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STATISTICAL QUALITY CONTROL

by

W.A.B. Janith

( SC/2007/6624 )

A Statistical thesis submitted to the Science faculty of Ruhuna University

in partial fulfillment of the requirements for the degree of

Bachelor of Science

Department of mathematics

University of Ruhuna

October 2010


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ACKNOWLEDGMENTS

I take this opportunity to express my profound sense of gratitude and respect to all

those who helped me throughout the duration of this report. I express my sincere

gratitude and thankfulness towards Prof. L.A.L.W.Jayasekara, Senior Lecturer in

Mathematics, University of Ruhuna for spent his valuable time for this lecture period. As

well as I offer my sincere thanks and sense of gratitude to Mr. B.G.S.A.Pradeep Department

of Mathematics, University of Ruhuna, who conducted SQC lecturer series so our Quality

control knowledge was improved

I am grateful to all our friends for providing critical feedback & support

whenever required.

W.A.B. Janith

SC/2007/6624

University of ruhuna


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Contents

1. Introduction statistical quality control....032. Useful Statistical distributions .04

2.1. Important continues distribution.042.2. Important discrete distribution 09

3. Statistical quality control method ..133.1. Control Charts For Variables.133.2. Control Charts For attributes ..203.3. Six Sigma Quality..25

4. Sampling techniques...255. Discussion 28


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Introduction statistical quality control

This is a modern definition of quality

Statistical Quality Control is a method by which companies gather and analyze data on the

variations which occur during production in order to determine if adjustments are needed

(Ebert & Griffin, 2005, p. 214). One of the most common methods used in order to achieve

this goal is the quality control chart. The charts are used to provide a visual graphic display

of instances when a process is beginning to go out of control. The purpose of the chart is toindicate this trend in order that the system may be brought back into control.

Statistica1 quality control (SQC) is the term used to describe the set of statistical tools

used by quality professionals. Statistical quality control can be divided into three broad

categories:

1. Descriptive statistics are used to describe quality characteristics and relationships.Included are statistics such as the mean, standard deviation, the range, and a

measure of the distribution of data.

2. Statistical process control (SPC) involves inspecting a random sample of theoutput from a process and deciding whether the process is producing products with

characteristics that fall within a predetermined range. SPC answers the question of

whether the process is functioning properly or not.

3. Acceptance sampling is the process of randomly inspecting a sample of goods anddeciding whether to accept the entire lot based on the results. Acceptance sampling

determines whether a batch of goods should be accepted or rejected.

Every product possesses a number of elements that jointly describe what the user or

consumer thinks of as quality. These parameters are often called quality characteristics.Sometimes these are called critical to quality (CTQ) characteristics. Quality characteristics

may be of several types;

1. Physical: length, weight, voltage, viscosity

2. Sensory: taste, appearance, color

3. Time Orientation: reliability, durability, serviceability

Since variability can only be described in statically terms, Statistical methods play a central

role in quality improvement efforts. In the application of statical methods to quality

engineering, it is fairly typical to classify data on quality characteristics as either attributes

or variables data are usually continuous measurements. Such as length, voltage, or


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viscosity. Attributes data, on the other hand, are usually discrete data, often taking the form

of counts. We will describe statistical-based quality control tools for dealing with both

types of data.

Useful Statistical distributions

A probability distribution is a mathematical model that relates the value of the variable

with the probability of occurrence of that value in the population. In other words, we might

visualize layer thickness as a random variable, because it take on different values in the

population according to same random mechanism, and then the probability distribution of

layer thickness describes the probability of occurrence of any value of layer thickness in

the population. There are two types of probability distributions.

Important continues distribution

The Normal Distribution


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Central Limit theorem

Practical interpretation the sum of independent random variables is approximately

normally distributed regardless of the distribution of each individual random variable in

the sum


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The Lognormal Distribution


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The Weibull Distribution

When = 1, the Weibull distribution reduces to the exponential distribution


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Important discrete distribution

The Hypergeometric Distribution


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The Binomial Distribution

Basis is in Bernoulli trials

The random variablexis the number of successes out ofn Bernoulli trials with constant

probability of successp on each trial

The Poisson distribution

Frequently used as a model for count data


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The Pascal Distribution

The random variablexis the number of Bernoulli trials upon which the rth success occurs

And the geometric distribution has many useful applications in SQC

probability plots

Determining if a sample of data might reasonably be assumed to come from a specificdistribution

Probability plots are available for various distributions

Easy to construct with computer software (MINITAB)

Subjective interpretation

Normal probability plots


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Other Probability Plots

What is a reasonable choice as a probability model for these data?

So we can chose convenient probability model using probability plot

Minimum Goodness of Fit is 0.724 so lognormal base e probability is a convenient model.


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Statistical quality control method

Statistical quality control methods extend the use of descriptive statistics to monitor thequality of the product and process. As we have learned so far, there are common and

assignable causes of variation in the production of every product. Using statistical process

control we want to determine the amount of variation that is common or normal. Then we

monitor the production process to make sure production stays within this normal range.

That is, we want to make sure the process is in a state of control. The most commonly used

tool for monitoring the production process is a control chart. Different types of control

charts are used to monitor different aspects of the production process. In this section I will

try to explain how to use control charts.

Control chart

The foundation for SQC (Statistical quality control) was laid by Dr. Walter Shewart

working in the Bell Telephone Laboratories in the 1920s conducting research on methods

to improve quality and lower costs. He developed the concept ofcontrolwith regard to

variation, and came up with SQC Charts which provide a simple way to determine if the

process is in control or not.

Dr. W. Edwards Deming built upon Shewarts work and took the concepts to Japan WWII.

There, Japanese industry adopted the concepts whole-heartedly. The resulting high quality

of Japanese products is world renowned. Dr. Deming is famous throughout Japan as a "God

of quality". Today, SQC is used in manufacturing facilities around the world. SQC is rapidly

becoming required in Healthcare and other service industries as well.

Shewhart [1931, p.6] defined control by saying:

a phenomenon will be said to be controlled when, through the use of pastexperience, we can predict, at least within limits, how the phenomenon maybe expected to vary in the future. Here it is understood that prediction withinlimits means that we can state, at least approximately, the probability thatthe observed phenomenon will fall within the given limits.

Control charts show the variance of the output of a process over time, such as the time it

takes for a patient to see a doctor in the immediate care facility. Control charts compare

this variance against upper and lower control limits to see if it fits within the expected,

specific, predictable and normalvariation levels.

CONTROL CHARTS FOR VARIABLES

Control charts for variables monitor characteristics that can be measured and have a

continuous scale, such as height, weight, volume, or width. When an item is inspected, the

variable being monitored is measured and recorded. For example, if we were producing

candles, height might be an important variable. We could take samples of candles and

measure their heights. Two of the most commonly used control charts for variables

monitor both the central tendency of the data (the mean) and the variability of the data

(either the standard deviation or the range). Note that each chart monitors a different type

of information. When observed values go outside the control limits, the process is assumed

not to be in control. Production is stopped, and employees attempt to identify the cause ofthe problem and correct it. Next we look at how these charts are created


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Mean (x-Bar) Charts

Subgroup Data with Unknown and A mean control chart is often referred to as anx-bar chart. It is used to monitor changes in

the mean of a process. To construct a mean chart we first need to construct the center line

of the chart. To do this we take multiple samples and compute their means. Usually these

samples are small, with about four or five observations. Each sample has its own mean, .

The center line of the chart is then computed as the mean of all sample means, where

the number of samples is:

To construct the upper and lower control limits of the chart, we use the following

Formulas:

UCL=

Center line =

LCL=

If we use as an estimator of and

as an estimator of , then the parameters of the chat are

Center line =

L

if we define


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The constant is tabulated for various sample size in following table A

n A2 D3 D4

2 1.88 0 3.27

3 1.02 0 2.57

4 0.73 0 2.28

5 0.58 0 2.11

6 0.48 0 2.00

7 0.42 0.08 1.92

8 0.37 0.14 1.86

9 0.34 0.18 1.82

10 0.31 0.22 1.78

11 0.29 0.26 1.74

12 0.27 0.28 1.72

13 0.25 0.31 1.69

14 0.24 0.33 1.67

15 0.22 0.35 1.65

16 0.21 0.36 1.64

17 0.20 0.38 1.62

18 0.19 0.39 1.61

19 0.19 0.40 1.60

20 0.18 0.41 1.59


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Example :

Maliban

Real value may be different because numerical method when run there is rounded but when we

calculate we may be not rounded.


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Range (R) Charts

Range (R) chartsare another type of control chart for variables. Whereas x-bar

charts measure shift in the central tendency of the process, range charts monitor the

dispersion or variability of the process. The method for developing and using R-charts is

the same as that for x-bar charts. The center line of the control chart is the average range,

and the upper and lower control limits are computed as follows:


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In the R chart, the center line will be . To determine the control limits, we need an

estimate of . Assuming that the quality characteristic is normally distributed. can be

found from the distribution of the relative range W=R/. The standard deviation of W,

say , is a known function of n, Thus, Since

the stranded deviation of R is

Since is unknown, we may estimate by

Above Equations reduces to equation

Considering Above example of maliban wafers product process


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CONTROL CHARTS FOR ATTRIBUTES

Control charts for attributes are used to measure quality characteristics that are

counted rather than measured. Attributes are discrete in nature and entail simple yes-or-

no decisions. For example, this could be the number of nonfunctioning light bulbs, the

proportion of broken eggs in a carton, the number of rotten apples, the number of scratcheson a tile, or the number of complaints issued. Two of the most common types of control

charts for attributes are p-charts and c-charts. P-charts are used to measure the

proportion of items in a sample that are defective. Examples are the proportion of broken

cookies in a batch and the proportion of cars produced with a misaligned fender. P-charts

are appropriate when both the number of defectives measured and the size of the total

sample can be counted. A proportion can then be computed and used as the statistic of

measurement.

C-charts count the actual number of defects. For example, we can count the number of

complaints from customers in a month, the number of bacteria on a Petri dish, or the

number of barnacles on the bottom of a boat. However, we cannotcompute the proportion

of complaints from customers, the proportion of bacteria on a Petri dish, or the proportion

of barnacles on the bottom of a boat.

Control Chart for Fraction Nonconforming-P charts

If p is not known, we estimate it from samples.

M: samples, each with n units (or observations )

Di: number of nonconforming units in sample i


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Average of all observations

Example

Data of p chart coca cola cans


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C-CHARTS

C-charts are used to monitor the number of defects per unit. Examples are the

number of returned meals in a restaurant, the number of trucks that exceed their weight

limit in a month, the number of discolorations on a square foot of carpet, and the number of

bacteria in a milliliter of water. Note that the types of units of measurement we are

considering are a period of time, a surface area, or a volume of liquid.

The average number of defects, is the center line of the control chart. The upper and lower

control limits are computed as follows:

Example :

The number of weekly customer complaints is monitored at a large hotel using a c-chart.

Complaints have been recorded over the past twenty weeks. Develop three-sigma control

limits using the following data:


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Source:Kangaroo Cabs service in sri lanka

The average number of complaints per week

is=

Therefore

As in the previous example, the LCL is

negative and should be rounded up to zero.

Following is the control chart for thisexample:

weeks No. of

Complaints

1

23

4

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20

3

23

1

3

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2

1

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1

3

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21

1

1

3

2

2

3


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Six Sigma Quality

Six Sigma is a business management strategy originally developed by Motorola, USA in

1981. As of 2010, it enjoys widespread application in many sectors of industry, although

its application is not without controversy.

Six Sigma seeks to improve the quality of process outputs by identifying and

removing the causes of defects (errors) and

minimizing variability in manufacturing and business processes. It uses a set of quality

management methods, including statistical methods, and creates a special infrastructure of

people within the organization ("Black Belts", "Green Belts", etc.) who are experts in these

methods. Each Six Sigma project carried out within an organization follows a defined

sequence of steps and has quantified financial targets (cost reduction or profit increase).

The term six sigma originated from terminology associated with manufacturing, specifically

terms associated with statistical modelling of manufacturing processes. The maturity of a

manufacturing process can be described by a sigma rating indicating its yield, or the

percentage of defect-free products it creates. A six-sigma process is one in which

99.99966% of the products manufactured are statistically expected to be free of defects

(3.4 defects per million). Motorola set a goal of "six sigmas" for all of its manufacturing

operations, and this goal became a byword for the management and engineering practices

used to achieve it.

Methods

Six Sigma projects follow two project methodologies inspired by Deming's Plan-Do-Check-Act Cycle.

These methodologies, composed of five phases each, bear the acronyms DMAIC and DMADV.

DMAIC is used for projects aimed at improving an existing business process. DMAIC is pronounced

as "duh-may-ick".

DMADV is used for projects aimed at creating new product or process designs. DMADV is

pronounced as "duh-mad-vee".

Origin and meaning of the term "six sigma process"


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The term "six sigma process" comes from the notion that if one has six standard deviations

between the process mean and the nearest specification limit, as shown in the graph,

practically no items will fail to meet specifications. This is based on the calculation method

employed in process capability studies.

Capability studies measure the number of standard deviations between the process meanand the nearest specification limit in sigma units. As process standard deviation goes up, or

the mean of the process moves away from the center of the tolerance, fewer standard

deviations will fit between the mean and the nearest specification limit, decreasing the

sigma number and increasing the likelihood of items outside specification.

From Wikipedia

Sampling techniques

Acceptance Sampling:

Inspection provides a means for monitoring quality. For example, inspection may be

performed on incoming raw material, to decide whether to keep it or return it to the

vendor if the quality level is not what was agreed on. Similarly, inspection can also be done

on finished goods before deciding whether to make the shipment to the customer or not.

However, performing 100% inspection is generally not economical or practical, therefore,

sampling is used instead. Acceptance Sampling is therefore a method used to make a

decision as to whether to accept or to reject lots based on inspection of sample(s). The

objective is not to control or estimate the quality of lots, only to pass a judgment on lots.

Using sampling rather than 100% inspection of the lots brings some risks both to the

consumer and to the producer, which are called the consumer's and the producer's risks,respectively. We encounter making decisions on sampling in our daily affairs.

Example:

statistical inference is made on the quality of the lot by inspecting only the small sample

drawn from the lot


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There are several Acceptance Sampling Plans:

- Single Sampling (Inference made on the basis of only one sample)

- Double Sampling (Inference made on the basis of one or two samples)

- Sequential Sampling (Additional samples are drawn until an inference can be

made)

etc.

Single Sampling Plans

A Single Sampling plan is characterized by n (the sample size) which is drawn from the lot

and inspected for defects. The number of defects (d) found are checked against c (the

acceptance number) and the procedure works as follows (clearly, d = 0, 1, 2, n):

Example:

Suppose n=100 and c=3, which means that if the number of defectives in the sample

(d) is equal to 0, 1, 2, or 3, then the lot will be accepted, and if d is 4 or more, then

the lot will be rejected.

As mentioned earlier, inherent in a sampling plan are producers and consumers

risk. These risks can be depicted by the following table:


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Formally, these risks are written as:

a : The producer's risk, is the probability that a lot with AQL will be rejected.

b : The consumer's risk, is the probability that a lot with LTPD will be accepted.

whereAcceptable Quality Level (AQL) = The quality level acceptable to the consumer

Lot Tolerance Percent Defective (LTPD) = The level of "poor' quality that the consumer

is willing to tolerate only a small percentage

of the time.

In general, both the producer and the consumer want to minimize their risks. The choice of

a well designed sampling plan can help both the producer and the consumer maintain their

respective risks at acceptable levels to both. For example, a = 5% for AQL of 0.02 and b =

10% for LTPD of 0.08.

Double Sampling

Double sampling (also called two-phase sampling - not to be confused with two-

stage sampling above) involves estimating two correlated variables. This method would be

used in cases where the primary variable of interest is expensive or difficult to measure,

but a secondary covariate is easily measurable. A small number of sample units are

randomly selected and both variables are measured at these locations. The secondary

variable only is then measured at a larger number of randomly selected points. The success

of a double-sampling sample design depends on how well correlated the primary and

secondary variables are.

Double-sampling is commonly used in estimation of above-ground biomass in rangelands.

Clipping and weighting of vegetation is expensive and tedious. With the double-sampling

method, ocular estimates of biomass are made for a small number of quadrats, and the

vegetation on those quadrats is then clipped and weighed. For the remaining quadrats, only

the ocular estimates are performed.

Advantages of double sampling are:

it can be much more efficient than directly sampling the primary variable if the

secondary variable can be measured quickly and it highly correlated with the primary

variable.

Disadvantages of double sampling are:

the formulas for data analysis and sample size estimation are much more complex than

for some other methods.


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Discussion

Types of Charts Available For The Data Gathered

Variable Data Charts Individual, Average and Range Charts

Variable data requires the use of variable charts. Variable charts are easy to understand

and use.

Attribute Data Charts


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References :

1. INTRODUCTION TO STATISTICAL QUALITY CONTROL, 5TH EDITION BY DOUGLAS C.MONTGOMERY. COPYRIGHT (C) 2005 JOHN WILEY & SONS, INC.

2. STATISTICAL QUALITY CONTROL , 7TH EDITION BY EUGENE L.GRANT RICHARD S.

LEAVENWORTH

RUHUNA MAIN LIBRARY CODE: 519.86 GRA

3. WADSWORTH, H. M., K. S. STEPHENS, AND A. B. GODFREY.MODERNMETHODS FOR QUALITY CONTROL AND IMPROVEMENT. NEW YORK: WILEY, 1986.

4. HTTP://WWW.WIKIPEDIA.ORG/

5. BASIC STATISTICS AND DATA ANALIYSIS BY LARRY J. KITCHENS APPALACHIAN

STATE UNIVERSITY

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