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Copyright 2004 Pearson Education, Inc.

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13-1 Overview

13-2 Control Charts for Variation and Mean

13-3 Control Charts for Attributes

Chapter 13

Statistical Process Control

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SEMANA 16

Esther Flores Ugarte ESTADSTICA II

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Created by Erin Hodgess, Houston, Texas

Section 13-1 and 13-2

Overview and Control

Charts for Variation and

Mean

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Overview

Chapter 2 Review Center: Measure of center

Variation: Measure of the amount

that scores vary among themselves Distribution: Nature or shape

of distribution of the data

Outliers: Sample values that are veryfar away from majority of other values

Time: Changing characteristics of

the data over time

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Slide 6


Chapter 13The main objective is to address the

changing characteristics over time.

By monitoring this characteristic, we arebetter able to control the production of

goods and services, thereby ensuring

better quality.

Overview

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Control Charts for

Variation and Mean

Definition

Process Data

These are data arranged according to

some time sequence.

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Slide 8


Definition

Process Data

These are data arranged according to

some time sequence.

Important characteristics of process data

can change over time.

Control Charts for

Variation and Mean

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Definition

Run Chart

A run chart is a sequential plot of

individual data values over time.

One axis (usually vertical) is used forthe data values, and the other axis

(usually the horizontal) is used for

the time sequence.

Control Charts for

Variation and Mean

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Slide 10


Example: MeasuringAircraft Altimeters

Treating the 80 altimeter errors in Table 13-1 as a string of

consecutive measurements, construct a run chart by

using a vertical axis for the errors and a horizontal axis to

identify the order of the sample data.

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Slide 11



Treating the 80 altimeter errors in Table 13-1 as a string of

consecutive measurements, construct a run chart by

using a vertical axis for the errors and a horizontal axis to

identify the order of the sample data.

We notice that the values on the right of the chart show

more fluctuations than those on the left of the chart. This

increased variation could mean that the altimeters are notmeeting FAA standards. The movement should be

investigated.

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Slide 12


Definition

A process is statistically stable (or withinstatistical control) if it has naturalvariation, with no patterns, cycles, or anyunusual points.

Only when a process is statistically stablecan its data be treated as if they came from

a population with a constant mean,standard deviation, distribution, and othercharacteristics.

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Slide 13


Figure 13-2 Process with Patterns

That Are Not Statistically Stable

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A common goal of many differentmethods of quality control is this:

Reduce variation in aproduct or a service.

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Definition

Random variation

Random variation is due to chance; it is thetype of variation inherent in any process

that is not capable of producing every goodor service exactly the same way every time.

Assignable variation

Assignable variation results from causesthat can be identified.

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Definition

A control chart of a process characteristic (such

as mean or variation) consists of value plotted

sequentially over time, and it includes a center

line as well as a lower control limit (LCL) and an

upper control limit (UCL). The center line

represents a central value of the characteristicmeasurements, whereas the control limits are

boundaries used to separate and identify any

points considered to be unusual .

Control Chart for Monitoring

Variation: The RChart

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An Rchart is a plot of the sample ranges

instead of individual values and is used

to monitor the variation in a process.



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The center line would be located at R,which denotes the mean of all sampleranges as well as another line for thelower control limit and a third line forthe upper control limit.

An Rchart is a Plot of the sample ranges

instead of individual values and is used

to monitor the variation in a process.



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Notation

n= size of each sample, or subgroup

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x= mean of the sample means, which isequivalent to the mean of all sample

values combined

Notation

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x= mean of the sample means, which isequivalent to the mean of all sample

values combined

R= mean of the sample ranges (that is,the

sum of the sample ranges divided bythe number of samples)

Notation

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Point plotted: Sample ranges

Monitoring Process

Variation: Control Chart for R

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Center line: R

Monitoring Process


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Center line: R

Upper Control Limit (UCL): D4R

Monitoring Process


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Center line: R


Lower Control Limit (LCL): D3R

Monitoring Process


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Center line: R


Lower Control Limit (LCL): D3

R

where the values of D4 and D3 are found inTable 13-2

Monitoring Process


T bl 13 2

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Table 13-2

Control Chart

Constants

E l M i

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Refer to the altimeter errors in Table 13-1. Using thesamples of size n= 4 collected each day of manufacturing,

construct a control chart for R.

R= 19 + 13 + ...+ 63 = 21.220

E l M i

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R= 19 + 13 + ...+ 63 = 21.220

D3 = 0.000

D4 = 2.282 from Table 13-2

E l M i

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D4R= (2.282)(21.2) = 48.4

D3R= (0.000)(21.2) = 0.0



R= 19 + 13 + ...+ 63 = 21.220

D3 = 0.000

D4 = 2.282 from Table 13-2

MINITAB Di l

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MINITAB Display

RChart for Errors

I t ti

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Interpreting

Control Charts

Upper and lower control limits of

a control chart are based on the

actual behavior of the process,not the desired behavior.

Criteria for Determining When a

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Slide 36


Criteria for Determining When a

Process Is Not Statistically Stable

(Out of Statistical Control)

1. There is a pattern, trend, or cycle that is obviously

not random (such as those depicted in Figure 13-2).

2. There is a point lying beyond the upper or lower

control limits.

3. Run of 8 Rule: There are eight consecutive points all

above or all below the center line. (With a

statistically stable process, there is a 0.5 probabilitythat a point will be above or below the center line, so

it is very unlikely that eight consecutive points will

all be above the center line or below it.)

W ill l th th t f t l

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We will use only the three out-of-control

criteria listed previously, but some

businesses use additional criteria such as these:

There are 6 consecutive points all increasing or all

decreasing.

There are 14 consecutive point alternating between up

and down (such as up, down, up, down, and so on).

Two out of 3 consecutive points are beyond control

limits that are 2 standard deviation away from

centerline.

Four out of 5 consecutive points are beyond control

limits that are 1 standard deviations away from the

centerline.

E l St ti ti l

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Example: StatisticalProcess Control

Examine the Rchart shown in the Minitab display for the

preceding example and determine whether the process

variation is within statistical control.

E l St ti ti l

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1. There is a pattern, trend, or cycle that is obviously notrandom: Going from left to right, there is a pattern of

upward trend.

2. There is a point (the rightmost point) that lies above

the upper control limit.

E l St ti ti l

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We conclude that the variation (not necessarily the mean)of the process is out of statistical control. Because the

variation appears to be increasing with time, immediate

corrective action must be taken to fix the variat ion among

the altimeter errors.

Monitoring Process

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Point plotted: Sample means

Monitoring Process

Mean: Control Chart for x

Monitoring Process

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Center line: x

Monitoring Process


Monitoring Process

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Center line: x

Upper Control Limit (UCL): x+ A2R

Monitoring Process


Monitoring Process

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Center line: x

Upper Control Limit (UCL): x + A2R

Lower Control Limit (LCL): xA2R

Monitoring Process


Monitoring Process

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Center line: x

Upper Control Limit (UCL): x+ A2R

Lower Control Limit (LCL): xA2R

where the values of A2 found in Table 13-2.

Monitoring Process


Example: Manufacturing

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Example: ManufacturingAircraft Altimeters


construct a control chart for x. Based on the control chart

for xonly, determine whether the process is within

statistical control.


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x = 2.50 + 2.75 + ...+ 9.75 = 6.45

20



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x = 2.50 + 2.75 + ...+ 9.75 = 6.45

20

A2 = 0.729



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x = 2.50 + 2.75 + ...+ 9.75 = 6.45

20

A2 = 0.729

UCL: x+ A2R= 6.45 + (0.729)(21.2) = 21.9

LCL: xA2R= 6.45

(0.729)(21.2) =

9.0



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Examination of the control chart shows that the process

mean is out of statistical control because at least one of

the three out-of-control criteria is not satisfied.

Specifically, the third criterion is not satisfied becausethere are 8 (or more) consecutive points all below the

center line. Also, there does appear to be a pattern of an

upward trend. Again, immediate corrective action is

required to fix the production process.


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SEMANA 17

Esther Flores Ugarte ESTADSTICA II

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Slide 53


Created by Erin Hodgess, Houston, Texas

Section 13-3

Control Charts forAttributes

Control Charts

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Control Charts

for Attributes

These charts monitor the qualitativeattributes of whether an item has some

particular characteristic.

In the previous section, the charts

monitored the quantitative characteristics.

The control chart for p(or pchart) isused to monitor the proportion pforsome attribute.

Notation

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p= pooled estimate of proportion ofdefective items in the process

Notation

Notation

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=

Notation

total number of defects found among all items sampled

total number of items sampled

Notation

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p= pooled estimate of proportion of

defective items in the process

=

q= pooled estimate of the proportion ofprocess items that are not defective

Notation



Notation

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=


= 1p

Notation



Notation

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=


= 1p

n= size of each sample (not the number of

samples)

Notation



Control Chart for p

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Control Chart for p

Control Chart for p

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Center line: p

Control Chart for p

Control Chart for p

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p q

Center line: p

Upper control limit: p+ 3

Control Chart for p

n

Control Chart for p

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p q

Center line: p


Lower control limit: p 3

Control Chart for p

n

p q

n

Control Chart for p

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p q

p q

Center line: p


Lower control limit: p 3

Control Chart for p

n

n

(If calculation for the lower control limit results in a

negative value, use 0 instead. If the calculation for the

upper control limit exceeds 1, use 1 instead.)

Example: In 13 consecutive years 100,000 were

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randomly selected and the number who died from

respiratory tract infections is reported below. Construct a

control chart for pand determine whether the process is

within statistical control.

Number of deaths:

25 24 22 25 27 30 31 30 33 32 33 32 31


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Number of deaths:

25 24 22 25 27 30 31 30 33 32 33 32 31

p= 25 + 24 + 22 + ...+ 31 = 375 = 0.000288

(13)(100,000) 1,300,000


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Number of deaths:

25 24 22 25 27 30 31 30 33 32 33 32 31

p= 25 + 24 + 22 + ...+ 31 = 375 = 0.000288

(13)(100,000) 1,300,000

q= 1

p= 0.999712


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Number of deaths:

25 24 22 25 27 30 31 30 33 32 33 32 31

p= 25 + 24 + 22 + ...+ 31 = 375 = 0.000288

(13)(100,000) 1,300,000

q= 1

p= 0.999712

n

p qp+ 3 = 0.000449

n

p qp 3 = 0.000127

P Chart for Death from

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Respiratory Tract Infections


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Slide 70randomly selected and the number who died from




Number of deaths:

25 24 22 25 27 30 31 30 33 32 33 32 31

Interpretation: Using the three out-of-control criteria listed

in Section 13-2, we conclude that this process is out of

statistical control since from the p-chart there appears to be

an upward trend, and there are eight consecutive points all

lying above the centerline (Run of 8 Rule). Based on thesedata, public health policies affecting respiratory tract

infections should be modified to cause a decrease in the

death rate.

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