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MG 1401 TOTAL QUALITY MANAGEMENT

     3. Statistical Process Control
                2010-2011
Statistical Process Control
SPC is an
  effective system for
     controlling the
        process parameters by
              comparing it with
  standards and take
     corrective action if there is any
         deviation by employing
  statistical methods.

SPC may be used to cover
  all uses of
      statistical techniques for the
         analysis of data that may be
              applied in the
  control of
      product
           quality
Statistical Process Control


Statistical techniques
1.   The seven tools of quality used by the quality circles
       SEVEN QUALITY CONTROL TOOLS
         (or) OLD SEVEN TOOLS
2. Control charts
    Control charts for Variables ( X and R charts) and
                                 Process Capability
    Control charts for Attributes
       - Defectives (p and np charts)
       - Defects    ( c and u charts )
3. Concept of Six Sigma
4. Management tools-
      New seven management tools
The Total Statistical Process Control (TSPC) System
The seven tools of Quality- (Old) Q- 7 tools

SEVEN QUALITY CONTROL TOOLS
       (or) OLD SEVEN TOOLS

1. Check Lists/ Check sheets
2. (Frequency) Histograms or Bar Graphs
3. Process flow diagrams / Charts
4. Cause and Effect, Fishbone, Ishikawa Diagram
5. Pareto Diagrams
6. Scatter Diagrams / Plots
7. Control Charts
The seven tools of Quality ( Q- 7 tools)
S.No
              Tools              Problem             Solving step
 1     Check Lists/ Check   How often it is      For finding faults
       sheets               done?
 2     (Frequency)          What do variations   For identifying
       Histograms or Bar    look like?           problems
       Graphs
 3     Process flow         What is done?        For understanding
       diagrams / Charts                         the “mess”
 4     Cause and Effect,    What cause the       For generating ideas
       Fishbone, Ishikawa   problem?
       Diagram
 5     Pareto Diagrams      Which are the big    For identifying
                            problems?            problems
 6     Scatter Diagrams /   What are the         For developing
           Plots            relationships        solutions
                            between factors?
 7     Control Charts       Which variations to For implementation
                            control and how?
Check sheets/tally sheet (Data collection sheet)

the intent and
   purpose of
      collecting data is to either
       control the
           production process, to see the
   relationship between
       cause-and-effect, or for the
           continuous improvement of those
   processes that
   produce any
        type of defect or
   nonconforming product.
Check sheets/tally sheet (Data collection sheet)
Check Sheet –
collecting data to
      compile in such a way as to be
         easily used, understood and

                  analyzed automatically.
   - as it is being completed, actually becomes
          a graphical representation of the data you are collecting,
   - thus
    you do NOT need any computer software, or spreadsheet
    to record the data.
   - it can be simply done with pencil and paper!
- is a
    data recording form that has been designed to
         readily interpret results from the
              form itself.
- needs to be designed for the specific data it is to gather.
Check sheets/tally sheet (Data collection sheet)


- used for the collection of
     quantitative or
         qualitative repetitive data.
- adaptable to
      different data gathering situations.
- minimal interpretation of results required.
- easy and quick to use.
- no control for various forms of bias –
   exclusion,
      interaction,
         perception,
            operational,
              non-response,
                   estimation.
CHECK SHEET (or)
        DEFECT CONCENTRATION DIAGRAM
DESCRIPTION                  WHEN TO USE
A check sheet is a           When
 structured,                  data can be
                              observed and
 prepared form for
 collecting and               collected repeatedly by the
 analyzing data.              same person or at the
This is a                     same location.
  generic tool that can be   When
   adapted for a              collecting data on the
                              frequency or
  wide variety of
                              patterns of
  purposes
                              events, problems,
                              defects, defect location,

                              defect causes, etc.
                             When
                              collecting data from a
Tqm3 ppt
Check sheet
Check sheet
(continuous data use)                                                                                                                                                                                               No.___________
                                                                                                                                                                                                                          741

                                                  PRODUCT ION CHECK SHEET

Product Name_________________________________
               Alternator Pulley                                                                                                              Date_________________________________
                                                                                                                                                          12- 02- 02
Usage________________________________________
              Pulley Bolt Torque                                                                                                              Factory_______________________________
                                                                                                                                                             Church Street
Specification__________________________________
                 2.2 +/- .5                                                                                                                   Section Name__________________________
                                                                                                                                                             SI Line
No. of Inspections______________________________
                        185                                                                                                                   Data Collector__________________________
                                                                                                                                                               Sam The Man
Total Number__________________________________
                     185                                                                                                                      Group Name___________________________
Lot Number___________________________________
                    1631                                                                                                                      Remarks:_____________________________

Dimensions         1.5     1.6      1.7          1.8         1.9        2.0            2.1             2.2           2.3        2.4           2.5               2.6           2.7         2.8           2.9         3.0          3.1      3.2




                                                                                                                                                                                              SpecUSL
                                       SpecLSL
              40



              35



              30



              25




                                                                                                                                                        XXXXX
              20
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              15                                                          X                                                      X
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              10




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               5                                                                                                                                                                                                     X
                           XX




                                                                                                                                                                                                                                 XX
               0   X                                                                                                                                                                                                                      X
    TOTAL
  FREQUENCY            1        2           7        13         10            16             19             17          12           16              20              17             13             8            5            6        2       1
Histogram or Bar Graph
The word histogram is derived from Greek:
     - histos 'anything set upright‘
          (as the masts of a ship, the bar of a loom, or the vertical bars of
           a histogram);
     - gramma 'drawing, record, writing'.
A generalization of the histogram is
     kernal smoothing techniques.
This will construct a very smooth
     probability density function from the supplied data
- is a graphic summary of variation in a set of data.
- it enables us to see
    patterns that are
    difficult to see in a simple table of numbers.
- can be analyzed to draw conclusions about the data set.
- continuous variable is clustered into categories and the value
    of each cluster is plotted to give a series of bars as above.
- without using some form of graphic this kind of
    problem can be
    difficult to analyze, recognize or identify.
Histogram
In statistics, a
    histogram is a
    graphical display of tabulated
    frequencies.
It shows what proportion of cases fall into
    each of several categories.
A histogram differs from a bar chart in that it is the
                            area of the bar that denotes the
                            value,
                             not the height, a
    crucial distinction when the categories are
                             not of uniform width
The categories are usually specified as
    non-overlapping intervals of some variable.
The categories (bars) must be adjacent.
Histogram or Bar Graph
•
    H or B
Histogram or Bar Graph
Histogram




Grouping a set of measurements into a Histogram
Histogram shapes

Low with gaps                  Isolated-peaked



High with
few bars                       Cog-toothed (or comb)


Skewed
(this is positive;
negative skew                  Plateau
has tail to right)



Exponential                    Edge-peaked

                     missing
                      bars


Dual-peaked
(bimodal)                      Truncated
Bar chart




Using the Bar Chart in problem-solving




       Numbers into bars
.




    Pie chart




                Band Chart
Flow Charts

• Pictures,
• symbols or
• text coupled with lines,
• arrows on lines show direction of flow.
• enables modeling of processes;
• problems/opportunities and decision points etc.
• develops a common understanding of a process by those
  involved.
• no particular standardization of symbology,
• so communication to a different audience may require
  considerable time and explanation.
.




                          Basic Flowchart elements




Decisions in Flowcharts
.




Continuing Flowcharts across pages




                                     Delay symbol
    Sub processes
.




    Example Flowcharts
Deployment Flowchart
Flow chart
Flow chart
Cause & Effect diagram
Cause and Effect ,
Fishbone,
Ishikawa Diagram

- is the brainchild of
    Kaoru Ishikawa, who pioneered
    quality management processes in the
    Kawasaki shipyards, and in the process
    became one of the
    founding fathers of                      DR. KAORU ISHIKAWA (1915–1989)
    modern management.
                                             Disciple of Juran & Feigenbaum.
                                             TQC in Japan, SPC,
- is used to                                 Cause &Effect Diagram, QC.

  explore all the potential or
  real causes (or inputs) that result in a
  single effect (or output).
Cause & Effect diagram
 (also called Ishikawa or fishbone chart )

DESCRIPTION                        WHEN TO USE
 - identifies                      - When identifying
   many possible causes                      possible causes for a
   for an                              problem.
   effect or                       - especially when a
   problem.                            team’s thinking tends to
-can be used to                        fall into
    structure a                        ruts
    brainstorming
    session.
  - immediately sorts
    ideas into         DR. KAORU ISHIKAWA (1915–1989)
    useful categories.
                        Disciple of Juran & Feigenbaum.
                        TQC in Japan, SPC,
                        Cause &Effect Diagram, QC.
Cause & Effect diagram
Causes are arranged according to their level of
   importance or detail, resulting in a
   depiction of relationships and
   hierarchy of events.
- This can help you search for
    root causes,
    identify areas where there may be
    problems, and
    compare the
                                                    DR. KAORU ISHIKAWA (1915–1989)
    relative importance of
    different causes.                               Disciple of Juran & Feigenbaum.
                                                    TQC in Japan, SPC,
- Causes in a                                       Cause &Effect Diagram, QC.
       cause & effect diagram are frequently arranged into
             four major categories.
                              While these categories can be anything, you
   will often see:
• manpower, methods, materials, and machinery
  (recommended for manufacturing)
• equipment, policies, procedures, and people
  (recommended for administration and service).
Cause & Effect diagram

• is a method for analyzing process dispersion.
• purpose is to relate causes and effects.
• three basic types:
1. Dispersion analysis,
2. Process classification and
3. Cause enumeration.
Effect = problem to be resolved, opportunity to be grasped,
   result to be achieved.
• excellent for capturing team brainstorming output and for
   filling in from the 'wide picture'.
• helps organize and relate factors, providing a sequential view.
• deals with time direction but not quantity.
• can become very complex.
• can be difficult to identify or
      demonstrate interrelationships.
Cause & Effect diagram
Cause & Effect diagram
Cause & Effect diagram
Cause & Effect diagram
Cause & Effect diagram




This fishbone diagram was drawn by a manufacturing team to try
  to understand the source of periodic iron contamination.
The team used the six generic headings to prompt ideas.
Layers of branches show thorough thinking about the causes of the problem.
Cause & Effect diagram
Cause & Effect diagram
Cause & Effect diagram
       (Standard )
Pareto diagrams

• Pareto diagrams are
   named after Vilfredo Pareto,
• an Italian sociologist and economist,
           who invented this method of informationAlfredo Pareto
                                                    (1848-1923) (Europe)
                presentation toward the end of the 19th century.
• The chart is similar to the histogram or bar chart, except
  that
• the bars are arranged in decreasing order from left to
  right along the abscissa.
• The fundamental idea behind the use of Pareto diagrams
• for quality improvement is that the first few (as presented
  on the diagram) contributing causes to a problem usually
  account for the majority of the result.
• Thus, targeting these "major causes" for elimination results
  in the most cost-effective improvement scheme
Pareto diagrams

Pareto Principle

• The Pareto principle suggests that
   most effects come from relatively few causes.   Alfredo Pareto
• In quantitative terms:                           (1848-1923) (Europe)
•   80% of the problems come from 20% of the causes
  (machines, raw materials, operators etc.);
•   80% of the wealth is owned by 20% of the people etc.
• Therefore
  effort aimed at the right 20% can solve 80% of the
  problems.
• Double (back to back) Pareto charts can be used to compare
  'before and after' situations.
• General use,
  to decide where to apply initial effort for maximum effect.
Pareto diagrams




                  Alfredo Pareto
                  (1848-1923) (Europe)
Pareto Chart (or)
                      Pareto diagram (or)
                        Pareto analysis

A Pareto chart is a bar graph.
   The lengths of the bars represent
   frequency or cost
   (time or money), and are arranged with              Alfredo Pareto
                                                       (1848-1923) (Europe)
   longest bars on the left and the
   shortest to the right.
To identify the ‘VITAL FEW FROM TRIVIAL MANY’ and to
   concentrate on the vital few for improvement.
A Pareto diagram indicates which problem we should solve first in
   eliminating defects and improving the operation.
The Pareto 80 / 20 rule
80 % of the problems are produced by

20 % of the causes.
Alfredo Pareto
(1848-1923) (Europe)
Tqm3 ppt
Pareto diagrams
The Pareto Chart
                                        Finding the right
                                        Pareto Chart



                                                         Convex Pareto




                                                        Concave or
                                                        'spiky' Pareto

                                    (clearly allows you to prioritize the action)




Prioritizing the action                    Alfredo Pareto
                                           (1848-1923) (Europe)
Before and After

                                                             .




The Sub-Pareto Chart




                       The Pareto Curve
Scatter diagrams

• are used to study

                  possible relationships between two
                        variables.
• Although these diagrams
  cannot prove that

                   one variable causes the other, they do
                       indicate the existence of a
  relationship, as well as the
                   strength of that relationship.
• is composed of a
    horizontal axis containing the
                   measured values of

                      one variable and
                                 a vertical axis representing the

                      measurements of the
Scatter Diagram (or) Scatter plot (or) X–Y graph
• The purpose of the scatter diagram is
     to display what happens to one variables when another variable is
  changed.
• is used to test a theory that the two variables are related.
• The type of relationship that exits is
   indicated by the slope of the diagram.

• The scatter diagram graphs
     pairs of numerical data, with
  one variable on each axis, to look for a
      relationship between them. If the
  variables are correlated, the points will
      fall along a line or curve.
  The better the correlation, the

        tighter the points will

        hug the line.
Scatter diagrams
Scatter diagrams
Tqm3 ppt
Scatter Diagram
       Types


        Positive




        Negative
                        Degrees of correlation



                            None                 High
         Curved




                             Low                 Perfect
          Part-linear
Tqm3 ppt
Control chart
• In statistical process control, the

     control chart, also known as the
    'Shewhart chart' or
    'process-behaviour chart' is a tool used to determine
    whether a manufacturing or
                     business process is in a state of
                    statistical control or not.


•    If the chart indicates that the
                                                     DR. WALTER A. SHEWHART
       process is                                    (1891–1967)
    currently under control then it can be used with -TQC &PDSA
               confidence to
         predict the
    future performance of the
    process.
Control chart
•   If the chart indicates that the
         process being monitored is
         not in control, the pattern it reveals can help

        determine the

       source of variation to be
       eliminated to
       bring the
       process back into control.
• A control chart is a specific kind of                DR. WALTER A. SHEWHART
  run chart that allows significant change to be       (1891–1967)
                                                       -TQC &PDSA
  differentiated from the
              natural variability of the process.
• This is key to
  effective process control and
  improvement.
Control chart
• They enable the
     control of distribution of variation rather than
       attempting to control each individual variation.
• Upper and
  lower control and
   tolerance limits are calculated for a
  process and
  sampled measures are regularly plotted about
           a central line between the two sets of limits.
• The plotted line corresponds to the
  stability/trend of the process.
• Action can be
  taken based on trend rather than on
      individual variation.
• This prevents
   over-correction/compensation for
         random variation, which would lead to many rejects.
                                                  DR. WALTER A. SHEWHART
                                                  (1891–1967)
                                                  -TQC &PDSA
Control charts

Mean and Control Limits




                           How control limits catch shifts




  Additional limit lines
Control chart
Control chart
Stratification Analysis
• Stratification Analysis determines the extent of the
  problem for relevant factors.
• o        Is the problem the same for all shifts? 
• o        Do all machines, spindles, fixtures have the same problem? 
• o        Do customers in various age groups or parts of the country have 
  similar problems? 
• The important stratification factors will vary with each
  problem, but most problems will have several factors.
• Check sheets can be used to collect data.
• Essentially this analysis seeks to develop a pareto diagram
  for the important factors.
Stratification Analysis

•    The hope is that the extent of the problem will not
    be the same across all factors.
•   The differences can then lead to identifying root cause.
•   When the 5W2H and Stratification Analysis are
    performed, it is important to consider a number of
    indicators.
•   For example, a customer problem identified by warranty
    claims may also be reflected by various in-plant indicators.
•   Sometimes, customer surveys may be able to define
    the problem more clearly.
•    In some cases analysis of the problem can be
    expedited by correlating different problem
    indicators to identify the problem clearly
Stratification (or) Flowchart (or) Run chart
• Stratification is a technique used in combination with other
  data analysis tools. When data from a variety of sources or
  categories have been lumped together, the meaning of the
  data can be impossible to see

• When to Use
• Before collecting data.
• When data come from several sources or conditions, such
  as shifts, days of the week, suppliers or population groups.
• When data analysis may require separating different
  sources or conditions.
Benefit from stratification.


• Always consider before collecting data whether
  stratification might be needed during analysis. Plan to
  collect stratification information. After the data are
  collected it might be too late.
• On your graph or chart, include a legend that identifies
  the marks or colors used.
Data Analysis
                             • What is Data Analysis
 • Data analysis is statistics + visualization + know-how
• Many of the methods for                                                                 
           data analysis are based on                                               
       multivariate statistics, which poses an additional 
   problem to the beginner: 
• multivariate statistics cannot be understood without 
   a profound knowledge of simple statistics. 
• Furthermore, several                                                                      
                  fields in science and engineering have 
   developed their own nomenclature                                                
                       assigning different names to the                             
                            same concepts. 
Data Analysis
•   Thus one has to                                                                            
    gather considerable knowledge and experience in 
    order to perform the analysis of data efficiently.
•    Possible applications of statistical methods can be in the fields 
    of medicine, engineering, quality inspection,
    election polling, analytical chemistry, physics,
    gambling

• Statistics and statistical methodology as the basis of 
  data analysis are concerned with two basic types of
  problems 
• (1) summarizing, describing, and exploring the data
  (2) using sampled data to infer the nature of the
  process which produced the data
• The first type of problems is covered by descriptive
  statistics, the second part is covered by inferential
  statistics.
• Another important aspect of data analysis is the data, which 
  can be of two different types: qualitative data, and 
  quantitative data .
Data Analysis

• Qualitative data does not contain quantitative information.
• Qualitative data can be classified into categories.
• In contrast, quantitative data represent an amount of something.
• A third distinction can be made according to the
     number of variables involved in the data analysis.
• If only one variable is used, the statistical procedures are
     summarized as univariate statistics.
• More than one variable result in multivariate statistics.
• A special case of multivariate statistics with                                   
            only two variables is sometimes called bivariate statistics.
Tqm3 ppt
Statistical Process Control (SPC)

Measures performance of a
   process
Uses mathematics (i.e., statistics)
Involves collecting,
         organizing, &
          interpreting data
 Objective:
   Regulate product quality
 Used to
   – Control the
         process as
          products are
         produced
   – Inspect samples of
         finished products
Statistical Process Control

                What is a process?




     Inputs          PROCESS             Outputs



A process can be described as a
  transformation of set of
  inputs into desired
  outputs.
WHY STATISTICS?
                  THE ROLE OF STATISTICS ………


                                                             µ




                                                   LSL   T       USL




 Statistics is the art of
  collecting, classifying, presenting, interpreting and
  analyzing numerical data, as well as
  making conclusions about the
  system from which the
  data was obtained.


January 9, 2013                                                        73
Descriptive Statistics
Descriptive Statistics is the
 branch of
 statistics which
 most people are
 familiar.

It characterizes and summarizes the
   most prominent features of a
  given set of data
  (means,
  medians,
  standard deviations,
  percentiles,
  graphs,
  tables and
  charts).

 January 9, 2013                              74
Inferential Statistics


Inferential Statistics is the

  branch of statistics that deals with

  drawing conclusions about

  a population based on

  information obtained from

  a sample drawn from that

  population.



 January 9, 2013                              75
Statistics
MEASURES OF CENTRAL TENDENCY
   Usually, the first topic in statistics is
   descriptive statistics.
The mean,
    median and
    mode are
    used to describe statistical data.
Variance and
    standard deviation help to
    understand how the data is spread out.
Central tendency is a
    typical or
    representative score.
The three measures of central tendency are the
    mode,
    median, and
    mean.
The term "measures of central tendency" refers to
.
                       finding the mean, median and mode.



    Mean:
              Average.
              The sum of a set of data divided by the number of data.
              (Do not round your answer unless directed to do so.)


    Median:
              The middle value, or the mean of the middle two values, when
              the data is arranged in numerical order. Think of a "median"
              being in the middle of a highway.


    Mode:
              The value ( number) that appears the most.
              It is possible to have more than one mode, and it is possible to
              have no mode. If there is no mode-write "no mode", do not write
              zero (0) .
Mode
Mode is the data
     value that
     occurs at a
     greater frequency than the others.
 Data: 1, 2, 3, 3, 3, 4, 4, 5

  Mode = 3
The mode, symbolized by
   Mo, is the most frequently occurring score value.
   If the scores for a given sample distribution are:

32 32  35  36  37  38  3 8  39 3 9 39  40  4 0  42  45 

- then the mode would be 39 because a score of 39 occurs 3 times,
    more than any other score.
- The mode may be seen on a frequency distribution as the score value
    which corresponds to the highest point.
- For example, the following is a frequency polygon of the data
    presented above:
Statistics - Mode



                3


2       2




                         1    1




32     36        39      42   45
            scores
Statistics - Mode
A distribution may have
   - more than one mode if the two most frequently occurring
    scores occur the same number of times.
 For example, if the earlier score distribution were modified as follows:
32 32 32  36  37  38  38  39 39 39  40  40  42  45 
- then there would be two modes, 32 and 39.
- Such distributions are called bimodal.
- The frequency polygon of a bimodal distribution is presented
    below.
Statistics - Mode

3               3




32             39
Statistics - Mode

In an extreme case there may be no unique mode, as in the case of
   a rectangular distribution.
The mode is not sensitive to extreme scores.
Suppose the original distribution was modified by changing the last
   number, 45, to 55 as follows:
     32  32  35  36  37  38  38  39 39 39  40  40  42  55
     The mode would still be 39.

In any case,
    the mode is a quick and dirty measure of central tendency.

Quick, because it is easily and quickly computed.

Dirty because it is not very useful; that is,
  - it does not give much information about the distribution.
Statistics - Mode
Statistics - Median
Median
The median is the
    exact middle value of a
    set of data values that have been sorted from the
    lowest value to
    highest.
If the number of data values
    even, then the
    median is the
    average of the
    two middle values.
Examples:
   Data: 1, 2, 3, 4, 5
   Median: 3

  Data: 1, 2, 3, 4, 5, 6
  Median: 3.5
Statistics - Median
The median, symbolized by
      Md, is the score value which
      cuts the distribution in half, such that half the scores fall above
     the median and half fall below it.
Computation of the median is relatively straightforward.
The first step is to
     rank order the scores from
     lowest to highest.
The procedure branches at the next step:
    - one way if there are an odd number of scores in the sample
     distribution,
    - another if there are an even number of scores.
If there is an odd number of scores as in the distribution below:
    32  32  35  36  36  37  38               
                                                  38               
                                                          39  39  39  40  40  45  46 
    - then the median is simply the middle number.
In the case above the median would be the number 38, because there
     are 15 scores all together with
      7 scores smaller and 7 larger.
Statistics - Median

• If there is an even number of scores, as in the distribution below: 
     32  35  36  36  37  38 
                                           38 39           
                                                           39  39  40  40  42  45
• then the median is the midpoint between the two middle scores: 
• in this case the value 38.5. 
• It was found by adding the two middle scores together and dividing 
     by two (38 + 39)/2 = 38.5.
•  If the
             two middle scores are the same value then the            
                           median is that value. 
• In the above system,
                       no account is paid to whether there is a                      
                    duplication of scores around the median. 
• In some systems a slight correction is performed to correct for 
     grouped data, but since the correction is slight and the data is 
     generally not grouped for computation in calculators or computers, it 
     is not presented here. 
Statistics - Median

The median, like the mode, is                                                               
        not effected by extreme scores, as the following 
     distribution of scores indicates: 
     32  35  36  36  37  38
                                           38 39              
                                                         39  39  40  40  42  55
The median is still the value of 38.5. 
The median is not as                                                                             
         quick and 
    dirty as the
    mode, but
    generally it is
    not the 
    preferred measure of 
    central tendency.
Statistics – The Mean


The mean, 
     symbolized by         ,
     is the sum of the scores divided by the number of scores. 
The following formula both defines and describes the procedure for 
     finding the mean:




 
where X is the sum of the scores and 
 N is the number of scores.
Statistics – The Mean
Mean
The mean most frequently used is the 
    arithmetic mean, which is the same as the

        average, although the                                                               
        geometric mean is 
        used also 
        at times. 
It is the arithmetic mean that is referred to when the word mean is 
     used by itself. 
        Expected value is another way of saying the mean.
        mean =    sum of data values / number of data values

         mean μ =    (1 + 2 + 3) / 3    =    6 / 3    =   2  

The lower case Greek letter μ is used to represent the mean.
If the mean is from a sample of data, then                                           
               x is used to represent the sample mean. 
Also, variables and Sigma notation is used to write the 
     general form of the mean.
Statistics – The Mean
The mean 
Application of this formula to the following data
  32  35  36  37  38  38  39  39  39  40  40  42  45 
yields the following results: 




Use of means as a way of describing a set of scores is fairly 
     common;
•      batting average,
•      bowling average, 
•      grade point average, and 
•      average points scored per game 
              are all means. 
Note the use of the word "average" in all of the above terms. 
In most cases when the term "average" is used, it refers to the
     mean, although not necessarily. 
When a politician uses the term "average income", for example, 
     he or she may be referring to the mean, median, or mode. 
Statistics – The Mean
The mean is sensitive to extreme scores.
           For example, the mean of the following data is 39.0,
     somewhat larger than the preceding example. 
32  35  36  37  38  38  39  39  39  40  40  42  55
In most cases the
     mean is the 
     preferred measure of
     central tendency, both as a
     description of the 
     data and as an
     estimate of the
     parameter.
 In order for the mean to be meaningful, however, the 
     acceptance of the
     interval property of 
     measurement is 
     necessary.
When this property is obviously violated, it is inappropriate and 
     misleading to compute a mean.
Kiwi Bird Problem

As is commonly known,                                                                             
                 KIWI-birds are native to                                                       
                                                         New Zealand.
They are born                                                                                 
   exactly one foot tall and grow                                                       
                     in one foot intervals. 
That is, one moment they are one foot tall and the next they are two 
   feet tall. They are also very rare.
 An investigator goes to New Zealand and                                           
               finds four birds.
 The                                                                                                            
                   mean of the heights of four birds is 4, the              
                                    median is 3, and the                                        
                                                   mode is 2.
                                                                      What are the heights
   of the four birds?
• Hint - examine the constraints of the mode first, the median second, 
   and the mean last.
Statistics
• Skewed Distributions and
  Measures of Central Tendency
• Skewness refers to the asymmetry of the distribution, such that 
  a symmetrical distribution exhibits                                             
            no skewness. In a symmetrical distribution the
                mean, median, and mode all fall at the same point, 
  as in the following distribution.
Statistics

•   An exception to this is the case of a
     bi-modal symmetrical distribution. 
•   In this case the mean and the median fall at the same point, 
    while the two modes correspond to the two highest points of 
    the distribution. An example follows:
Statistics
•   A positively skewed distribution is asymmetrical and points in
    the positive direction. If a test was very difficult and almost 
    everyone in the class did very poorly on it, the resulting distribution 
    would most likely be positively skewed. 




•   In the case of a positively skewed distribution, the mode is
    smaller than the median, which is smaller than the
    mean. This relationship exists because the mode is the point on 
    the x-axis corresponding to the highest point, that is the score 
    with greatest value, or frequency. The median is the point on the 
    x-axis that cuts the distribution in half, such that 50% of the 
    area falls on each side.                    Mo
Statistics
•   The mean is the balance point of the distribution. Because 
    points further away from the balance point change the center of 
    balance, the mean is pulled in the direction the distribution is 
    skewed. For example, if the distribution is positively skewed,
    the mean would be pulled in the direction of the skewness, or 
    be pulled toward larger numbers.




•    One way to remember the order of the mean, median, and
     mode in a skewed distribution is to remember that the mean is 
     pulled in the direction of the extreme scores. In a 
     positively skewed distribution, the extreme scores are larger, thus 
     the mean is larger than 
     the median.  
Statistics
Statistics
•   A negatively skewed distribution is asymmetrical and points 
    in the negative direction, such as would result with a very easy test. 
    On an easy test, almost all students would perform well and only a 
    few would do poorly. 




•   The order of the measures of central tendency would be the 
    opposite of the positively skewed distribution, with the mean
    being smaller than the median, which is smaller than
    the mode.  
Statistics
MEASURES OF VARIABILITY
• Variability refers to the spread or dispersion of scores. 
• A distribution of scores is said to be highly variable if the 
     scores differ widely from one another. 
• Three statistics will be discussed which measure variability: the
     range, the variance, and the standard deviation. 
• The latter two are very closely related. 
Range
Range is the highest data value minus the lowest data
     values.    
      Data: 1, 2, 3, 4, 5, 6, 7
      Highest Data Value: 7
      Lowest Data Value: 1       Range: 7 - 1 = 6
• It is a quick and dirty measure of variability, although when a test 
     is given back to students they very often wish to know the range of 
     scores. 
• Because the range is greatly affected by extreme scores,
     it may give a distorted picture of the scores. 
Statistics
•   The following two distributions have the same range, 13, yet appear 
    to differ greatly in the amount of variability. 
•   Distribution 1 32 35 36 36 37 38 40 42 42 43 43 45
•   Distribution 2 32 32 33 33 33 34 34 34 34 34 35 45
•   For this reason, among others, the range is not the most
    important measure of variability. 

Variance
• Variance is used to measure
                       how far the data is away from the mean.
• The distance of the data point from the
                       mean is a deviation.
•  The deviations are added together to get a value 
     representing all the deviations together. 
• However, since some deviations can be negative, the
     total could be zero.
• To account for this, the deviations are squared and then 
     added together.
• When divided by the number of
               deviations,
               the result is
               variance.
Statistics
 Standard Deviation
• The standard deviation is just
            the square root of the variance. 
• A statistic is an algebraic expression combining scores into a single 
     number. 
• Statistics serve two functions:
• they estimate parameters in
              population models and they describe the
              data.
• i.e., Population variance, standard deviation
             Sample variance,      standard deviation

• The variance, symbolized by
                "s2", is a measure of variability. Th
             standard deviation, symbolized by
                "s", is the
                      positive square root of the variance.
Statistics
• Variance :formula: 




•    Note that the variance could almost be the 
                      average squared deviation around
         the mean if the expression were divided by N rather than N-1.
•  It is divided by N-1, called the degrees of freedom (df), for 
     theoretical reasons.
•  If the mean is known, as it must be to compute the numerator of the 
     expression, then only N-1 scores that are free to vary. 
• That is if the mean and N-1 scores are known, then it is possible to 
     figure out the Nth score. 
• One needs only recall the KIWI-bird problem to convince oneself 
     that this is in fact true. 
Statistics
•    The formula for the
      variance presented above is a definitional formula, it defines 
     what the variance means. 
•    The variance may be computed from this formula, but in 
     practice this is rarely done. 
•    The computation is performed in a number of steps, which are 
     presented below:

Steps
1. Find the mean of the scores.
2. Subtract the mean from every score.
3. Square the results of step 2.
4. Sum the results of step 3.
5. Divide the results of step 4 by N-1.
6. Take the square root of step 5.
7.       The result at step 5 is the sample variance,
                          at step 6, the sample standard deviation.
Measures of dispersion
Quartiles
        If we divide a cumulative frequency curve into 
        quarters, 
             the value at the lower quarter is referred to as 
        the lower quartile, 
             the value at the middle gives the median and the 
        value at the upper quarter is the upper quartile.
A set of numbers may be as follows: 
         8, 14, 15, 16, 17, 18, 19, 50. 
        The mean of these numbers is 19.625 . However, 
        the extremes in this set (8 and 50) distort the range. 
        The inter quartile range is a method of measuring 
        the spread of the numbers by finding the middle 
        50% of the values. 
         It is useful since it ignore the extreme values. It is a 
        method of measuring the spread of the data.
The lower quartile is (n+1)/4 th value
    (n is the cumulative frequency, ie 157 in this case) 
        and 
        the upper quartile is the 3(n+1)/4 the value. 
        The difference between these two is the inter 
        quartile range (IQR).
In the above example, 
         the upper quartile is the 118.5th value and 
         the lower quartile is the 39.5th value. 
 If we draw a cumulative frequency curve, we see that 
        the lower quartile, therefore, is about 17 and   
        the upper quartile is about 37. 
Therefore the IQR is 20 (bear in mind that this is a rough 
        sketch- if you plot the values on graph paper you 
        will get a more accurate value). 
Measures of dispersion measure how spread out a set of data is.
                                  Variance and Standard Deviation
                                  The formulae for the variance and standard deviation are given below.
                                   m means the mean of the data.
   .
                                  Variance = s2 = S (xr – m)2
                                                        n
                                  The standard deviation, s, is the square root of the variance.
                                  What the formula means:
                                  (1) xr - m means take each value in turn and subtract the mean from each value.
                                  (2) (xr - m)² means square each of the results obtained from step (1).
                                                This is to get rid of any minus signs.
                                  (3) S(xr - m)² means add up all of the results obtained from step (2).
   Example:                       (4) For variance divide step (3) by n, which is the number of numbers
      Find the variance and standard deviation of the following numbers:  the answer to step (4).
                                  (5) For the standard deviation, square root
            1, 3, 5, 5, 6, 7, 9, 10 .                   The mean = m = 46/ 8 = 5.75
 x                 (Step 1):     (Step 2):  
                  xr - m             
 1           (1 - 5.75)  -4.75 (xr - m)²                    (Step 4):  n = 46, therefore
 3           (3 - 5.75)  -2.75  22.563                            variance = 61.504/ 46 = 1.34 (3sf)
 5           (5 - 5.75)   -0.75     7.563                   (Step 5):   
 5           (5 - 5.75)   -0.75     0.563                          standard deviation = 1.16 (3sf)
 6           (6 - 5.75)    0.25     0.563 
 7           (7 - 5.75)    1.25     0.063 
 9           (9 - 5.75)    3.25     1.563
10          (10 - 5.75)   4.25  10.563 
46                                18.063
(n)            (Step 3): S(xr - m)² 61.504
Grouped Data
 There are many ways of writing the formula for the standard deviation. The one above is for a
 population of numbers. The formula for the standard deviation when the data is grouped is:
   .




Example:
   The table shows marks (out of 10) such questions, it is often easiest to
                                    In
   obtained by 20 people in a test set your working out in a table:
   Mark (x)      Frequency (f)      fx             fx²
    1               0               0               0
    2               1                                                    Sf = 20
                                    2               4
    3               1               3               9
                                                                         Sfx = 118
    4               3               12              48                   Sfx² = 764
    5               2               10              50                   variance = Sfx² - ( Sfx )²
    6               5               30              180                               Sf ( Sf )
    7               5               35              245                   = 764 - (118)²
    8               2               16              128                      20    ( 20 )
    9               0               0               0                     = 38.2 - 34.81 = 3.39
   10               1               10              100
Work out the variance of this data.
                    20              118            764
Population Vs. Sample (Certainty Vs. Uncertainty)


      A sample is just a subset of all possible values

                                  sample
                  population

      Since the sample does not contain all the possible values, there
  is some uncertainty about the population.
   Hence any statistics, such as mean and standard
  deviation, are just estimates of the true population
  parameters.



January 9, 2013                                                          107
Tqm3 ppt
Population and sample
Population is the entire (complete) collection of all the 
         measurements of an observed quality characteristic
        -variation pattern is not known
A Sample is a collection of measurements selected from some large 
         source or population  - i.e., a part of the population
Population:                                                     Smooth curve

      (‘) prime symbol is used to identify parameters
Parameters:            mean (μ),
       population standard deviation (σ ).

- has finite number of items  e.g., production of shafts in a day  
- it is impossible to measure all the population
  - the conclusion about the population is derived from the 
         mean and standard deviation of the sample
Types:    Finite population             - finite number
               Infinite population           - infinite number
               Existent population         - concrete individuals
               Hypothetical population  - possible ways- population of head 
                and tail obtained by tossing a coin an infinite number of times
Population and sample
Sample:
Statistic – average (x), and sample standard deviation (s)
                             Histogram




To analyze  and draw  conclusion about the universe, a sample  is 
   selected at random to represent the population
- small section selected is            - sample
- process of such selection is - sampling
Population and sample (e.g.,)
Normal curve

-the normal curve is the most important frequency curve
- is also known as Gaussian curve and probability curve
- is symmetrical
- is unimodal
- is bell shaped distribution with mean, median, and mode having  
   the same value.
-The normal distribution is fully defined by the population mean
  and population
                               50.00%
  standard deviation
                                 68.26%




                            95.45%

                            99.73%
-∞                                                                +∞
     -3σ      -2σ       -1σ                 +1σ       +2σ   +3σ
                      -0.6745σμ       +0.6745σ
           Area under the normal distribution curve
Normal curve
The mean (μ), and standard deviation (σ ) are the
     population parameters
The mean (x), and standard deviation (s) are for the               
     sample quantity drawn from the population
For practical usage
It is necessary to convert from mean values and standard deviations 
     other than zero and one respectively
This procedure called normalizing involves substituting 
 z = x – μ       thus the values read from the table represent the area 
        σ
                                 ∞
under normal curve from –        to  z = x - μ
                                             σ


                                             σ= 1.5 
Normal curves with different standard
deviations but identical means


                                                  σ= 3.0

                                                            σ= 4.5 

     5       8      11     14      17   20       23    26   29        32   35
Normal curve
Tqm3 ppt
Variance Shown in a Probability Distribution
Control Chart (or) Statistical process control

VARIATIONS
• Different types of control charts can be used, depending upon the
  type of data.
• The two broadest groupings are for
     variable data and
     attribute data.
• Variable data are measured on a continuous scale.
   For example: time, weight, distance or temperature

                     can be measured in fractions or decimals.
  The possibility of measuring to
                     greater precision defines variable data.
Attribute data
      are counted and
      cannot have fractions or decimals.
Attribute data arise when you are determining only the
      presence or absence of something:
      success or failure,
      accept or reject,
      correct or not correct.

For example,
      a report can have
      four errors or five errors,
      but it cannot have
      four and a half errors.
Control Chart
• is a graphical representation of the collected information
                 - information may be measured quality characteristics
• detects the variation in the process and
• warns if there is any deviation from the specification

                      Essential features of a control chart



                                     Upper Control Limit
    Variable Values




                                        Central Line

                                     Lower Control Limit



                              Time
Control Chart
Purposes
• Show changes in data pattern
   – e.g., trends
       • Make corrections before process is out of control
• Show causes of changes in data
   – Assignable causes
       • Data outside control limits or trend in data
   – Natural causes
       • Random variations around averag

 In the charts,
• If all the points (sample averages and ranges) are
     within the control limits,
  - then the process is said to be in “Statistical control”
• If any one point or more in the control charts go
  outside the control limits,
  - then the process is said to be “out of control”
Quality Characteristics



             Variables                        Attributes
1. Characteristics that you        1. Characteristics for which you
    measure,                          focus on defects
     e.g., weight, length

2. May be in whole or
                                   2. Classify products as either
   in fractional numbers              ‘good’ or ‘bad’, or
                                       count # defects
                                       – e.g., radio works or not

3. Continuous random variables     3. Categorical or discrete random
                                      variables
Control Chart Types




                            Control Charts



        Variables                                 Attributes
         Charts                                    Charts



 R                    X                  p & np            c&u
Chart               Chart                Chart             Chart
Control Chart
Classification:
For variables- X and R charts
• Measures where the metric consists of
                  a number which indicates a precise value is called
   Variable data.
    – Time
    – Miles/Hr
For variables
    Sample average
    Range
    Grant average
    Grant range
Control limits

X chart                           R chart
- upper limit                       - upper limit
- lower limit                       - lower limit
Both the charts should be plotted together
If the sub group size is 6 or less LCLR = 0
Variables charts


– X and R chart (also called averages and range chart)
– X and s chart
– chart of individuals (also called X chart, X-R chart, IX-
  MR chart, Xm R chart, moving range chart)
– moving average–moving range chart (also called MA–
  MR chart)
– target charts (also called difference charts, deviation
  charts and nominal charts)
– CUSUM (also called cumulative sum chart)
– EWMA (also called exponentially weighted moving
  average chart)
– multivariate chart (also called Hotelling T2)
X Chart


Type of variables control chart
   – Interval or ratio scaled numerical data
• Shows sample means over time
• Monitors process average and tells whether changes have occurred.
  These changes may due to
           1. Tool wear
           2. Increase in temperature
           3. Different method used in the
                 second shift
           4. New stronger material

• Example: Weigh samples of coffee & compute means of
  samples; Plot
R Chart

Type of variables control chart
   – Interval or ratio scaled numerical data
• Shows sample ranges over time
   – Difference between smallest & largest values in
                                       inspection sample
• Monitors variability in process,
• it tells us the loss or gain in dispersion.
 This change may be due to:
                 1. Worn bearing
                 2. A loose tool
                 3. An erratic flow of lubricant to machine
                 4. Sloppiness of machine operator
• Example:
      Weigh samples of coffee &
      compute ranges of samples;
Construction of X and R Charts
•   Step 1: Select the Characteristics for applying a control chart.
•   Step 2: Select the appropriate type of control chart.
•   Step 3: Collect the data.
•   Step 4: Choose the rational sub-group i.e Sample
•   Step 5: Calculate the average ( X) and range R for each sample.
•   Step 6: Cal Average of averages of (X) and average of range (R)
•   Steps 7:Cal the limits for X and R Charts.
•   Steps 8: Plot Centre line (CL), UCL and LCL on the chart
•   Steps 9: Plot individual X and R values on the chart.
•   Steps 10: Check whether the process is in control (or) not.
•   Steps 11: Revise the control limits if the points are outside.
X Chart
                Control Limits




UCL = x + A R                                      From
   x       2                                       Tables
LCL = x − A R
   x       2

 Sub group average X = (x1 + x2 +x3 +x4 +x5 ) /5

 Sub group range R = Max Value – Min value
R Chart Control Limits




 UCL R = D 4 R
                                                 From Tables

 LCL R = D 3 R


Problem8.1 from TQM by V.Jayakumar Page No 8.5
Control Chart
Description Control charts for individual measurements
(e.g., the sample size = 1) use the moving range of two successive
observations to measure the process variability.
The combination of the X Chart for Individuals and the Moving Range chart
is often called an X and Rm or XmR Chart.
X-Bar R Chart (Mean-Range Chart)
X-Bar Sigma Chart (Mean-Sigma Chart)
I-R Chart (Individual Range)
Median-Range Chart
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X-Bar Sigma Chart with Variable Subgroup Sample Size
EWMA (Exponentially Weighted Moving Average Chart)
MA Chart (Moving Average Chart)
CuSum Chart (Tabular Cumulative Sum Chart )
Types of Control Charts for Attribute Data
    Measures where the metric is composed of a classification in
     one or two (or more) categories is called Attribute data.
           Description                  Type        Sample Size


Control Chart for (defectives)       p Chart     may change
proportion non conforming units
Control Chart for (defectives)       np Chart    must be constant
no. of non conforming units in a
sample
Control Chart for (defects)          c Chart     must be constant
no. of non conformities in a
sample
Control Chart for (defects)          u Chart     may Change
no. of non conformities per unit
p Chart for Attributes


(also called proportion chart)


Type of attributes control chart
   – Nominally scaled categorical data
       • e.g., good-bad, Yes/No
• Shows % of nonconforming items
Example: Count # defective chairs &
              divide by total chairs inspected;
               Plot
   – Chair is either defective or not defective
p Chart
                (also called proportion chart)


p = np / n
  where p = Fraction of Defective
       np = no of Defectives
       n = No of items inspected in sub group
        p = Average Fraction Defective = ∑np/ ∑n = CL


                           p (1 −p )
      UCL p =p +z
                               n


                           p (1 −p )
       LCL p =p −z
                               n
p Chart
                 Control Limits



              p (1 − p )
UCL p = p + z
                  n           z = 3 for
                              99.7% limits
              p (1 − p )
LCL p = p − z
                  n
Purpose of the p Chart


      Identify and correct causes of bad quality
      The average proportion of defective articles submitted for
      inspection, over a period.
      To suggest where X and R charts to be used.
      Determine average Quality Level.




•   Problem 9.1 Page no 9.3 TQM by V.Jayakumar
np CHART


    P and np are quiet same
    Whenever subgroup size is variable, p chart is used.

           If sub group size is constant, then np is used.
    FORMULA: Central Line CLnp = n p
    Upper Control Limit, UCLnp = n p +3√ n p (1- p )
    Lower Control Limit, LCLnp = n p -3 √ n p (1- p )
    Where p = ∑ np/∑n =Average Fraction Defective
          n = Number of items inspected in subgroup.
•   Problem No 9.11 page No 9.11 in TQM by V.Jayakumar
c Chart

• (also called count chart)
• Type of attributes control chart
   – Discrete quantitative data
• Shows number of nonconformities (defects) in a unit
   – Unit may be chair, steel sheet, car etc.
   – Size of unit must be constant
• Example:
      Count no of defects (scratches, chips etc.) in

      each chair of a sample of 100 chairs;

       Plot
c Chart
        Control Limits


UCLc = c + 3 c           Use 3 for
                         99.7% limits

LCLc = c − 3 c
Control Chart


• Attribute Control Chart Templates
• p-Chart (Fraction or Percent of Defective Parts, Fraction or
  Percent Non-Conforming),
• np-Chart (Number of Defective Parts, Number of Non-
  Conforming),
• c-Chart (Number of Defects, Number of Non-Conformities ) and
• u-Chart (Number of Defects per Unit, Number of Non-Conformities
  Per Unit ).
• The p-Chart and u-Chart templates come in versions which
  support variable subgroup sample sizes.
p-Chart (Percent Defective Parts, or Percent Defective Non-Conforming)
np-Chart (Number Defective Parts, or Number Non-Conforming
c-Chart (Number Defects, or Number Non-Conformities)
u-Chart (Number of Defects Per Unit, or Number of Non-Conformities per Unit)
u-Chart with Variable Subgroup Sample Size
SPC control limits at the 1-, 2- and 3-sigma levels
Process capability
• Control limits- as a function of the averages
• Specifications-
    permissible variation in the
      size of the part, and are therefore, for

          individual values
• The specification or
    tolerance limits are established by
       design engineers to meet a particular
          function
• The specifications have an

     optional location
• The control limits,
     process spread ( process capability),
        distribution of averages, and

             distribution of individual values are
        interdependent and
      determined by the
Process capability
• Even the process (average value of items) is in control, the
     individual item may
         not be within the limits
• So it is necessary to see whether the

     process is capable of producing the items

        within the specified limits
• This can be achieved by
     carrying out the process capability
• Process capability is an industrial term that characterizes how
     tolerance specification of a product relates to the

        centering (bias) and
 - variation (Process Capability standard deviation, SD or s)
                       of the process.
• High capability means that the

      process can readily produce a product
Process capability

• Low capability means that the
       process will likely produce products
         outside the tolerance specifications
             (i.e., defective products or defects).
• Process capability may be defined as the
  minimum spread of the
  specific quality characteristic measurements
• Quality characteristic will have a normal distribution with
  mean μ and standard deviation σ
• The upper normal tolerance limit μ + 3 σ
• The lower normal tolerance limit μ – 3 σ
• The spread of the normal distribution between the
  natural tolerance limits 6 σ, is the process capability
• If the process capability 6 σ is less than the
  specification limits (USL-LSL), the process is capable,
  otherwise not
Process Capability
• The process capability ratio (PCR or Cp) is the ratio between
   the specification limits and the
          process capability
• PCR or Cp = (USL-LSL) / 6 σ
If Cp is >1.00, the process is capable of meeting the specifications
If Cp is < 1.00, the process is not capable of meeting the
   specifications
• One common measure of process capability is called
    process capability index Cpk, which is calculated as
   Cpk = (Tolerance specification - bias)/3SD.
   - upper capability index = CpU = (USL - μ) / 3 σ
   - lower capability index = CpL = (μ - LSL ) / 3 σ
   - process capability index Cpk = { min. CpU, CpL }
• If the tolerance specification were 12%,
• SD 2%, and
• Bias 0.0%,
   Cpk would be 2.00, which is considered the
      ideal capability,
• i.e., a six-sigma process because
       six multiples of the SD fit within the tolerance specification.
Process Capability
•  If the tolerance were 12%, SD 4%, and Bias 0.0%,
   Cpk would be 1.00,
  - which is considered the minimum capability for a production
   process and corresponds to a three-sigma process.
• If the tolerance were 12%, SD 2%, and Bias 3.0%,
    Cpk would be 1.50.
   Although this initially starts out as
             a six-sigma process when there is no bias,
   - the effect of a bias of 1.5 sigma actually reduces the process
   capability and makes this equivalent to a four-point-five
   process.
  This would still be considered a good production process if
   adequately controlled,
  - but it would still be desirable to eliminate the bias if possible
Process Capability
Parts per Million
• Many companies now measure defects in parts per million.
• We will recall that
          3 sigma deviations each side of the process mean will
  encompass 99.73% of the population.
• We have been looking at
  Process Capability using +- 3 sigma so we are really
  looking at 99.73% of the population.
• To give us some safety, we wanted the
          +- 3 sigma to fall within 75% of the tolerance.
• This equates to +- 4 sigma at 100% of the tolerance.
• If the +- 3 sigma had covered the total
      tolerance 0.27% would not be encapsulated in the spread that
  we were using.
• This would equate to 2,700 defective parts per million,
  1,350 exceeding top limit and
     1,350 failing to reach bottom limit.
• Many companies now try for figures much less than this.
• If you use +-5 sigma instead of +- 3 sigma in your calculations
  you will be fairly close to 1 part per million defects provided
  the process remains centralized and in control.
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Capability Ratio % Total = 6 sigma / Total Tolerance
Cm = Total Tolerance / 6 sigma
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Process Capability Analysis
.
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Process capability
Six sigma
• Six Sigma was intended to
     improve the quality of processes that are already
        under control –
  - major special causes of process problems have been removed.
  The output of these process usually follows a
    Normal distribution with the process capability defined as
   ± 3 sigma.
  The process mean will vary each time a process is executed
  using different equipment, different personnel, different
  materials, etc.
  The observed variation in the
  process mean was ± 1.5 sigma.
Motorola, one of the world’s leading manufacturers of electronic
  equipments introduced in 1980s, the concept of
          6 sigma process quality to enhance the quality and
  reliability of the products by the then CEO, Bob Galvin.
Motorola decided
    a design tolerance (specification width) of ± 6 sigma
  was needed so that there will be only
      3.4 ppm defects -- measurements outside the design
  tolerance.
  This was defined as Six Sigma quality.
1
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Six sigma
• Since shifts or biases equivalent to 1.5s are difficult to detect
  by statistical QC, a six-sigma process provides
   - better guarantee that products will be produced within the desired
  specifications and
   - with a low defect rate.
• Another way of looking at this is that a six-sigma process can
  be monitored with any QC procedure,
• e.g., with 3 SD limits and low N, and any important
  problems or errors will be detected and can be corrected.
• As process capability decreases to
• five-sigma to
• four-sigma to
• three-sigma,
• the choice of QC procedure becomes more and more
  important in order to detect important problems.
• Processes with lower capability may not even be controllable to a
  defined level of quality!
Allowable total error, TEa,
•   This latter situation is illustrated in the accompanying figure,
    - where the tolerance specification is replaced by a Total
    Error specification,
    - which is a common form of a quality specification for a
    laboratory test.
For example,
    - the CLIA criteria for acceptable performance in proficiency testing
    events are given in the form of an allowable total error, TEa,
    - thus there is a published list of TEa specifications for regulated
    analytes.
In terms of TEa,
    - Six Sigma Quality Management sets
      a precision goal of TEa/6 and
      an accuracy goal of 1.5(TEa/6) or TEa/4.
    - In terms of the industrial process capability,
      the combination of the six-sigma
      precision and
      accuracy goals results in a Cpk of 1.5.
Laboratory TE Criteria vs Process
                 Capability
• Laboratories evaluate process capability when they perform method
  validation studies.
• They don't calculate an index such as Cpk,
• they do combine the effects of inaccuracy and imprecision for
  comparison with the allowable total error.
• Commonly used TE criteria include
• TEa > bias + 4SD,
• TEa > bias + 3SD, and
• TEa > bias + 2SD, all of which are used on a decision-making
  tool called the Method Decision Chart.
• If the criterion requires that TEa > bias + 4SD,
• this corresponds to a four-sigma process if there is no bias,
• e.g., if TEa is 12%,
   bias is 0%, and
   SD is 3%,
  Cpk would be 1.33,
  which is a good production process that should be controllable to
  the desired quality.
Laboratory TE Criteria vs Process
                 Capability
• If the criterion requires that TEa > bias + 3SD,
  this corresponds to a three-sigma process if there is no bias,
  e.g., if TEa is 12%,
  bias is 0%, and
 SD is 4%,
 Cpk would be 1.00, which is the minimal capability needed for a
  production process.
• If the criterion requires that TEa > bias + 2SD,
  this corresponds to a two-sigma process if there is no bias,
  e.g., if TEa is 12%,
   bias is 0%, and
  SD is 6%,
  Cpk would be 0.67, which is unacceptable for production according
  to industrial guidelines.
• Process performance, as evaluated by commonly used laboratory
  TE criteria, does not approach the six-sigma capability desired for
  industrial processes. Improvements in laboratory methods are still
  needed to achieve five-sigma to six-sigma capability.
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Stages or fundamentals in six sigma
       the DMAIC Roadmap
DMAIC and Lean tools deployed in the
         Shewhart Cycle
DMAIC six sigma approach.


•   The six sigma approach for projects is
        DMAIC
     (define, measure, analyze, improve and control).
•   These steps are the most common
          six sigma approach to project work.
•   Some organizations omit the D in
    DMAIC because it is really management work.
•   With the D dropped from
     DMAIC the Black Belt is charged with
          MAIC only in that six sigma approach.
•    We believe define is
       too important be left out and sometimes management
    does not do an adequate job of defining a project.
•   Our six sigma approach is the full DMAIC
Define (DMAIC)
• Define is the first step in our six sigma approach of DMAIC
• DMAIC first asks leaders to define our core processes.
• It is important to define the selected
    project scope,
    expectations,
    resources and
    timelines.
• identifies specifically
   what is part of the project and
   what is not, and
   explains the scope of the project.
• Many times- process documentation are at a general level.
• Additional work is often required to adequately
  understand and correctly document the processes.
• As the saying goes “The devil is in the details.”
Measure (DMAIC)
    - most important thing to know is where we are going.
    - some of the first information you need before starting any
     journey is your current location.
•    The six sigma approach asks - to quantify and benchmark
     the process using actual data.
•    At a minimum
     consider the
       mean or average performance and some estimate of the
                                                 dispersion or
     variation
     (maybe even calculate the standard deviation).
     Trends and cycles can also be very revealing.
•    The two data points and
     extrapolate to infinity is
        not a six sigma  approach.
•    Process capabilities can be calculated
       once there is performance data
Analyze (DMAIC )
-   project is understood - baseline performance documented - verified
    that there is real opportunity,
-   then - do an analysis of the process.
-   the six sigma approach applies
    statistical tools to validate
      root causes of problems.
-   any number of tools and tests can be used.
       objective is to understand the process at a level sufficient

     to be able to formulate options for improvement.
• compare the various options with each other to determine the
  most promising alternatives.
• as with many activities, balance must be achieved.
• superficial analysis and understanding will lead to
  unproductive options being selected,
  forcing recycle through the process to make improvements.
• at the other extreme is the paralysis of analysis.
• striking the appropriate balance is what makes the
        six sigma highly valuable.
Improve (DMAIC )
• six sigma approach ideas and
   solutions are put to work.
  - discovered and validated all known
           root causes for the existing opportunity.
• The six sigma approach requires to
   identify  solutions.
• Few ideas or opportunities are so good that all are
        an instant success.
 - there must be checks to assure that the
   desired results are being achieved.
 - some experiments and
       trials may be required in order to find the best solution.
• When making trials and experiments
    it is important that all project associates understand that
    these are trials and
   really are
         part of the
              six sigma approach.
Control (DMAIC )
•   Many people believe the
        best performance you can ever get from a process is at
    the very beginning
•    Over time there is an expectancy that slowly things will get a little
    worse until finally it is time for another major effort towards
    improvement.
•   Contrasted with this is the
    Kaizen approach that seeks to make everything
    incrementally better on a continuous basis.
•    The sum of all
     these incremental improvements can be quite large.
•    As part of the
               six sigma approach performance tracking
    mechanisms and measurements are in place to assure,
    at a minimum, that the
         gains made in the project are not lost over a period of
    time.
•   As part of the control step we encourage
         sharing with others in the organization.
•    With this the six sigma approach really
         starts to create phenomenal returns, ideas and
•    projects in one part of the organization are
    translated in a very rapid fashion to implementation
           in another part of the organization.
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Dr.Kaoru Ishikawa, Professor at Tokyo University
            & Father of Q C in Japan.

• CAUSE ANALYSIS TOOLS are Cause and Effect diagram,
  Pareto analysis & Scatter diagram.
• EVALUATION AND DECISION MAKING TOOLS are
  decision matrix and multivoting
• DATA COLLECTION AND ANALYSIS TOOLS are check sheet,
  control charts, DOE, scatter diagram, stratification, histogram,
  survey.
• IDEA CREATION TOOLS are Brainstorming, Benchmarking,
  Affinity diagram, Normal group technique.
• PROJECT PLANNING AND IMPLEMENTATIONTOOLS are Gantt
  chart and PDCA Cycle.
Second seven tools
•   In the quality improvement movement in Japan in the
    latter half of the 20th century, the Japanese Union of
    Scientists and Engineers (JUSE) were influential in defining a
    set of basic tools that could be used for improving
    processes. These came to be known as the first seven tools.
•   These mostly were useful for quantitative problems, so a
    second set of seven tools was defined for the more
    qualitative problems that arise, such as around customer
    needs. These are:
•   Relations Diagram
•   Affinity Diagram
•   Tree Diagram
•   Matrix Diagram
•   Matrix Data Analysis Chart
•   Process Decision Program Chart
•   Activity Network
•   Just to complicate things, the Matrix Data Analysis Chart,
    which is somewhat complex to use, is often replaced with the
    Prioritization Matrix. And for further fun, alternative names are
    used, for example the Relations Diagram is sometimes called
    the 'Interrelationship Digraph'
Relations Diagram
•   In many problem situations, there are multiple complex
    relationships between the different elements of the problem,
    which cannot be organized into familiar structures such as
    hierarchies or matrices. The Relations Diagram addresses
    these situations by showing relationships between items with a
    network of boxes and arrows.
•   The most common use of the Relations Diagram is to show the
    relationship between one or more problems and their
    causes, although it can also be used to show any complex
    relationship between problem elements, such as
    information flow within a process.
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Affinity Diagram ('KJ' diagram)

•   A diagram that is used as a method of sorting qualitative data,
    which usually comes in the form of short phrases or sentences
    (eg. 'Customers are unhappy with delivery delays').
•   It is often done with Post-it Notes, although the original
    method used 3" x 5" cards.
•   It is a great method of working as a group to sort out issues
    and fuzzy situations.
•    It is also useful for sorting such as customer comments from
    surveys.
•   Building an Affinity Diagram is often known as 'doing a KJ',
    after its originator, Kawakita Jiro (this is in order of surname,
    given name, as in the Japanese tradition
Affinity diagram




Affinity diagram (KJ)
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Affinity Diagram components
Affinity Diagram -Moving the cards
.

Affinity Diagram

Exit interview comments
from checkout operators
Affinity diagram
Tree diagram
   • is used to find feasible measures for problem solving to clarify the
     content of an area to be improved through branching at each
     node or view point
   Creating a Tree diagram
   • Establish the basic objective, Think primary means, Deploy
     secondary means and beyond, Conform the relationship between
     the objectives and means and complete the tree diagram, Evaluate
     the basic means
   To minimize Gear box replacement Time Measures Development type tree
      diagram                                       Purchase
                            By machining
                                                     Fabricate




                                                                          Ideas to be produced by Brain storming
                                                     Reduce men
                      Improving co-ordination
                                                     Motivate

To minimize                                     Service training
 Gear box                 Improve expertise
Replacement                                     In house training
    time
                                                      Plan
                        Making RC/new units
                         Readily available      Collect in advance

                                                 Engage men for
                                                     cleaning
                       Improve house keeping
                                                Practice cleanliness
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Tree diagram
          or Systematic Diagram or Dendrogram




            Using the Tree Diagram in Problem Solving




Horizontal tree diagram                    Vertical tree diagram
Different
shapes
for Trees
Tree diagram
Why-Why Diagram
Tree diagram
How-How Diagram
Matrix Diagram
• The Matrix Diagram allows a many-to-many comparison of two lists,
    by turning the second list on its side to form a matrix.
 - indicated in the cell where the row and column of the two items cross.
 - use of different symbols to indicate different comparison levels and
    the weighting of the items being compared.
- different shapes of matrix for comparing more than the basic two
    lists. - the L-Matrix, C-Matrix, T-Matrix, X-Matrix and Y-Matrix.
• The Matrix Diagram is a core tool in Quality Function Deployment
    (QFD).
The C-Matrix is a variation on the Matrix Diagram, which is one of the
    second seven tools.
• The C-matrix compares three lists simultaneously, such as the
    people, products and processes in a factory.
• Being three-dimensional, it is difficult and complex to produce and
    draw.
• It becomes easier if there are few relationships to map.
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Matrix Diagram
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Tqm3 ppt

  • 1. MG 1401 TOTAL QUALITY MANAGEMENT 3. Statistical Process Control 2010-2011
  • 2. Statistical Process Control SPC is an effective system for controlling the process parameters by comparing it with standards and take corrective action if there is any deviation by employing statistical methods. SPC may be used to cover all uses of statistical techniques for the analysis of data that may be applied in the control of product quality
  • 3. Statistical Process Control Statistical techniques 1. The seven tools of quality used by the quality circles SEVEN QUALITY CONTROL TOOLS (or) OLD SEVEN TOOLS 2. Control charts Control charts for Variables ( X and R charts) and Process Capability Control charts for Attributes - Defectives (p and np charts) - Defects ( c and u charts ) 3. Concept of Six Sigma 4. Management tools- New seven management tools
  • 4. The Total Statistical Process Control (TSPC) System
  • 5. The seven tools of Quality- (Old) Q- 7 tools SEVEN QUALITY CONTROL TOOLS (or) OLD SEVEN TOOLS 1. Check Lists/ Check sheets 2. (Frequency) Histograms or Bar Graphs 3. Process flow diagrams / Charts 4. Cause and Effect, Fishbone, Ishikawa Diagram 5. Pareto Diagrams 6. Scatter Diagrams / Plots 7. Control Charts
  • 6. The seven tools of Quality ( Q- 7 tools) S.No Tools Problem Solving step 1 Check Lists/ Check How often it is For finding faults sheets done? 2 (Frequency) What do variations For identifying Histograms or Bar look like? problems Graphs 3 Process flow What is done? For understanding diagrams / Charts the “mess” 4 Cause and Effect, What cause the For generating ideas Fishbone, Ishikawa problem? Diagram 5 Pareto Diagrams Which are the big For identifying problems? problems 6 Scatter Diagrams / What are the For developing Plots relationships solutions between factors? 7 Control Charts Which variations to For implementation control and how?
  • 7. Check sheets/tally sheet (Data collection sheet) the intent and purpose of collecting data is to either control the production process, to see the relationship between cause-and-effect, or for the continuous improvement of those processes that produce any type of defect or nonconforming product.
  • 8. Check sheets/tally sheet (Data collection sheet) Check Sheet – collecting data to compile in such a way as to be easily used, understood and analyzed automatically. - as it is being completed, actually becomes a graphical representation of the data you are collecting, - thus you do NOT need any computer software, or spreadsheet to record the data. - it can be simply done with pencil and paper! - is a data recording form that has been designed to readily interpret results from the form itself. - needs to be designed for the specific data it is to gather.
  • 9. Check sheets/tally sheet (Data collection sheet) - used for the collection of quantitative or qualitative repetitive data. - adaptable to different data gathering situations. - minimal interpretation of results required. - easy and quick to use. - no control for various forms of bias – exclusion, interaction, perception, operational, non-response, estimation.
  • 10. CHECK SHEET (or) DEFECT CONCENTRATION DIAGRAM DESCRIPTION WHEN TO USE A check sheet is a When structured, data can be observed and prepared form for collecting and collected repeatedly by the analyzing data. same person or at the This is a same location. generic tool that can be When adapted for a collecting data on the frequency or wide variety of patterns of purposes events, problems, defects, defect location, defect causes, etc. When collecting data from a
  • 13. Check sheet (continuous data use) No.___________ 741 PRODUCT ION CHECK SHEET Product Name_________________________________ Alternator Pulley Date_________________________________ 12- 02- 02 Usage________________________________________ Pulley Bolt Torque Factory_______________________________ Church Street Specification__________________________________ 2.2 +/- .5 Section Name__________________________ SI Line No. of Inspections______________________________ 185 Data Collector__________________________ Sam The Man Total Number__________________________________ 185 Group Name___________________________ Lot Number___________________________________ 1631 Remarks:_____________________________ Dimensions 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 SpecUSL SpecLSL 40 35 30 25 XXXXX 20 XXXX XX XX XXXXX XXXXX XXXXX XXXXX XXXXX XXXXX 15 X X XXX XXX XX XXXXX XXXXX XXXXX XXXXX XXXXX XXXXX XXXXX XXXXX XXXXX XXXXX 10 XXX XX XXXXX XXXXX XXXXX XXXXX XXXXX XXXXX XXXXX XXXXX XXXXX XXXXX XXXXX XXXXX XXXXX XXXXX 5 X XX XX 0 X X TOTAL FREQUENCY 1 2 7 13 10 16 19 17 12 16 20 17 13 8 5 6 2 1
  • 14. Histogram or Bar Graph The word histogram is derived from Greek: - histos 'anything set upright‘ (as the masts of a ship, the bar of a loom, or the vertical bars of a histogram); - gramma 'drawing, record, writing'. A generalization of the histogram is kernal smoothing techniques. This will construct a very smooth probability density function from the supplied data - is a graphic summary of variation in a set of data. - it enables us to see patterns that are difficult to see in a simple table of numbers. - can be analyzed to draw conclusions about the data set. - continuous variable is clustered into categories and the value of each cluster is plotted to give a series of bars as above. - without using some form of graphic this kind of problem can be difficult to analyze, recognize or identify.
  • 15. Histogram In statistics, a histogram is a graphical display of tabulated frequencies. It shows what proportion of cases fall into each of several categories. A histogram differs from a bar chart in that it is the area of the bar that denotes the value, not the height, a crucial distinction when the categories are not of uniform width The categories are usually specified as non-overlapping intervals of some variable. The categories (bars) must be adjacent.
  • 16. Histogram or Bar Graph • H or B
  • 18. Histogram Grouping a set of measurements into a Histogram
  • 19. Histogram shapes Low with gaps Isolated-peaked High with few bars Cog-toothed (or comb) Skewed (this is positive; negative skew Plateau has tail to right) Exponential Edge-peaked missing bars Dual-peaked (bimodal) Truncated
  • 20. Bar chart Using the Bar Chart in problem-solving Numbers into bars
  • 21. . Pie chart Band Chart
  • 22. Flow Charts • Pictures, • symbols or • text coupled with lines, • arrows on lines show direction of flow. • enables modeling of processes; • problems/opportunities and decision points etc. • develops a common understanding of a process by those involved. • no particular standardization of symbology, • so communication to a different audience may require considerable time and explanation.
  • 23. . Basic Flowchart elements Decisions in Flowcharts
  • 24. . Continuing Flowcharts across pages Delay symbol Sub processes
  • 25. . Example Flowcharts
  • 29. Cause & Effect diagram Cause and Effect , Fishbone, Ishikawa Diagram - is the brainchild of Kaoru Ishikawa, who pioneered quality management processes in the Kawasaki shipyards, and in the process became one of the founding fathers of DR. KAORU ISHIKAWA (1915–1989) modern management. Disciple of Juran & Feigenbaum. TQC in Japan, SPC, - is used to Cause &Effect Diagram, QC. explore all the potential or real causes (or inputs) that result in a single effect (or output).
  • 30. Cause & Effect diagram (also called Ishikawa or fishbone chart ) DESCRIPTION WHEN TO USE - identifies - When identifying many possible causes possible causes for a for an problem. effect or - especially when a problem. team’s thinking tends to -can be used to fall into structure a ruts brainstorming session. - immediately sorts ideas into DR. KAORU ISHIKAWA (1915–1989) useful categories. Disciple of Juran & Feigenbaum. TQC in Japan, SPC, Cause &Effect Diagram, QC.
  • 31. Cause & Effect diagram Causes are arranged according to their level of importance or detail, resulting in a depiction of relationships and hierarchy of events. - This can help you search for root causes, identify areas where there may be problems, and compare the DR. KAORU ISHIKAWA (1915–1989) relative importance of different causes. Disciple of Juran & Feigenbaum. TQC in Japan, SPC, - Causes in a Cause &Effect Diagram, QC. cause & effect diagram are frequently arranged into four major categories. While these categories can be anything, you will often see: • manpower, methods, materials, and machinery (recommended for manufacturing) • equipment, policies, procedures, and people (recommended for administration and service).
  • 32. Cause & Effect diagram • is a method for analyzing process dispersion. • purpose is to relate causes and effects. • three basic types: 1. Dispersion analysis, 2. Process classification and 3. Cause enumeration. Effect = problem to be resolved, opportunity to be grasped, result to be achieved. • excellent for capturing team brainstorming output and for filling in from the 'wide picture'. • helps organize and relate factors, providing a sequential view. • deals with time direction but not quantity. • can become very complex. • can be difficult to identify or demonstrate interrelationships.
  • 33. Cause & Effect diagram
  • 34. Cause & Effect diagram
  • 35. Cause & Effect diagram
  • 36. Cause & Effect diagram
  • 37. Cause & Effect diagram This fishbone diagram was drawn by a manufacturing team to try to understand the source of periodic iron contamination. The team used the six generic headings to prompt ideas. Layers of branches show thorough thinking about the causes of the problem.
  • 38. Cause & Effect diagram
  • 39. Cause & Effect diagram
  • 40. Cause & Effect diagram (Standard )
  • 41. Pareto diagrams • Pareto diagrams are named after Vilfredo Pareto, • an Italian sociologist and economist, who invented this method of informationAlfredo Pareto (1848-1923) (Europe) presentation toward the end of the 19th century. • The chart is similar to the histogram or bar chart, except that • the bars are arranged in decreasing order from left to right along the abscissa. • The fundamental idea behind the use of Pareto diagrams • for quality improvement is that the first few (as presented on the diagram) contributing causes to a problem usually account for the majority of the result. • Thus, targeting these "major causes" for elimination results in the most cost-effective improvement scheme
  • 42. Pareto diagrams Pareto Principle • The Pareto principle suggests that most effects come from relatively few causes. Alfredo Pareto • In quantitative terms: (1848-1923) (Europe) • 80% of the problems come from 20% of the causes (machines, raw materials, operators etc.); • 80% of the wealth is owned by 20% of the people etc. • Therefore effort aimed at the right 20% can solve 80% of the problems. • Double (back to back) Pareto charts can be used to compare 'before and after' situations. • General use, to decide where to apply initial effort for maximum effect.
  • 43. Pareto diagrams Alfredo Pareto (1848-1923) (Europe)
  • 44. Pareto Chart (or) Pareto diagram (or) Pareto analysis A Pareto chart is a bar graph. The lengths of the bars represent frequency or cost (time or money), and are arranged with Alfredo Pareto (1848-1923) (Europe) longest bars on the left and the shortest to the right. To identify the ‘VITAL FEW FROM TRIVIAL MANY’ and to concentrate on the vital few for improvement. A Pareto diagram indicates which problem we should solve first in eliminating defects and improving the operation. The Pareto 80 / 20 rule 80 % of the problems are produced by 20 % of the causes.
  • 48. The Pareto Chart Finding the right Pareto Chart Convex Pareto Concave or 'spiky' Pareto (clearly allows you to prioritize the action) Prioritizing the action Alfredo Pareto (1848-1923) (Europe)
  • 49. Before and After . The Sub-Pareto Chart The Pareto Curve
  • 50. Scatter diagrams • are used to study possible relationships between two variables. • Although these diagrams cannot prove that one variable causes the other, they do indicate the existence of a relationship, as well as the strength of that relationship. • is composed of a horizontal axis containing the measured values of one variable and a vertical axis representing the measurements of the
  • 51. Scatter Diagram (or) Scatter plot (or) X–Y graph • The purpose of the scatter diagram is to display what happens to one variables when another variable is changed. • is used to test a theory that the two variables are related. • The type of relationship that exits is indicated by the slope of the diagram. • The scatter diagram graphs pairs of numerical data, with one variable on each axis, to look for a relationship between them. If the variables are correlated, the points will fall along a line or curve. The better the correlation, the tighter the points will hug the line.
  • 55. Scatter Diagram Types Positive Negative Degrees of correlation None High Curved Low Perfect Part-linear
  • 57. Control chart • In statistical process control, the control chart, also known as the 'Shewhart chart' or 'process-behaviour chart' is a tool used to determine whether a manufacturing or business process is in a state of statistical control or not. • If the chart indicates that the DR. WALTER A. SHEWHART process is (1891–1967) currently under control then it can be used with -TQC &PDSA confidence to predict the future performance of the process.
  • 58. Control chart • If the chart indicates that the process being monitored is not in control, the pattern it reveals can help determine the source of variation to be eliminated to bring the process back into control. • A control chart is a specific kind of DR. WALTER A. SHEWHART run chart that allows significant change to be (1891–1967) -TQC &PDSA differentiated from the natural variability of the process. • This is key to effective process control and improvement.
  • 59. Control chart • They enable the control of distribution of variation rather than attempting to control each individual variation. • Upper and lower control and tolerance limits are calculated for a process and sampled measures are regularly plotted about a central line between the two sets of limits. • The plotted line corresponds to the stability/trend of the process. • Action can be taken based on trend rather than on individual variation. • This prevents over-correction/compensation for random variation, which would lead to many rejects. DR. WALTER A. SHEWHART (1891–1967) -TQC &PDSA
  • 60. Control charts Mean and Control Limits How control limits catch shifts Additional limit lines
  • 63. Stratification Analysis • Stratification Analysis determines the extent of the problem for relevant factors. • o        Is the problem the same for all shifts?  • o        Do all machines, spindles, fixtures have the same problem?  • o        Do customers in various age groups or parts of the country have  similar problems?  • The important stratification factors will vary with each problem, but most problems will have several factors. • Check sheets can be used to collect data. • Essentially this analysis seeks to develop a pareto diagram for the important factors.
  • 64. Stratification Analysis • The hope is that the extent of the problem will not be the same across all factors. • The differences can then lead to identifying root cause. • When the 5W2H and Stratification Analysis are performed, it is important to consider a number of indicators. • For example, a customer problem identified by warranty claims may also be reflected by various in-plant indicators. • Sometimes, customer surveys may be able to define the problem more clearly. • In some cases analysis of the problem can be expedited by correlating different problem indicators to identify the problem clearly
  • 65. Stratification (or) Flowchart (or) Run chart • Stratification is a technique used in combination with other data analysis tools. When data from a variety of sources or categories have been lumped together, the meaning of the data can be impossible to see • When to Use • Before collecting data. • When data come from several sources or conditions, such as shifts, days of the week, suppliers or population groups. • When data analysis may require separating different sources or conditions.
  • 66. Benefit from stratification. • Always consider before collecting data whether stratification might be needed during analysis. Plan to collect stratification information. After the data are collected it might be too late. • On your graph or chart, include a legend that identifies the marks or colors used.
  • 67. Data Analysis • What is Data Analysis • Data analysis is statistics + visualization + know-how • Many of the methods for                                                                          data analysis are based on                                                    multivariate statistics, which poses an additional  problem to the beginner:  • multivariate statistics cannot be understood without  a profound knowledge of simple statistics.  • Furthermore, several                                                                                      fields in science and engineering have  developed their own nomenclature                                                                     assigning different names to the                                                       same concepts. 
  • 68. Data Analysis • Thus one has to                                                                             gather considerable knowledge and experience in  order to perform the analysis of data efficiently. •  Possible applications of statistical methods can be in the fields  of medicine, engineering, quality inspection, election polling, analytical chemistry, physics, gambling • Statistics and statistical methodology as the basis of  data analysis are concerned with two basic types of problems  • (1) summarizing, describing, and exploring the data (2) using sampled data to infer the nature of the process which produced the data • The first type of problems is covered by descriptive statistics, the second part is covered by inferential statistics. • Another important aspect of data analysis is the data, which  can be of two different types: qualitative data, and  quantitative data .
  • 69. Data Analysis • Qualitative data does not contain quantitative information. • Qualitative data can be classified into categories. • In contrast, quantitative data represent an amount of something. • A third distinction can be made according to the      number of variables involved in the data analysis. • If only one variable is used, the statistical procedures are summarized as univariate statistics. • More than one variable result in multivariate statistics. • A special case of multivariate statistics with                                           only two variables is sometimes called bivariate statistics.
  • 71. Statistical Process Control (SPC) Measures performance of a process Uses mathematics (i.e., statistics) Involves collecting, organizing, & interpreting data Objective: Regulate product quality Used to – Control the process as products are produced – Inspect samples of finished products
  • 72. Statistical Process Control What is a process? Inputs PROCESS Outputs A process can be described as a transformation of set of inputs into desired outputs.
  • 73. WHY STATISTICS? THE ROLE OF STATISTICS ……… µ LSL T USL Statistics is the art of collecting, classifying, presenting, interpreting and analyzing numerical data, as well as making conclusions about the system from which the data was obtained. January 9, 2013 73
  • 74. Descriptive Statistics Descriptive Statistics is the branch of statistics which most people are familiar. It characterizes and summarizes the most prominent features of a given set of data (means, medians, standard deviations, percentiles, graphs, tables and charts). January 9, 2013 74
  • 75. Inferential Statistics Inferential Statistics is the branch of statistics that deals with drawing conclusions about a population based on information obtained from a sample drawn from that population. January 9, 2013 75
  • 76. Statistics MEASURES OF CENTRAL TENDENCY Usually, the first topic in statistics is descriptive statistics. The mean, median and mode are used to describe statistical data. Variance and standard deviation help to understand how the data is spread out. Central tendency is a typical or representative score. The three measures of central tendency are the mode, median, and mean.
  • 77. The term "measures of central tendency" refers to . finding the mean, median and mode. Mean: Average. The sum of a set of data divided by the number of data. (Do not round your answer unless directed to do so.) Median: The middle value, or the mean of the middle two values, when the data is arranged in numerical order. Think of a "median" being in the middle of a highway. Mode: The value ( number) that appears the most. It is possible to have more than one mode, and it is possible to have no mode. If there is no mode-write "no mode", do not write zero (0) .
  • 78. Mode Mode is the data value that occurs at a greater frequency than the others. Data: 1, 2, 3, 3, 3, 4, 4, 5 Mode = 3 The mode, symbolized by Mo, is the most frequently occurring score value. If the scores for a given sample distribution are: 32 32  35  36  37  38  3 8  39 3 9 39  40  4 0  42  45  - then the mode would be 39 because a score of 39 occurs 3 times, more than any other score. - The mode may be seen on a frequency distribution as the score value which corresponds to the highest point. - For example, the following is a frequency polygon of the data presented above:
  • 79. Statistics - Mode 3 2 2 1 1 32 36 39 42 45 scores
  • 80. Statistics - Mode A distribution may have - more than one mode if the two most frequently occurring scores occur the same number of times. For example, if the earlier score distribution were modified as follows: 32 32 32  36  37  38  38  39 39 39  40  40  42  45  - then there would be two modes, 32 and 39. - Such distributions are called bimodal. - The frequency polygon of a bimodal distribution is presented below.
  • 82. Statistics - Mode In an extreme case there may be no unique mode, as in the case of a rectangular distribution. The mode is not sensitive to extreme scores. Suppose the original distribution was modified by changing the last number, 45, to 55 as follows: 32  32  35  36  37  38  38  39 39 39  40  40  42  55 The mode would still be 39. In any case, the mode is a quick and dirty measure of central tendency. Quick, because it is easily and quickly computed. Dirty because it is not very useful; that is, - it does not give much information about the distribution.
  • 84. Statistics - Median Median The median is the exact middle value of a set of data values that have been sorted from the lowest value to highest. If the number of data values even, then the median is the average of the two middle values. Examples: Data: 1, 2, 3, 4, 5 Median: 3 Data: 1, 2, 3, 4, 5, 6 Median: 3.5
  • 85. Statistics - Median The median, symbolized by Md, is the score value which cuts the distribution in half, such that half the scores fall above the median and half fall below it. Computation of the median is relatively straightforward. The first step is to rank order the scores from lowest to highest. The procedure branches at the next step: - one way if there are an odd number of scores in the sample distribution, - another if there are an even number of scores. If there is an odd number of scores as in the distribution below: 32  32  35  36  36  37  38                                                                  38                                                                          39  39  39  40  40  45  46  - then the median is simply the middle number. In the case above the median would be the number 38, because there are 15 scores all together with 7 scores smaller and 7 larger.
  • 86. Statistics - Median • If there is an even number of scores, as in the distribution below:       32  35  36  36  37  38                                             38 39                                                                       39  39  40  40  42  45 • then the median is the midpoint between the two middle scores:  • in this case the value 38.5.  • It was found by adding the two middle scores together and dividing  by two (38 + 39)/2 = 38.5. •  If the two middle scores are the same value then the                                   median is that value.  • In the above system, no account is paid to whether there is a                                      duplication of scores around the median.  • In some systems a slight correction is performed to correct for  grouped data, but since the correction is slight and the data is  generally not grouped for computation in calculators or computers, it  is not presented here. 
  • 87. Statistics - Median The median, like the mode, is                                                                   not effected by extreme scores, as the following  distribution of scores indicates:       32  35  36  36  37  38                                            38 39                                                                        39  39  40  40  42  55 The median is still the value of 38.5.  The median is not as                                                                                  quick and      dirty as the     mode, but     generally it is     not the  preferred measure of      central tendency.
  • 88. Statistics – The Mean The mean,       symbolized by         ,      is the sum of the scores divided by the number of scores.  The following formula both defines and describes the procedure for  finding the mean:   where X is the sum of the scores and   N is the number of scores.
  • 89. Statistics – The Mean Mean The mean most frequently used is the  arithmetic mean, which is the same as the average, although the                                                                        geometric mean is          used also          at times.  It is the arithmetic mean that is referred to when the word mean is  used by itself.  Expected value is another way of saying the mean. mean =    sum of data values / number of data values          mean μ =    (1 + 2 + 3) / 3    =    6 / 3    =   2   The lower case Greek letter μ is used to represent the mean. If the mean is from a sample of data, then                                                      x is used to represent the sample mean.  Also, variables and Sigma notation is used to write the       general form of the mean.
  • 90. Statistics – The Mean The mean  Application of this formula to the following data   32  35  36  37  38  38  39  39  39  40  40  42  45  yields the following results:  Use of means as a way of describing a set of scores is fairly  common; •      batting average, •      bowling average,  •      grade point average, and  •      average points scored per game                are all means.  Note the use of the word "average" in all of the above terms.  In most cases when the term "average" is used, it refers to the mean, although not necessarily.  When a politician uses the term "average income", for example,  he or she may be referring to the mean, median, or mode. 
  • 91. Statistics – The Mean The mean is sensitive to extreme scores. For example, the mean of the following data is 39.0, somewhat larger than the preceding example.  32  35  36  37  38  38  39  39  39  40  40  42  55 In most cases the mean is the       preferred measure of central tendency, both as a description of the       data and as an estimate of the parameter.  In order for the mean to be meaningful, however, the       acceptance of the interval property of       measurement is       necessary. When this property is obviously violated, it is inappropriate and  misleading to compute a mean.
  • 92. Kiwi Bird Problem As is commonly known,                                                                                            KIWI-birds are native to                                                                                                              New Zealand. They are born                                                                                  exactly one foot tall and grow                                                                          in one foot intervals.  That is, one moment they are one foot tall and the next they are two  feet tall. They are also very rare.  An investigator goes to New Zealand and                                                        finds four birds.  The                                                                                                                             mean of the heights of four birds is 4, the                                                median is 3, and the                                                                                         mode is 2. What are the heights of the four birds? • Hint - examine the constraints of the mode first, the median second,  and the mean last.
  • 93. Statistics • Skewed Distributions and Measures of Central Tendency • Skewness refers to the asymmetry of the distribution, such that  a symmetrical distribution exhibits                                                        no skewness. In a symmetrical distribution the mean, median, and mode all fall at the same point,  as in the following distribution.
  • 94. Statistics • An exception to this is the case of a bi-modal symmetrical distribution.  • In this case the mean and the median fall at the same point,  while the two modes correspond to the two highest points of  the distribution. An example follows:
  • 95. Statistics • A positively skewed distribution is asymmetrical and points in the positive direction. If a test was very difficult and almost  everyone in the class did very poorly on it, the resulting distribution  would most likely be positively skewed.  • In the case of a positively skewed distribution, the mode is smaller than the median, which is smaller than the mean. This relationship exists because the mode is the point on  the x-axis corresponding to the highest point, that is the score  with greatest value, or frequency. The median is the point on the  x-axis that cuts the distribution in half, such that 50% of the  area falls on each side.  Mo
  • 96. Statistics • The mean is the balance point of the distribution. Because  points further away from the balance point change the center of  balance, the mean is pulled in the direction the distribution is  skewed. For example, if the distribution is positively skewed, the mean would be pulled in the direction of the skewness, or  be pulled toward larger numbers. • One way to remember the order of the mean, median, and mode in a skewed distribution is to remember that the mean is  pulled in the direction of the extreme scores. In a  positively skewed distribution, the extreme scores are larger, thus  the mean is larger than       the median.  
  • 98. Statistics • A negatively skewed distribution is asymmetrical and points  in the negative direction, such as would result with a very easy test.  On an easy test, almost all students would perform well and only a  few would do poorly.  • The order of the measures of central tendency would be the  opposite of the positively skewed distribution, with the mean being smaller than the median, which is smaller than the mode.  
  • 99. Statistics MEASURES OF VARIABILITY • Variability refers to the spread or dispersion of scores.  • A distribution of scores is said to be highly variable if the  scores differ widely from one another.  • Three statistics will be discussed which measure variability: the range, the variance, and the standard deviation.  • The latter two are very closely related.  Range Range is the highest data value minus the lowest data values.           Data: 1, 2, 3, 4, 5, 6, 7  Highest Data Value: 7  Lowest Data Value: 1 Range: 7 - 1 = 6 • It is a quick and dirty measure of variability, although when a test  is given back to students they very often wish to know the range of  scores.  • Because the range is greatly affected by extreme scores, it may give a distorted picture of the scores. 
  • 100. Statistics • The following two distributions have the same range, 13, yet appear  to differ greatly in the amount of variability.  • Distribution 1 32 35 36 36 37 38 40 42 42 43 43 45 • Distribution 2 32 32 33 33 33 34 34 34 34 34 35 45 • For this reason, among others, the range is not the most important measure of variability.  Variance • Variance is used to measure                        how far the data is away from the mean. • The distance of the data point from the mean is a deviation. •  The deviations are added together to get a value  representing all the deviations together.  • However, since some deviations can be negative, the total could be zero. • To account for this, the deviations are squared and then  added together. • When divided by the number of deviations, the result is variance.
  • 101. Statistics Standard Deviation • The standard deviation is just the square root of the variance.  • A statistic is an algebraic expression combining scores into a single  number.  • Statistics serve two functions: • they estimate parameters in population models and they describe the               data. • i.e., Population variance, standard deviation Sample variance, standard deviation • The variance, symbolized by                 "s2", is a measure of variability. Th              standard deviation, symbolized by                 "s", is the positive square root of the variance.
  • 102. Statistics • Variance :formula:  • Note that the variance could almost be the  average squared deviation around          the mean if the expression were divided by N rather than N-1. •  It is divided by N-1, called the degrees of freedom (df), for  theoretical reasons. •  If the mean is known, as it must be to compute the numerator of the  expression, then only N-1 scores that are free to vary.  • That is if the mean and N-1 scores are known, then it is possible to  figure out the Nth score.  • One needs only recall the KIWI-bird problem to convince oneself  that this is in fact true. 
  • 103. Statistics • The formula for the variance presented above is a definitional formula, it defines  what the variance means.  • The variance may be computed from this formula, but in  practice this is rarely done.  • The computation is performed in a number of steps, which are  presented below: Steps 1. Find the mean of the scores. 2. Subtract the mean from every score. 3. Square the results of step 2. 4. Sum the results of step 3. 5. Divide the results of step 4 by N-1. 6. Take the square root of step 5. 7. The result at step 5 is the sample variance,                           at step 6, the sample standard deviation.
  • 104. Measures of dispersion Quartiles If we divide a cumulative frequency curve into  quarters,               the value at the lower quarter is referred to as  the lower quartile,               the value at the middle gives the median and the  value at the upper quarter is the upper quartile. A set of numbers may be as follows:           8, 14, 15, 16, 17, 18, 19, 50.          The mean of these numbers is 19.625 . However,  the extremes in this set (8 and 50) distort the range.          The inter quartile range is a method of measuring  the spread of the numbers by finding the middle  50% of the values.           It is useful since it ignore the extreme values. It is a  method of measuring the spread of the data. The lower quartile is (n+1)/4 th value     (n is the cumulative frequency, ie 157 in this case)  and          the upper quartile is the 3(n+1)/4 the value.          The difference between these two is the inter  quartile range (IQR). In the above example,           the upper quartile is the 118.5th value and           the lower quartile is the 39.5th value.   If we draw a cumulative frequency curve, we see that  the lower quartile, therefore, is about 17 and            the upper quartile is about 37.  Therefore the IQR is 20 (bear in mind that this is a rough  sketch- if you plot the values on graph paper you  will get a more accurate value). 
  • 105. Measures of dispersion measure how spread out a set of data is. Variance and Standard Deviation The formulae for the variance and standard deviation are given below. m means the mean of the data. . Variance = s2 = S (xr – m)2 n The standard deviation, s, is the square root of the variance. What the formula means: (1) xr - m means take each value in turn and subtract the mean from each value. (2) (xr - m)² means square each of the results obtained from step (1). This is to get rid of any minus signs. (3) S(xr - m)² means add up all of the results obtained from step (2). Example: (4) For variance divide step (3) by n, which is the number of numbers Find the variance and standard deviation of the following numbers:  the answer to step (4). (5) For the standard deviation, square root          1, 3, 5, 5, 6, 7, 9, 10 .                 The mean = m = 46/ 8 = 5.75  x          (Step 1):  (Step 2):         xr - m         1  (1 - 5.75)  -4.75 (xr - m)² (Step 4):  n = 46, therefore  3  (3 - 5.75)  -2.75  22.563       variance = 61.504/ 46 = 1.34 (3sf)  5  (5 - 5.75)   -0.75     7.563 (Step 5):     5  (5 - 5.75)   -0.75     0.563         standard deviation = 1.16 (3sf)  6  (6 - 5.75)    0.25     0.563   7  (7 - 5.75)    1.25     0.063   9  (9 - 5.75)    3.25     1.563 10 (10 - 5.75)   4.25  10.563  46               18.063 (n) (Step 3): S(xr - m)² 61.504
  • 106. Grouped Data There are many ways of writing the formula for the standard deviation. The one above is for a population of numbers. The formula for the standard deviation when the data is grouped is: . Example: The table shows marks (out of 10) such questions, it is often easiest to In obtained by 20 people in a test set your working out in a table: Mark (x) Frequency (f) fx fx² 1 0 0 0 2 1 Sf = 20 2 4 3 1 3 9 Sfx = 118 4 3 12 48 Sfx² = 764 5 2 10 50 variance = Sfx² - ( Sfx )² 6 5 30 180 Sf ( Sf ) 7 5 35 245 = 764 - (118)² 8 2 16 128 20 ( 20 ) 9 0 0 0 = 38.2 - 34.81 = 3.39 10 1 10 100 Work out the variance of this data. 20 118 764
  • 107. Population Vs. Sample (Certainty Vs. Uncertainty) A sample is just a subset of all possible values sample population Since the sample does not contain all the possible values, there is some uncertainty about the population. Hence any statistics, such as mean and standard deviation, are just estimates of the true population parameters. January 9, 2013 107
  • 109. Population and sample Population is the entire (complete) collection of all the  measurements of an observed quality characteristic         -variation pattern is not known A Sample is a collection of measurements selected from some large  source or population  - i.e., a part of the population Population: Smooth curve       (‘) prime symbol is used to identify parameters Parameters: mean (μ), population standard deviation (σ ). - has finite number of items  e.g., production of shafts in a day   - it is impossible to measure all the population   - the conclusion about the population is derived from the  mean and standard deviation of the sample Types:    Finite population             - finite number                Infinite population           - infinite number                Existent population         - concrete individuals                Hypothetical population  - possible ways- population of head                  and tail obtained by tossing a coin an infinite number of times
  • 110. Population and sample Sample: Statistic – average (x), and sample standard deviation (s) Histogram To analyze  and draw  conclusion about the universe, a sample  is  selected at random to represent the population - small section selected is - sample - process of such selection is - sampling
  • 112. Normal curve -the normal curve is the most important frequency curve - is also known as Gaussian curve and probability curve - is symmetrical - is unimodal - is bell shaped distribution with mean, median, and mode having      the same value. -The normal distribution is fully defined by the population mean and population 50.00%   standard deviation 68.26% 95.45% 99.73% -∞ +∞ -3σ -2σ -1σ +1σ +2σ +3σ -0.6745σμ +0.6745σ Area under the normal distribution curve
  • 113. Normal curve The mean (μ), and standard deviation (σ ) are the population parameters The mean (x), and standard deviation (s) are for the                sample quantity drawn from the population For practical usage It is necessary to convert from mean values and standard deviations  other than zero and one respectively This procedure called normalizing involves substituting   z = x – μ thus the values read from the table represent the area  σ ∞ under normal curve from –        to  z = x - μ σ σ= 1.5  Normal curves with different standard deviations but identical means σ= 3.0 σ= 4.5  5 8 11 14 17 20 23 26 29 32 35
  • 116. Variance Shown in a Probability Distribution
  • 117. Control Chart (or) Statistical process control VARIATIONS • Different types of control charts can be used, depending upon the type of data. • The two broadest groupings are for variable data and attribute data. • Variable data are measured on a continuous scale. For example: time, weight, distance or temperature can be measured in fractions or decimals. The possibility of measuring to greater precision defines variable data.
  • 118. Attribute data are counted and cannot have fractions or decimals. Attribute data arise when you are determining only the presence or absence of something: success or failure, accept or reject, correct or not correct. For example, a report can have four errors or five errors, but it cannot have four and a half errors.
  • 119. Control Chart • is a graphical representation of the collected information - information may be measured quality characteristics • detects the variation in the process and • warns if there is any deviation from the specification Essential features of a control chart Upper Control Limit Variable Values Central Line Lower Control Limit Time
  • 120. Control Chart Purposes • Show changes in data pattern – e.g., trends • Make corrections before process is out of control • Show causes of changes in data – Assignable causes • Data outside control limits or trend in data – Natural causes • Random variations around averag In the charts, • If all the points (sample averages and ranges) are within the control limits, - then the process is said to be in “Statistical control” • If any one point or more in the control charts go outside the control limits, - then the process is said to be “out of control”
  • 121. Quality Characteristics Variables Attributes 1. Characteristics that you 1. Characteristics for which you measure, focus on defects e.g., weight, length 2. May be in whole or 2. Classify products as either in fractional numbers ‘good’ or ‘bad’, or count # defects – e.g., radio works or not 3. Continuous random variables 3. Categorical or discrete random variables
  • 122. Control Chart Types Control Charts Variables Attributes Charts Charts R X p & np c&u Chart Chart Chart Chart
  • 123. Control Chart Classification: For variables- X and R charts • Measures where the metric consists of a number which indicates a precise value is called Variable data. – Time – Miles/Hr For variables Sample average Range Grant average Grant range Control limits X chart R chart - upper limit - upper limit - lower limit - lower limit Both the charts should be plotted together If the sub group size is 6 or less LCLR = 0
  • 124. Variables charts – X and R chart (also called averages and range chart) – X and s chart – chart of individuals (also called X chart, X-R chart, IX- MR chart, Xm R chart, moving range chart) – moving average–moving range chart (also called MA– MR chart) – target charts (also called difference charts, deviation charts and nominal charts) – CUSUM (also called cumulative sum chart) – EWMA (also called exponentially weighted moving average chart) – multivariate chart (also called Hotelling T2)
  • 125. X Chart Type of variables control chart – Interval or ratio scaled numerical data • Shows sample means over time • Monitors process average and tells whether changes have occurred. These changes may due to 1. Tool wear 2. Increase in temperature 3. Different method used in the second shift 4. New stronger material • Example: Weigh samples of coffee & compute means of samples; Plot
  • 126. R Chart Type of variables control chart – Interval or ratio scaled numerical data • Shows sample ranges over time – Difference between smallest & largest values in inspection sample • Monitors variability in process, • it tells us the loss or gain in dispersion. This change may be due to: 1. Worn bearing 2. A loose tool 3. An erratic flow of lubricant to machine 4. Sloppiness of machine operator • Example: Weigh samples of coffee & compute ranges of samples;
  • 127. Construction of X and R Charts • Step 1: Select the Characteristics for applying a control chart. • Step 2: Select the appropriate type of control chart. • Step 3: Collect the data. • Step 4: Choose the rational sub-group i.e Sample • Step 5: Calculate the average ( X) and range R for each sample. • Step 6: Cal Average of averages of (X) and average of range (R) • Steps 7:Cal the limits for X and R Charts. • Steps 8: Plot Centre line (CL), UCL and LCL on the chart • Steps 9: Plot individual X and R values on the chart. • Steps 10: Check whether the process is in control (or) not. • Steps 11: Revise the control limits if the points are outside.
  • 128. X Chart Control Limits UCL = x + A R From x 2 Tables LCL = x − A R x 2 Sub group average X = (x1 + x2 +x3 +x4 +x5 ) /5 Sub group range R = Max Value – Min value
  • 129. R Chart Control Limits UCL R = D 4 R From Tables LCL R = D 3 R Problem8.1 from TQM by V.Jayakumar Page No 8.5
  • 130. Control Chart Description Control charts for individual measurements (e.g., the sample size = 1) use the moving range of two successive observations to measure the process variability. The combination of the X Chart for Individuals and the Moving Range chart is often called an X and Rm or XmR Chart.
  • 131. X-Bar R Chart (Mean-Range Chart)
  • 132. X-Bar Sigma Chart (Mean-Sigma Chart)
  • 136. X-Bar Sigma Chart with Variable Subgroup Sample Size
  • 137. EWMA (Exponentially Weighted Moving Average Chart)
  • 138. MA Chart (Moving Average Chart)
  • 139. CuSum Chart (Tabular Cumulative Sum Chart )
  • 140. Types of Control Charts for Attribute Data Measures where the metric is composed of a classification in one or two (or more) categories is called Attribute data. Description Type Sample Size Control Chart for (defectives) p Chart may change proportion non conforming units Control Chart for (defectives) np Chart must be constant no. of non conforming units in a sample Control Chart for (defects) c Chart must be constant no. of non conformities in a sample Control Chart for (defects) u Chart may Change no. of non conformities per unit
  • 141. p Chart for Attributes (also called proportion chart) Type of attributes control chart – Nominally scaled categorical data • e.g., good-bad, Yes/No • Shows % of nonconforming items Example: Count # defective chairs & divide by total chairs inspected; Plot – Chair is either defective or not defective
  • 142. p Chart (also called proportion chart) p = np / n where p = Fraction of Defective np = no of Defectives n = No of items inspected in sub group p = Average Fraction Defective = ∑np/ ∑n = CL p (1 −p ) UCL p =p +z n p (1 −p ) LCL p =p −z n
  • 143. p Chart Control Limits p (1 − p ) UCL p = p + z n z = 3 for 99.7% limits p (1 − p ) LCL p = p − z n
  • 144. Purpose of the p Chart Identify and correct causes of bad quality The average proportion of defective articles submitted for inspection, over a period. To suggest where X and R charts to be used. Determine average Quality Level. • Problem 9.1 Page no 9.3 TQM by V.Jayakumar
  • 145. np CHART P and np are quiet same Whenever subgroup size is variable, p chart is used. If sub group size is constant, then np is used. FORMULA: Central Line CLnp = n p Upper Control Limit, UCLnp = n p +3√ n p (1- p ) Lower Control Limit, LCLnp = n p -3 √ n p (1- p ) Where p = ∑ np/∑n =Average Fraction Defective n = Number of items inspected in subgroup. • Problem No 9.11 page No 9.11 in TQM by V.Jayakumar
  • 146. c Chart • (also called count chart) • Type of attributes control chart – Discrete quantitative data • Shows number of nonconformities (defects) in a unit – Unit may be chair, steel sheet, car etc. – Size of unit must be constant • Example: Count no of defects (scratches, chips etc.) in each chair of a sample of 100 chairs; Plot
  • 147. c Chart Control Limits UCLc = c + 3 c Use 3 for 99.7% limits LCLc = c − 3 c
  • 148. Control Chart • Attribute Control Chart Templates • p-Chart (Fraction or Percent of Defective Parts, Fraction or Percent Non-Conforming), • np-Chart (Number of Defective Parts, Number of Non- Conforming), • c-Chart (Number of Defects, Number of Non-Conformities ) and • u-Chart (Number of Defects per Unit, Number of Non-Conformities Per Unit ). • The p-Chart and u-Chart templates come in versions which support variable subgroup sample sizes.
  • 149. p-Chart (Percent Defective Parts, or Percent Defective Non-Conforming)
  • 150. np-Chart (Number Defective Parts, or Number Non-Conforming
  • 151. c-Chart (Number Defects, or Number Non-Conformities)
  • 152. u-Chart (Number of Defects Per Unit, or Number of Non-Conformities per Unit)
  • 153. u-Chart with Variable Subgroup Sample Size
  • 154. SPC control limits at the 1-, 2- and 3-sigma levels
  • 155. Process capability • Control limits- as a function of the averages • Specifications- permissible variation in the size of the part, and are therefore, for individual values • The specification or tolerance limits are established by design engineers to meet a particular function • The specifications have an optional location • The control limits, process spread ( process capability), distribution of averages, and distribution of individual values are interdependent and determined by the
  • 156. Process capability • Even the process (average value of items) is in control, the individual item may not be within the limits • So it is necessary to see whether the process is capable of producing the items within the specified limits • This can be achieved by carrying out the process capability • Process capability is an industrial term that characterizes how tolerance specification of a product relates to the centering (bias) and - variation (Process Capability standard deviation, SD or s) of the process. • High capability means that the process can readily produce a product
  • 157. Process capability • Low capability means that the process will likely produce products outside the tolerance specifications (i.e., defective products or defects). • Process capability may be defined as the minimum spread of the specific quality characteristic measurements • Quality characteristic will have a normal distribution with mean μ and standard deviation σ • The upper normal tolerance limit μ + 3 σ • The lower normal tolerance limit μ – 3 σ • The spread of the normal distribution between the natural tolerance limits 6 σ, is the process capability • If the process capability 6 σ is less than the specification limits (USL-LSL), the process is capable, otherwise not
  • 158. Process Capability • The process capability ratio (PCR or Cp) is the ratio between the specification limits and the process capability • PCR or Cp = (USL-LSL) / 6 σ If Cp is >1.00, the process is capable of meeting the specifications If Cp is < 1.00, the process is not capable of meeting the specifications • One common measure of process capability is called process capability index Cpk, which is calculated as Cpk = (Tolerance specification - bias)/3SD. - upper capability index = CpU = (USL - μ) / 3 σ - lower capability index = CpL = (μ - LSL ) / 3 σ - process capability index Cpk = { min. CpU, CpL } • If the tolerance specification were 12%, • SD 2%, and • Bias 0.0%, Cpk would be 2.00, which is considered the ideal capability, • i.e., a six-sigma process because six multiples of the SD fit within the tolerance specification.
  • 159. Process Capability • If the tolerance were 12%, SD 4%, and Bias 0.0%, Cpk would be 1.00, - which is considered the minimum capability for a production process and corresponds to a three-sigma process. • If the tolerance were 12%, SD 2%, and Bias 3.0%, Cpk would be 1.50. Although this initially starts out as a six-sigma process when there is no bias, - the effect of a bias of 1.5 sigma actually reduces the process capability and makes this equivalent to a four-point-five process. This would still be considered a good production process if adequately controlled, - but it would still be desirable to eliminate the bias if possible
  • 160. Process Capability Parts per Million • Many companies now measure defects in parts per million. • We will recall that 3 sigma deviations each side of the process mean will encompass 99.73% of the population. • We have been looking at Process Capability using +- 3 sigma so we are really looking at 99.73% of the population. • To give us some safety, we wanted the +- 3 sigma to fall within 75% of the tolerance. • This equates to +- 4 sigma at 100% of the tolerance. • If the +- 3 sigma had covered the total tolerance 0.27% would not be encapsulated in the spread that we were using. • This would equate to 2,700 defective parts per million, 1,350 exceeding top limit and 1,350 failing to reach bottom limit. • Many companies now try for figures much less than this. • If you use +-5 sigma instead of +- 3 sigma in your calculations you will be fairly close to 1 part per million defects provided the process remains centralized and in control.
  • 162. Capability Ratio % Total = 6 sigma / Total Tolerance Cm = Total Tolerance / 6 sigma
  • 166. .
  • 169. Six sigma • Six Sigma was intended to improve the quality of processes that are already under control – - major special causes of process problems have been removed. The output of these process usually follows a Normal distribution with the process capability defined as ± 3 sigma. The process mean will vary each time a process is executed using different equipment, different personnel, different materials, etc. The observed variation in the process mean was ± 1.5 sigma. Motorola, one of the world’s leading manufacturers of electronic equipments introduced in 1980s, the concept of 6 sigma process quality to enhance the quality and reliability of the products by the then CEO, Bob Galvin. Motorola decided a design tolerance (specification width) of ± 6 sigma was needed so that there will be only 3.4 ppm defects -- measurements outside the design tolerance. This was defined as Six Sigma quality.
  • 170. 1
  • 174. Six sigma • Since shifts or biases equivalent to 1.5s are difficult to detect by statistical QC, a six-sigma process provides - better guarantee that products will be produced within the desired specifications and - with a low defect rate. • Another way of looking at this is that a six-sigma process can be monitored with any QC procedure, • e.g., with 3 SD limits and low N, and any important problems or errors will be detected and can be corrected. • As process capability decreases to • five-sigma to • four-sigma to • three-sigma, • the choice of QC procedure becomes more and more important in order to detect important problems. • Processes with lower capability may not even be controllable to a defined level of quality!
  • 175. Allowable total error, TEa, • This latter situation is illustrated in the accompanying figure, - where the tolerance specification is replaced by a Total Error specification, - which is a common form of a quality specification for a laboratory test. For example, - the CLIA criteria for acceptable performance in proficiency testing events are given in the form of an allowable total error, TEa, - thus there is a published list of TEa specifications for regulated analytes. In terms of TEa, - Six Sigma Quality Management sets a precision goal of TEa/6 and an accuracy goal of 1.5(TEa/6) or TEa/4. - In terms of the industrial process capability, the combination of the six-sigma precision and accuracy goals results in a Cpk of 1.5.
  • 176. Laboratory TE Criteria vs Process Capability • Laboratories evaluate process capability when they perform method validation studies. • They don't calculate an index such as Cpk, • they do combine the effects of inaccuracy and imprecision for comparison with the allowable total error. • Commonly used TE criteria include • TEa > bias + 4SD, • TEa > bias + 3SD, and • TEa > bias + 2SD, all of which are used on a decision-making tool called the Method Decision Chart. • If the criterion requires that TEa > bias + 4SD, • this corresponds to a four-sigma process if there is no bias, • e.g., if TEa is 12%, bias is 0%, and SD is 3%, Cpk would be 1.33, which is a good production process that should be controllable to the desired quality.
  • 177. Laboratory TE Criteria vs Process Capability • If the criterion requires that TEa > bias + 3SD, this corresponds to a three-sigma process if there is no bias, e.g., if TEa is 12%, bias is 0%, and SD is 4%, Cpk would be 1.00, which is the minimal capability needed for a production process. • If the criterion requires that TEa > bias + 2SD, this corresponds to a two-sigma process if there is no bias, e.g., if TEa is 12%, bias is 0%, and SD is 6%, Cpk would be 0.67, which is unacceptable for production according to industrial guidelines. • Process performance, as evaluated by commonly used laboratory TE criteria, does not approach the six-sigma capability desired for industrial processes. Improvements in laboratory methods are still needed to achieve five-sigma to six-sigma capability.
  • 179. Stages or fundamentals in six sigma the DMAIC Roadmap
  • 180. DMAIC and Lean tools deployed in the Shewhart Cycle
  • 181. DMAIC six sigma approach. • The six sigma approach for projects is DMAIC (define, measure, analyze, improve and control). • These steps are the most common six sigma approach to project work. • Some organizations omit the D in DMAIC because it is really management work. • With the D dropped from DMAIC the Black Belt is charged with MAIC only in that six sigma approach. • We believe define is too important be left out and sometimes management does not do an adequate job of defining a project. • Our six sigma approach is the full DMAIC
  • 182. Define (DMAIC) • Define is the first step in our six sigma approach of DMAIC • DMAIC first asks leaders to define our core processes. • It is important to define the selected project scope, expectations, resources and timelines. • identifies specifically what is part of the project and what is not, and explains the scope of the project. • Many times- process documentation are at a general level. • Additional work is often required to adequately understand and correctly document the processes. • As the saying goes “The devil is in the details.”
  • 183. Measure (DMAIC) - most important thing to know is where we are going. - some of the first information you need before starting any journey is your current location. • The six sigma approach asks - to quantify and benchmark the process using actual data. • At a minimum consider the mean or average performance and some estimate of the dispersion or variation (maybe even calculate the standard deviation). Trends and cycles can also be very revealing. • The two data points and extrapolate to infinity is not a six sigma  approach. • Process capabilities can be calculated once there is performance data
  • 184. Analyze (DMAIC ) - project is understood - baseline performance documented - verified that there is real opportunity, - then - do an analysis of the process. - the six sigma approach applies statistical tools to validate root causes of problems. - any number of tools and tests can be used. objective is to understand the process at a level sufficient to be able to formulate options for improvement. • compare the various options with each other to determine the most promising alternatives. • as with many activities, balance must be achieved. • superficial analysis and understanding will lead to unproductive options being selected, forcing recycle through the process to make improvements. • at the other extreme is the paralysis of analysis. • striking the appropriate balance is what makes the six sigma highly valuable.
  • 185. Improve (DMAIC ) • six sigma approach ideas and solutions are put to work. - discovered and validated all known root causes for the existing opportunity. • The six sigma approach requires to identify  solutions. • Few ideas or opportunities are so good that all are an instant success. - there must be checks to assure that the desired results are being achieved. - some experiments and trials may be required in order to find the best solution. • When making trials and experiments it is important that all project associates understand that these are trials and really are part of the six sigma approach.
  • 186. Control (DMAIC ) • Many people believe the best performance you can ever get from a process is at the very beginning • Over time there is an expectancy that slowly things will get a little worse until finally it is time for another major effort towards improvement. • Contrasted with this is the Kaizen approach that seeks to make everything incrementally better on a continuous basis. • The sum of all these incremental improvements can be quite large. • As part of the six sigma approach performance tracking mechanisms and measurements are in place to assure, at a minimum, that the gains made in the project are not lost over a period of time. • As part of the control step we encourage sharing with others in the organization. • With this the six sigma approach really starts to create phenomenal returns, ideas and • projects in one part of the organization are translated in a very rapid fashion to implementation in another part of the organization.
  • 189. Dr.Kaoru Ishikawa, Professor at Tokyo University & Father of Q C in Japan. • CAUSE ANALYSIS TOOLS are Cause and Effect diagram, Pareto analysis & Scatter diagram. • EVALUATION AND DECISION MAKING TOOLS are decision matrix and multivoting • DATA COLLECTION AND ANALYSIS TOOLS are check sheet, control charts, DOE, scatter diagram, stratification, histogram, survey. • IDEA CREATION TOOLS are Brainstorming, Benchmarking, Affinity diagram, Normal group technique. • PROJECT PLANNING AND IMPLEMENTATIONTOOLS are Gantt chart and PDCA Cycle.
  • 190. Second seven tools • In the quality improvement movement in Japan in the latter half of the 20th century, the Japanese Union of Scientists and Engineers (JUSE) were influential in defining a set of basic tools that could be used for improving processes. These came to be known as the first seven tools. • These mostly were useful for quantitative problems, so a second set of seven tools was defined for the more qualitative problems that arise, such as around customer needs. These are: • Relations Diagram • Affinity Diagram • Tree Diagram • Matrix Diagram • Matrix Data Analysis Chart • Process Decision Program Chart • Activity Network • Just to complicate things, the Matrix Data Analysis Chart, which is somewhat complex to use, is often replaced with the Prioritization Matrix. And for further fun, alternative names are used, for example the Relations Diagram is sometimes called the 'Interrelationship Digraph'
  • 191. Relations Diagram • In many problem situations, there are multiple complex relationships between the different elements of the problem, which cannot be organized into familiar structures such as hierarchies or matrices. The Relations Diagram addresses these situations by showing relationships between items with a network of boxes and arrows. • The most common use of the Relations Diagram is to show the relationship between one or more problems and their causes, although it can also be used to show any complex relationship between problem elements, such as information flow within a process.
  • 193. Affinity Diagram ('KJ' diagram) • A diagram that is used as a method of sorting qualitative data, which usually comes in the form of short phrases or sentences (eg. 'Customers are unhappy with delivery delays'). • It is often done with Post-it Notes, although the original method used 3" x 5" cards. • It is a great method of working as a group to sort out issues and fuzzy situations. • It is also useful for sorting such as customer comments from surveys. • Building an Affinity Diagram is often known as 'doing a KJ', after its originator, Kawakita Jiro (this is in order of surname, given name, as in the Japanese tradition
  • 198. . Affinity Diagram Exit interview comments from checkout operators
  • 200. Tree diagram • is used to find feasible measures for problem solving to clarify the content of an area to be improved through branching at each node or view point Creating a Tree diagram • Establish the basic objective, Think primary means, Deploy secondary means and beyond, Conform the relationship between the objectives and means and complete the tree diagram, Evaluate the basic means To minimize Gear box replacement Time Measures Development type tree diagram Purchase By machining Fabricate Ideas to be produced by Brain storming Reduce men Improving co-ordination Motivate To minimize Service training Gear box Improve expertise Replacement In house training time Plan Making RC/new units Readily available Collect in advance Engage men for cleaning Improve house keeping Practice cleanliness
  • 202. Tree diagram or Systematic Diagram or Dendrogram Using the Tree Diagram in Problem Solving Horizontal tree diagram Vertical tree diagram
  • 206. Matrix Diagram • The Matrix Diagram allows a many-to-many comparison of two lists, by turning the second list on its side to form a matrix. - indicated in the cell where the row and column of the two items cross. - use of different symbols to indicate different comparison levels and the weighting of the items being compared. - different shapes of matrix for comparing more than the basic two lists. - the L-Matrix, C-Matrix, T-Matrix, X-Matrix and Y-Matrix. • The Matrix Diagram is a core tool in Quality Function Deployment (QFD). The C-Matrix is a variation on the Matrix Diagram, which is one of the second seven tools. • The C-matrix compares three lists simultaneously, such as the people, products and processes in a factory. • Being three-dimensional, it is difficult and complex to produce and draw. • It becomes easier if there are few relationships to map.

Editor's Notes

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