”.

Control limits are calculated using the laws of probability.

SPCSPCRandom variations are acceptable but in the new age

competition lower variation has become an important tool.

Steps in setting up a control chart:1. Choose the characteristic to be charted. (a pareto

analysis of high occuring defects or Identifying process variables and conditions which contribute to the end product characteristics).

2. Choosing the type of control chart3. Deciding the central line4. A system for collecting data5. Calculating control limits6. Plotting data and interpreting the results.

SPCSPCSome of the control charts are

- R Charts (amount of dispersion – Range)

- X¯ Charts (process average)- P Charts (Proportion non-conforming)- C charts (No. of non-conforming)

SPCSPC• X¯ and R¯ Charts

For a normal distribution scenario, the control limits at ±3бUCL = X¯+A2R¯LCL = X¯ -A2R¯Short cut is UCL = D4R¯LCL = D3R¯

X¯ = Grand average = average of the sample averagesR¯ = Average of the sample rangesA2 = constant found from the tablesD3 & D4 = Again constants that can be obtained from tablesn = No. of readings in each sample

SPCSPCConstants for X and R charts

n A2 D3 D4 d2

2 1.88 0 3.268 1.128

3 1.023 0 2.574 1.693

4 0.729 0 2.282 2.059

5 0.577 0 2.114 2.326

6 0.483 0 2.004 2.534

7 0.419 0.076 1.924 2.704

8 0.373 0.136 1.864 2.847

9 0.337 0.184 1.816 2.97

10 0.308 0.223 1.777 3.078

SPCSPCProcess Capability :Most widely adopted is the ±3б (a total of 6

sigma)Cp Capability ratio = Specification Range

process capability= UCL-LCL

6sS=R¯/d2

S = an estimate of б

SPCSPC

• Capability Index Cpk

Cp refers to variation in a process

Cpk reflects both variation and location of the process average

Cpk = min (X¯-LCL , UCL-X¯)

3s 3s

SPCSPCProcess capability and Product outside specification limits

Process capability Total product outside 2 sided spec. limits

0.5 13.36%

0.67 4.55%

1 0.3%

1.33 64 ppm

1.63 1 ppm

2 0

Limits % total area within specified limits

X¯± 0.6745 s 50

X¯± s 68.26

X¯± 2s 95.46

X¯± 3s 99.73

SPCSPCFor exampleUCL = 20 LCL = 8X¯=16 s=2

Capability ratio = UCL-LCL/6s=(20-8)/12= 1.0

Capability index=min of (16-8)/6, (20-16)/6= 0.67

SPCSPCP-Chart:With a p-chart, a sample is taken periodically from the production or

service process and the proportion of defective items in the sample is determined to see if the proportion falls within the control limits on the chart.

UCL = p¯+zбp

LCL = p¯+zбp

z = no. of standard deviations from the process averagep¯= the sample proportion defective; an estimate of the process

averageБp = the standard deviation of the sample proportion

The sample standard deviation is computed as

Бp = √[p¯(1-p¯)/n]

Where n is the sample size

SPCSPCC-chart:A c-chart is used when it is not possible to

compute a proportion defective and the actual no. of defects must be used

UCL = c¯+zбc

LCL = c¯-zбc

бc = √c¯

Acceptance SamplingAcceptance Sampling• Need of sampling

Acceptance SamplingAcceptance Sampling• Need of sampling• Cost to justify inspection – 100%

inspection is justified when the cost of a loss incurred by not inspecting is greater than inspection

– Eg.: fault of item results in Rs.10 loss and average defects are 3%, then the expected cost of faulty item is ).03 x Rs.10 = Rs.0.30 each. So if cost of inspection of each item is less than Rs.0.30 then only it makes sense to inspect 100%

Acceptance SamplingAcceptance Sampling• Need of sampling• Cost to justify inspection – 100% inspection is

justified when the cost of a loss incurred by not inspecting is greater than inspection


• Sampling is needed only when the quantity of output is very high.

Acceptance SamplingAcceptance Sampling• Need of sampling• Cost to justify inspection – 100% inspection is justified when

the cost of a loss incurred by not inspecting is greater than inspection



• Objective (of the producer) is to ensure that the sampling plan has a low probability of rejecting good lots. Lots are classified as good if they contain no more than a specified level of defectives (known as acceptable quality level – AQL).

Acceptance SamplingAcceptance Sampling• Need of sampling• Cost to justify inspection – 100% inspection is justified when

the cost of a loss incurred by not inspecting is greater than inspection



• Objective (of the producer) is to ensure that the sampling plan has a low probability of rejecting good lots. Lots are classified as good if they contain no more than a specified level of defectives (known as acceptable quality level – AQL). Eg.: HP’s insistence on 2% AQL from a Japanese vendor

Acceptance SamplingAcceptance Sampling• Need of sampling• Cost to justify inspection – 100% inspection is justified when the

cost of a loss incurred by not inspecting is greater than inspection


• Sampling is needed only when the quantity of output is very high. • Objective (of the producer) is to ensure that the sampling plan has

a low probability of rejecting good lots. Lots are classified as good if they contain no more than a specified level of defectives (known as acceptable quality level – AQL). Eg.: HP’s insistence on 2% AQL from a Japanese vendor

• Objective (from consumer’s side) is to ensure that the sampling plan has a low probability of accepting a bad lot. Lots are classified as low quality if the percentage of defectives is more than a specified amount called Lot tolerance percent defective (LTPD)

7. Process Capability & SQC

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