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Quality

Statistical Process Control:

Control Charts

Process Capability

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SPC

Traditional view:

Statistical Process Control (SPC) is a statistical method of separating variation

resulting from special causes from natural variation in order to eliminate the

special causes, and to establish and maintain consistency in the process,

enabling process improvement .

A broader view is (Goetsch):

SPC does not refer to a particular technique, algorithm or procedure

SPC is an optimisation philosophy concerned with continuous process

improvements, using a collection of (statistical) tools for:

data and process analysis

making inferences about process behaviour

decision making

SPC is a key component of Total Quality initiatives

In either view, Variation is a key issue

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SPC

Ultimately, SPC seeks to maximise profit by:

improving product quality

improving productivity

streamlining process

reducing waste

reducing emissions

improving customer service, etc.

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SPC-Tools

Commonly used tools in SPC include:

Flow charts

Run charts

Pareto charts and analysis

Cause-and-effect diagrams

Frequency histograms

Scatter diagrams

Control charts

Process capability studies

…

These tools are usually used to complement each other, rather than employed as stand-alone techniques.

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Statistical control

Processes that are not in a state of statistical control:

show excessive variations

exhibit variations that change with time

A process in a state of statistical control is said to be statistically

stable.

Control charts are used to detect whether a process is statistically

stable or not.

Control charts differentiates between variations: that are normally expected in the process due chance or common

causes (many, small, unavoidable, chance causes jointly affecting the

system)

that change over time due to special causes (assignable cause,

unexpected).

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Variation

Variations due to common causes:

have (usually) small effect on the process

are inherent to the process because of:

the nature of the system

the way the system is managed

the way the process is organized and operated

can only be removed by

making modifications to the process

changing the process hence its removal are mainly the responsibility of higher

management (e.g., equipment/technology,…)

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Variation

Variations due to special causes are:

localised in nature

exceptions to the system

considered as abnormalities

often specific to a:

certain operator

certain machine

certain batch of material, etc.

Investigation and removal of variations due to special causes are key to process improvement

Sometimes, the frontier between common and special causes may not be very clear due the complexity of the systems/process.

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1-Control Charts: how they work

The principles behind the application of control charts are very

simple and are based on the combined use of:

Run Charts

Hypothesis testing

The procedure is:

sample the process at regular intervals.

plot the statistic (or some measure of performance), e.g.

mean, range, number of defects, etc.

check (graphically) if the process is under statistical control

(if the process is not under statistical control, do something

about it).

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Control Charts

Control charts make assumptions about the plotted

statistic, namely:

it is independent, i.e., a value is not influenced by its

past value and will not affect future values.

it is normally distributed, i.e., the data has a normal

probability density function.

The assumptions of normality and independence

enable predictions to be made about the data.

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Control charts provide a graphical means for testing hypotheses about the data being monitored.

Consider the commonly used Shewhart Chart as an example.

.

Common (chance) and special (assignable) causes

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General model for a control chart

General model for a control chart

UCL = μ + kσ

CL = μ

LCL = μ – kσ

where μ is the mean of the variable, and σ is the standard deviation

of the variable. UCL=upper control limit; LCL = lower control limit;

CL = center line where k is the distance of the control limits from the

center line, expressed in terms of standard deviation units. When k

is set to 3, we speak of 3-sigma control charts. Historically, k = 3 has

become a standard* in industry .

*some industries use higher values; a trend is the 6-sigma approach.

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Types of control charts

Different charts are used depending on the nature of the charted data. Commonly used charts are:

for Variables

The data is measured along a continuous scale such as length,time, weight, viscosity, breaking strength, temperature, etc.

Provide more information than discrete data.

for Attributes (countable)

An attribute is an intrinsic property of a given item that either does or does not exist. Hence, the data assumes only two values: good-bad, pass-fail, etc, and is measured by counting.

Being of qualitative nature, discrete data do not provide the degree to which a quality characteristic is nonconforming.

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I-Control Charts for Variables

Of the several CC, we will address the

Shewhart sample mean ( --Chart)

Shewhart sample range (R-Chart)

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I-Control Charts for Variables

X -Chart: deals with the average value in a

process (monitors the central tendency)

R-Chart: takes into count the range of the

values (reflects process variability).

Both are used together

R-chart usually applies when the number of

observations is less or equal to 10. For larger

number observations the S-chart (using the

standart deviation instead) may provide better

information.

Control charts for the mean and the range

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Step 1: Using a preselected sampling scheme and sample size

(number of samples and number of instances per sample), record

measurements of the selected quality characteristic on the

appropriated forms.

Step 2: For each sample calculate the sample mean and the range

Step 3: Obtain the center line and trial control limits (and perhaps

the warning limits)

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Step 2

n

X

X

n

i

i

1

)..1,()..1,( niXMINniXMAXR ii

For a Xbar-chart:

Averages

of the quality characteristic

over a sample of n units

Y- axis of a control chart Formulas

For a R-chart:

Ranges

of measures in a sample of n

units

Step 3

Instead of computing σx from the raw data we can use therelation between the process standard deviation σ and the meanof the ranges.

Multiplying factors (A2, D4, D3, d3,.., see table …) can be used to calculate the center line and the control limits.

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Central line Control limits

Upper limit Lower limit

m

X

X

m

j

j

1

m

R

R

m

j

j

1

Xbar-

chart

R-chart

Definition Formula

Average

on m samples of the quality

characteristic averages over n

units tested per sample

Average

on m samples of the ranges

calculated for each sample of n

units

RAXUCLX

2

RDUCLR 4

RAXLCLX

2

RDLCLR 3

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Factors for computing the Control limits,

X-Charts, R-Charts(example of a partial table)

n A2 D3 D4 d2

___________________________________________

2 1.880 0.000 3.267 1.128

3 1.023 0.000 2.574 1.693

4 0.729 0.000 2.282 2.059

5 0.577 0.000 2.115 2.326

6 0.483 0.000 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 2.970

10 0.308 0.223 1.777 3.078

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X and R chartsCOEFFICIENTS

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Run Rules

Run Rules are rules that are used to indicate out-of-statistical

control situations. Typical run rules for Shewhart X-charts with

control and warning limits are:

a point lying beyond the control limits.

2 consecutive (or 2 out 3 consecutive) points lying

beyond the 2-sigma warning limits

(0.025x0.025x100 = 0.06% chance of occurring).

4 out of 5 outside 1-sigma limit.

7 or more consecutive points lying on one side of the

mean ( 0.57x100 = 0.8% chance of occurring and

indicates a shift in the mean of the process).

5 or 6 consecutive points going in the same direction

(indicates a trend).

Other run rules can be formulated using similar principles.

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Examples of ooc: Points outside, cycles, trends, shifts

(clockwise)

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II-Control Charts for Attributes

Inlude:

p-Chart: a chart of the proportion defective in each sample set.

np-Chart: a chart of the number of defectives in each sample.

A defect is a non-conformance to a specific standard.

A defective part or product is one that exhibits one or more non-

conformances (it has one or more defects).

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p-Chart

A process control chart that measures a proportion of defective ornonconforming items within a sample or population.

An element or item under inspection may have one or more definable attributes (an attribute is an intrinsic property of a given item that either does or does not exist).

If any one of the inspected attributes is nonconforming, the entire item is counted as nonconforming.

The number of items in the sample that are determined to benonconforming are summed and a proportion of the total is evaluated.

P-Chart

Although the distribution of sample information follows abinomial distribution, that distribution can be approximated by anormal distribution (the sample size is large, n >= 25) with a:

mean of p

standard deviation of p

The 3s control limits are

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s p = npp /)]1)([(

UCLp = p + 3 s p

LCLp = p - 3 s p

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p-Chart Example

Control Chart

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.400

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Observations (Sample Number)

Pro

po

rtio

n

Proportion p-bar (0.199) UCL (0.368) LCL (0.030)

UCL (0.368)

LCL (0.030)

p-Chart (Example –Krishnamoorthi)

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Collecting a dataset for a p-Chart

The data required for a p-Chart should meet the following criteria:

Subgroup Sample Size (n) ≥ 50

Sample size may be up to 100 or more, but between 50 and

100 is often adequate.

Number of subgroups (number of samples taken) ≥ 25

When gathering data in the subgroup samples, it is preferable

(but not mandatory) that the sample sizes be the same.

If sample sizes are not the same, a different calculation will be

required.

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p-Chart with equal sample sizes

With equal sample sizes, the first step requires calculating the mean

subgroup proportion. This is accomplished by averaging all of the proportions

calculated from each sample set:

Mean Subgroup Proportion (Equal Sample Sizes)

where: Pi = Sample proportion for subgroup i

k = Number of samples of size n

k

P

p

k

i

i 1

Business Statistics, 5th Edition, Groebner, et al

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Example dataset for a p-Chart(Equal Sample Sizes; partial data:15 samples shown only)

The proportion of

defective or

nonconforming items in

each sample is

calculated by dividing

the number defective by

the sample size

Sample Nonconforming

Subgroup Sample

Size Proportion

1 10 50 0.200

2 11 50 0.220

3 10 50 0.200

4 9 50 0.180

5 8 50 0.160

6 11 50 0.220

7 10 50 0.200

8 9 50 0.180

9 10 50 0.200

10 9 50 0.180

11 11 50 0.220

12 13 50 0.260

13 9 50 0.180

14 8 50 0.160

15 9 50 0.180

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Example dataset for a p-Chart(Equal Sample Sizes)

For this example, there are 50 subgroups/samples (k) (only 15 shown

on previous slides…)

Applied Formula:

Estimate of the sample error for subgroup proportions :

where

= Mean subgroup proportion

n = Common Sample Sizep

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Example dataset for a p-Chart(Equal Sample Sizes)

The standard error will be used to calculate the upper and lower control

limits in the next step

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Example dataset for a p-Chart(Equal Sample Sizes)

With the Mean Subgroup Proportion, standard error, and upper /

lower control limits determined, fill out the table with the calculated

data:Partial table:

Sample Nonconforming Sample Size Proportion UCL (0.359) p-bar (0.192) LCL (0.025)

1 10 50 0.200 0.359 0.192 0.025

2 11 50 0.220 0.359 0.192 0.025

3 10 50 0.200 0.359 0.192 0.025

4 9 50 0.180 0.359 0.192 0.025

5 8 50 0.160 0.359 0.192 0.025

6 11 50 0.220 0.359 0.192 0.025

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Example dataset for a p-Chart (cont.)

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Evaluating the p-Chart

Four conditions or trends which warrant immediate attention:

Five sample means in a row above or below the target or

reference line.

Six sample means in a row that are steadily increasing or

decreasing (trending in one direction).

Fourteen sample means in a row alternating above and below the

target or reference line.

Fifteen sample means in a row within 1 standard error of the

target or reference line.

Summary on control charts (in Krishnamoorthi)

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2-Process Capability

Process capability–represents the performance of a process in a state

of statistical control. It is determined by the total variability that exists

because of al common causes present in the system. In some sense, is

a measure of the uniformity of a quality characteristic of interest.

It is an index of the uniformity of the output. A common measure is

given by 6σ, also called the process spread. If a normal distribution for

the output quality characteristic can be assumed, 99,74% of the

distribution will be 3σ to either side of the mean.

Process capability can also be viewed as the variation in the product

quality characteristic that remains after all special causes have been

removed: the product’s performance is predictable. This allows

determining the ability of the product to meet customer’s expectattions.

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Capability vs Control

Control

Capability

Capable

Not Capable

In Control Out of Control

IDEAL

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Process Capability Analysis (PCA)

Process capability analysis estimates process capability. It

involves estimating the process mean and standard deviation of

the quality characteristic.

Also, it allows estimating the form of the relative frequency

distribution of the characteristic of interest.

If the specification limits are known a PCA will also estimate the

proportion of of nonconforming product.

Natural tolerance limits (or process capability limits)

are influenced or established by the process itself.

represent the inherent variation in the quality characteristic

of the individual items produced by a process in control.

Are estimated based on the population values or, more

commonly from large representative samples.

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PCA

Assuming a normal distribution of the quality characteristic, the

upper and lower natural limits are, respectively:

UNTL= µ + 3σ,

LNTL= µ - 3σ,

where

µ is the process mean,

σ the process standard deviation.

Technically, there might not any relationship between the

process capability limits and the specification limits (USL

and LSL). The former is inherent to the process; the latter

is influenced or determined by the customer.

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PCA

Three cases may occur:

Process spread < specification spread, the proceses is

capable.

Process spread = specification spread, we have an

acceptable or adequate situation (in which there is no

room for error!). If the process goes out of control (change

of mean, and/or of the standard deviation), then a

proportion of the product will be nonconforming.

Process spread > specification spread, even though

the process is under control, this is an undesirable

situation. The process is not capable.

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Process capability index Cp

The process capability index is an easily measure of the

goodeness of the process performance. The following indices

are nondimensional (do not depend on a specific process).

The Cp index: Cp = (USL-LSL) / 6σ when σ is unknown it is replaced by its estimate. The sample

standard deviation s is one estimate of σ (standart deviation of

the process).

Cp > 1, the process is quite capable.

Cp = 1, the process is adequate

Cp < 1, the process is not capable

---------------------------------- ‘Sigma’ is a capability estimate typically used with attribute

data (i.e., with defect rates). Motorola’s ‘number of sigmas (6σ)

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Process capability index Cpk

However, the Cp index does not tackles the lack of centering. It only

measures the process variability in comparison to the specification

spread. It represents the process potential.

The Cpk index takes into account the lack of centering in the process.

It represents the actual capability of the process with existing

parameters values; it measures the process performance.

Cpk= Min {(USL- µ ) / 3σ, (µ - LSL) / 3σ}

Cpk > 1.33 (quite capable)

Cpk = 1.00 to 1.33 (capable)

Cpk < 1.00 (not capable)

Desirable values are Cpk > 1 (usually, customers require 1.33 or above to cover possible drift in the process center)

Example Cp,Cpk (Krishnamoorthi)

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Notes on Accuracy and Precision (Measurement Systems Analysis)

The measurement system analysis (MSA) is done to make sure

adequate measuring instruments are available for measurements to be

made. And to make certain that the measuring instruments will

measure the product characteristics and process parameters truthfully.

Definitions:

Bias: Difference between the average of readings and the true

value of the measurement. Small bias, good Accuracy

Precision: Measured by the std. dev. of the readings, which

measures the lack of Precision.

-----------------------------------------------------------------------------------

Accuracy: ‘exatidão’,‘grau de proximidade c/ o valor real

Precision: ‘grau de variação’

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Concepts of accuracy and precision(a note)

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Main references

Read:

First Course in Quality Engineering-Krishnamoorthi

(Chaps 4,5)

Quality Mgt-Goetsch/Davies (Chap 18)

---------

The Certified Manager of Quality Handbook-Westcott

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