2.Measured Data and Relations to Quality

Lecture 3: Measured data and Statistics

Introduces the concept of measurement variation and statistical methods for

measuring and describing variation Compiled by Ramdziah Md.Nasir

2 Clear Vision Customer Satisfaction

Leadership Process Orientation

Focus on Quality

Employee Involvement

Supplier Partnering

Continuous Improvement

Business Growth

The Road to Business Growth

Recap Lecture 2: TQM

3

Begins with the Senior Managements and the CEOs

commitment

Involvement is required

Requires the education of Senior Management in TQM

concepts

Timing of the implementation process can be very

important

Formation of the Quality Council

Development of Core Values, Vision Statement,

Mission Statement, Quality Policy Statement

TQM Implementation

4

Quality Council:

Composed of: CEO, the Senior

Managers of the functional areas, such

as design, marketing, finance,

production, and quality; and a

coordinator or consultant

The coordinator will ensure that the team

members are empowered and know their

responsibilities

TQM Implementation

5

Quality Council Duties:

1. Develop the core values, vision,

mission, and quality policy statements

2. Develop the strategic long-term plan

with goals and the annual quality

improvement program with objectives

3. Create the total education and training

plan

4. Determine and continually monitor the

cost of poor quality

TQM Implementation

6

Quality Council Duties:

5. Determine the performance measures for

the organization

6. Determine projects that improve the

processes

7. Establish multifunctional project and

departmental or work group team

8. Establish or revise the recognition and

reward system

TQM Implementation

7

Core Values for the Malcolm Baldrige

National Quality Award:

Visionary Leadership

Customer-driven Excellence

Organizational & Personal Learning

Valuing Employees & Partners

TQM Implementation

8


National Quality Award Contd:

Agility

Focus on the Future

Management for Innovation

Management by Fact

TQM Implementation

9


National Quality Award Contd:

Social Responsibility

Focus on Results and Creating Value

Systems Perspective

TQM Implementation

10

Quality Statements:

Include the Vision Statement, Mission

Statement, and Quality Policy

Statement

They are part of the strategy planning

process, which includes goals and

objectives

Develop with input from all personnel

TQM Implementation

11

Seven Steps to Strategy Planning:

Customer Needs

Customer Positioning

Predict the Future

Gap Analysis

Closing the Gap

Alignment

Implementation

TQM Implementation

Continuous Process Improvement/Development (CPD)

PROCESS People Equipment Method Procedures Environment Materials

FEEDBACK

OUTPUT Information Data Product Service, etc.

OUTCOMES

INPUT Materials Money Information Data, etc

CONDITIONS

Figure 2-3 Input/output process model

Unacceptable

Poor

Good

Best

High loss

Loss (to producing organization, customer, and society)

Low loss

Frequency

Lower Target Upper

Specification

Target-oriented quality yields more product in the best category

Target-oriented quality brings product toward the target value

Conformance-oriented quality keeps products within 3 standard deviations

L = D2C where

L = loss to society

D = distance from target value

C = cost of deviation

Taguchis Loss Function

Learning Objectives

Understand the errors related to measurement

Know the round-off rules

Able to distinguish between two types of variations special cause and common cause

Know what statistic is and its applications

Know what distributions are and how they are used in SPC

Able to calculate the mean, median, mode, range and standard deviation for a set of numbers

Able to draw a histogram for a set of numbers

Measurement

Any measurement is only as good as the measuring device/technique or the persons using it.

Measurement error always exist, measured value is an estimation.

Accuracy is the smallest unit on the measuring device

Maximum error of a measurement is half the accuracy

Distribution is an ordered set of numbers that are grouped in some manners. The distribution maybe in a table, graph or picture form

Example:

2.34cm is accurate to the nearest hundredth of a cm (0.01), therefore maximum error is 0.01/2 =0.005

When a measurement is written, its accuracy is implied by the number of place value, e.g. 2.34cm and 2.340cm are significantly different!. The 2.34cm would lie between 2.335 and 2.344, and 2.340 would lie between 2.3395 and 2.3404!

Round-off Rules:

1. If number to the right is half of that

place value, round UP to next digit, e.g. 23.472 is 23.5 (nearest tenth)

2. If number to the right is half of that place value, truncate to that place value, e.g. 23.414 to nearest tenth is 23.4

Variation

2 types:- (a) Special-cause, (b) common cause

Individual measurements are different, but when grouped together, they form a predictable pattern called distribution

Every distribution has measurable characteristics such as:

Location - Position of middle value (or average value)

Spread width of distribution curve Shape the way measurements stack

up

Special cause variation

(a) Special cause variation (or assignable-cause variation)

unpredictable variation that do not normally occur due to worn parts, improper allignment, etc

- A derived variation

- Can be eliminated by local action on a particular segment of the process

- Local action can handle ~15% of process problem

Common cause variation

(b) Common cause variation

- It is inherent (built-in) in the process, not a derived variation

- Approximately 85% of process problems are due to common cause variation

- Require process changes to remove the built-in variation decision by management

Variation (within or between subgroup)

Under ideal condition, e.g. for a manufacturing process, only common cause variation occur within subgroup (batch), and special cause variation occurs between subgroup.

Special cause should occur between batch, not within a batch.

Care must be taken so that differences in operators, machines, batches of raw materials in production lines do not show up within subgroups.

Do different control charts for different operators if operators make a difference

Effect of Variation

Day 1 Day 2 Day 3 Target

Day 1 Day 2 Day 3

Target

(a) When Special cause variation is present unpredictable distribution

(b) When Special cause variation is eliminated leaving only common cause variation predictable (process is in statistical control)

Histogram

Graphic representation of the frequencies of observed values, usually plotted using rectangles.

Vertical axis is the frequency, horizontal axis is the category.

Category

Fre

qu

en

cy

Mid-point

Interval, i

Upper boundary

Lower boundary

Steps to construct a histogram

Collect data and construct a tally sheet Determine the range, R = Xh Xl Determine Cell interval, i = R / (1+3.322 log n ) Determine cell mid-point, using the lowest data value, MPl = Xl+

(0.5i)

Determine the cell boundary for either upper or lower boundary Plot the histogram

Sample Tabulation Frequency

1

2

3

4

III

IIII

II

I

3

4

2

1

Tally sheet

What is Statistics? Statistics is the science of data handling Data types: (a) Variable data quality characteristics that are

measureable and normally continuous (may take on any values);

(b) Attribute data quality characteristics that are either present or not present, conforming or non-conforming, countable, normally discrete values (integers)

Its applications normally involve using sample information to make decision about a population of measurement

A population is the set of all possible data values of interest, while a sample is only a subset of a part of the population

Four steps in application of statistics: (a) collection of data; (b) Organization of data; (c) Analysis of data; (d) Interpretation of data

Population

Sample 1

Sample 2

Sample 3

Descriptive vs Inductive Statistics

Descriptive or deductive statistics attempts to describe and analyze a subject or group

Inductive statistics is trying to determine from a limited amount of data (sample) an important conclusion about a much larger amount of data (population). Since the conclusions or inferences cannot be made with absolute certainty, the language of probability is often used

Measure of central tendency- describes the center position of the data (mean, median, mode)

Measure of dispersion describe the spread of data (range, variance, standard deviation)

Mean, where Xi is one observation, N is number of sample

Median is the middle point of a data series (observation in the middle of sorted data

Mode the most frequently occuring value

Descriptive Statistics

i

N

i XN

X 11

100 91 85 84 75 72 72 69 65

Mode Median

Mean = 79.22

Descriptive Statistics Measure of dispersion (range, variance and standard deviation) The range is calculated by taking the maximum value and

subtracting the minimum value.

Variance is the squared of the summation of the difference between each value and the mean divided by number of samples

1, 3, 5, 7, 9, 11

Range = 11-1 = 10 n

xn

ii

1(

Std deviation is the square-root of variance. Measures spreading tendency of the data

n

xn

ii

1

(

If is small, high probability of

getting the values close to mean

value

If is large, high probability of getting

the values away from mean value

= population mean


Other measure of dispersion (skewness, kurtosis, coefficient of variation)

Skewness - lack of symmetry of data distribution. A negative values indicate skewed to the left, positive indicates skewed to the right.

0 right

3

3

13

/)(

s

nxxfa

n

iii

Note: S = = std dev

See examples 4.6, p147 & 4.7, p149


Measure of Dispersion - Kurtosis Kurtosis Measure the peakness of the data. It is a dimensionless

value. The value must be compared to a normal distribution to determined if it more peaked or flatter peaked distribution.

Platykurtic (flatter) Mesokurtic (normal) Leptokurtic (more peaked)

4

4

14

/)X(fa

s

nxn

iii

See example 4.8, p151

Note: S = = std dev


Measure of Dispersion Coefficient of variation CV measure how much variation exist relative to the mean. Unit in %.

See example 4.9, p152

XCV

%100

Normal Distribution (Gaussian distribution)

Always symmetrical, unimodal, bell-shaped distribution with mean, median, mod having the same value

Much variation in nature and in industry follow the normal distribution curves.

Offer good description of variations occuring in most quality characteristics in industry it becomes the basis of many techniques

x scale

z scale

-3 +3 +2 + - -2

-3 +3 +2 +1 -1 -2 0

iX

Z

Population, sample, reading (notations used)

Population

Sample 1

Sample 2

Sample 3

X-bar = Average value for a sample (which has a few

readings)

s = standard deviation of a sample

(Greek letter mu) = Mean (also equivalent to Average) value for a population (which has a few samples and

readings)

(Greek letter sigma) = standard deviation for a population

n = number of readings from a sample

N = number of samples/groups in a population

Use of Statistics in Quality Changing data into

information

Statistical Process Control (SPCs) Historical Background

Walter Shewhart suggested that every process exhibits some degree of variation and therefore is expected. identified two types of variation (chance cause) and

(assignable cause)

proposed first control chart to separate these two types of variation.

SPC was applied during World War II to ensure interchangeability of parts for weapons/ equipment.

Resurgence of SPC in the 1980s in response to Japanese manufacturing success.

Product Control And Process Control Philosophy

The product control view:

measures quality of a product in terms of its acceptability as measured by conformance to engineering specifications.

emphasizes detection and containment of defective material through inspection/screening, therefore making

quality and productivity opposing rather than supportive forces.

The process control view:

emphasizes the prevention of defective material from being made in the first place by seeking the root cause of the problem and eliminating it altogether.

makes quality and productivity enhancement possible simultaneously by continually seeking ways to reduce

variation, thereby eliminating waste and inefficiency in the process and variation in performance of the product.

Mean ,standard deviation

Example 1

The scores on a test given to students in a large class are normally distributed with a mean of 57 and a standard deviation of 10. The passing score for the exam is 30. If a student is randomly selected, what is the probability that he or she passed the exam? A score of 75 or greater is needed to obtain an A on the exam. What percentage of the students received an A?

Mean ,standard deviation

Given: X = 57 and X = 10, and process represented is normal distribution.

To calculate the probability of passed, we need to find P (X30). Use z transformation to determine the values of Z associated with the values of X

z = (30 - 57) / 10 = -2.7 Therefore, we need to find P (Z -2.7). Using Table A.1 in the Appendix,

P (Z -2.7) = 1 - P (Z -2.7) = 1 - 0.0035 = 0.9965 Therefore, probability of student passing is 99.65 %

To calculate the probability of randomly selected student has score A,

Need to find P (X 75). Use Z transformation to determine the values of Z associated with the values of X.

z = (75 - 57) / 10 = 1.8 Therefore, we need to find P (Z 1.8). Using Table A.1 in the Appendix, P (Z 1.8) = 1 - P (Z 1.8)

= 1 0.9641 = 0.0359 Therefore, probability of student scoring an A is 3.59 %

The accompanying table represent the weight in gram of moulded instrument display panels. The samples were collected at half hour intervals . Prepare a tally sheet, of the individual measurements and then prepare a frequency histogram of the data, clearly labelling the cell boundaries. Comment on the shape of the distribution.

Sample X1 X2 X3 X4 X5

1 14 15 13 14 13

2 20 18 14 17 8

3 14 17 14 11 14

4 15 16 11 18 14

5 9 17 18 13 12

6 19 15 14 15 16

7 16 13 14 13 17

8 14 17 9 16 15

9 14 14 12 13 13

10 15 13 17 14 16

11 18 18 16 15 11

12 20 12 13 17 14

13 1 8 9 12 7

14 12 14 16 14 20

15 18 17 12 19 18

16 19 17 16 16 17

17 14 13 15 16 18

18 14 17 12 16 11

19 18 15 16 15 12

20 15 9 12 13 20

Answer To create a frequency histogram is to select the

number of cells and the cell boundaries. Since the data are integer values, the cell boundaries can be set at xx.5, xx+ cell width + 0.5, etc. The range of the data is 20- 7=13 so using 13 cells might work well. Thus, the integer values are the cell midpoints and the cell boundaries are set at 6.5, 7.5, 8.5, ... , 20.5. For example, the boundaries for the cell with midpoint 13 are 12.5 and 13.5.

The next step is to make a tally of the frequency of occurrence of the data appear within each cell, as follows:

Cell Midpoint Frequency per Cell frequency, fi

7 1

8 11 2

9 1111 4

10 0

11 11111 5

12 111111111 9

13 11111111111 11

13.5 - 14.0 - 14.5 111111111111111111 18

15 11111111111 11

16 111111111111 12

17 11111111111 11

18 111111111 9

19 111 3

20 1111 4

Total, fi = 100

the cell interval, i= R/1+3.322 log n

cell midpoint MPl= Xl+i/2 (normally odd value)

cell boundaries = 0.1-i =-0.4 (lower boundary), 3.5+i (upper boundary)

frequencies.= from tally sheet

Next, this information is used to plot the histogram:

8 12 16 20

Frequency

4

8

12

16

Weight in gram

From the shape of the histogram, it appears that the data came from a process that might be considered as a

candidate for representation by a normal distribution

(excel form)

What is Statistics?

What is it Not

Has Something to Do With Data.

Objectives of Data Collection

Understanding, insights, illumination

An Inexact Science Given Industrial Realities

Probabilities in Manufacturing

Examples with objectives

classifying parts as being defective or non-defective -- reducing number of defectives

studying the number of monthly orders received better adjusting inventory levels to match orders

measuring gas output when acid concentrations are changed --better predicting and controlling gas

levels

Statistical Thinking & Modelling Engineers Think

Deterministically

Deterministic Models Do Not Explain Variability

Deterministic Models Do Not Account for Variability

Engineering Education/Practice Blames

Factors That Remain a Mystery

Limitations in Measurement Process

Engineering Method Depends on Data

Real Data Exhibits Variability

Obscures Ability to Make Sound Decisions

Engineers Must Learn to Think Statistically

Understanding of Risk and Uncertainty

Key is Discovering Sources of Variability

Data Collection

What is the fundamental purpose? What important questions need answers?

What is the characteristic of interest? How will it be measured? Issues

What is known about the measurement process?

How does engineering model impact data collection? What data does the model require?

How robust is the model to data error?

How do model parameters support problem solution?

Are there physical constraints that impede ability to collect data?

Statistic types

Deductive statistics describe a complete data

set

Inductive statistics deal with a limited amount

of data

Types of data `

Variables data - quality characteristics that are

measurable values.

Measurable and normally continuous; may take on

any value.

Attribute data - quality characteristics that are

observed to be either present or absent,

conforming or nonconforming.

Countable and normally discrete; integer

Within vs. between subgroup

variation Under ideal conditions: only common

cause variation occurs within subgroups

and special cause variation occurs among

between subgroups

Special Causes Should Occur

Between Batches not Within

Care must be taken so that differences in operators, machines, lots of raw materials,

production lines do not show up within

subgroups.

Do different control charts for different operators if operators make a difference.

Descriptive statistics

Measures of Central Tendency

Describes the center position of the data

Mean Median Mode

Measures of Dispersion

Describes the spread of the data

Range Variance Standard deviation

Measures of central

tendency: Mean Arithmetic mean x =

where xi is one observation, means add up what follows and N is the number of observations

So, for example, if the data are : 0,2,5,9,12 the

mean is (0+2+5+9+12)/5 = 28/5 = 5.6

N

i

ixN 1

1

Measures of central

tendency: Median - mode Median = the observation in the middle of

sorted data

Mode = the most frequently occurring value

Median and mode

100 91 85 84 75 72 72 69 65

Mean = 79.22

Median

Mode

Measures of dispersion:

range The range is calculated by taking the maximum

value and subtracting the minimum value.

2 , 4 ,6 ,8 ,10, 12 , 14

Range = 14 - 2 = 12


variance Calculate the deviation from the mean for

every observation.

Square each deviation

Add them up and divide by the number of

observations

n

xn

ii

1(


standard deviation The standard deviation is the square root of

the variance. The variance is in square units so the standard deviation is in the same units

as x.

n

xn

ii

1

(

Standard deviation and

curve shape If is small, there is a high probability for

getting a value close to the mean.

If is large, there is a correspondingly higher

probability for getting values further away from

the mean.

Chebyshevs theorem

If a probability distribution has the mean and

the standard deviation , the probability of

obtaining a value which deviates from the

mean by at least k standard deviations is at

most 1/k2.

2

1(

kkxP

As a result

Probability of obtaining a value beyond x standard deviations is at most::

2 standard deviations

1/22 = 1/4 = 0.25


1/32 = 1/9 = 0.11


1/42 = 1/16 = 0.0625

Other measures of

dispersion: skewness When a distribution lacks symmetry, it is

considered skewed.

0 right

3

3

13

/)(

s

nxxfa

n

iii

Other measures of

dispersion: kurtosis suggests peak-ness of the data

a can be used to compare distributions

4

4

14

/)X(fa

s

nxn

iii

The normal frequency

distribution 22 2/)(

2

1)(

xexf

The normal curve

A normal curve is symmetrical about The mean, mode, and median are equal

The curve is uni-modal and bell-shaped

Data values concentrate around the mean

Area under the normal curve equals 1

The normal curve

If x follows a bell-shaped (normal) distribution,

then the probability that x is within

1 standard deviation of the mean is 68%

2 standard deviations of the mean is 95 %

3 standard deviations of the mean is 99.7%

The standardized normal

x scale

z scale

-3 +3 +2 + - -2

-3 +3 +2 +1 -1 -2 0

= 0

= 1

Test Statistic and Decision Rule

Critical Region, Critical Value,

and Significance Level

Type I Error

A Type I error is the decision error when the researcher incorrectly rejects the null hypothesis

(when the null is true).

The probability of that error is a..

a. is the probability that the test statistic lies in them critical region when the null hypothesis is true.

When the null is rejected, we say that the test is statistically significant at a 100 a % significance level.

The p-Value

A p-value is the lowest level (of significance) at which the observed value

of test statistic is significant.

The p-value gives researcher an alternative to merely rejecting or not

rejecting the null.

A small p-value clearly refutes Ho

Summary For Hypothesis Testing

State the null hypothesis H0: q = q0 Choose an appropriate alternate hypothesis Ha: q < q0,

q > q0, or q ,q0,

Chose a significant level of size a

Select the appropriate test statistic and critical region (if the decision is based on a p-value, the critical region is not necessary) and state the decision rule in terms of the test statistic

Compute the value of test statistic from the sample data

Reject H0 based on the decision rule (if the test statistic is in the critical region or if the p-value is less than a): otherwise do not reject H0

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2.Measured Data and Relations to Quality