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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
Core Values for the Malcolm Baldrige
National Quality Award Contd:
Agility
Focus on the Future
Management for Innovation
Management by Fact
TQM Implementation
9
Core Values for the Malcolm Baldrige
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
Descriptive Statistics
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
Descriptive Statistics
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
Descriptive Statistics
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.
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
Measures of dispersion:
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(
Measures of dispersion:
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
3 standard deviations
1/32 = 1/9 = 0.11
4 standard deviations
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