Transcript Chapter 4 Statistics

Agenda

Review Homework
•

Chapter 4 – 2, 3
•
Statistics in Quality
• Central Tendency
• Dispersion
• Shape
• Standardized Normal
Curve
• Central Limit Theorem
Statistical Process Control
•
Week 9 assignment
•
Lecture/discussion
•


Kurt Manufacturing
Statistics
Week 8
Read chapter 5
Homework
•
Problems: Chapter 4 - 8, 9,
24, 26, 36, 44, 49
The Use of Statistics
in Quality
Chapter Four
Statistics
A few notes on SPC’s
historical background

Walter Shewhart (Bell Labs 1920s) - 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 successfully applied during World War II as
a means of insuring interchangeability of parts for
weapons/ equipment.
Resurgence of SPC in the 1980s in response to
Japanese manufacturing success.
Statistics
The basics



“Don’t inspect the product, inspect the
process.”
“You can’t inspect it in, you’ve got to build
it in.”
“If you can’t measure it, you can’t
manage it.”
Statistics
Barriers to process control


Tendency to focus on volume of output
rather than quality of output.
Tendency to measure products against a
set of internal conformance specifications
that may or may not relate to customer
expectations.
Statistics
The SPC approach

The SPC approach is designed to identify
underlying cause of problems which
cause process variations that are outside
predetermined tolerances and to
implement controls to fix the problem.
Statistics
The SPC steps
Basic approach:
 Awareness that a problem exists.
 Determine the specific problem to be
solved.
 Diagnose the causes of the problem.
 Determine and implement remedies.
 Implement controls to hold the gains
achieved by solving the problem.
Statistics
SPC requires the use of
statistics


Quality improvement efforts have their
foundation in statistics.
Statistical process control involves the
collection
• tabulation
• analysis
• interpretation
• presentation
of numerical data.
•
Statistics
Statistic types


Deductive statistics describe a complete
data set
Inductive statistics deal with a limited
amount of data
Statistics
Statistics
Parameters: 2
Inferential
Statistics
POPULATION
Deductive
SAMPLE
Statistics: x, s, s2
Inductive
Statistics
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
Statistics
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
Statistics
Measures of central
tendency: Mean
N
Arithmetic mean x =
1
xi

N i 1
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

Statistics
Measures of central
tendency: Median - mode


Median = the observation in the ‘middle’
of sorted data
Mode = the most frequently occurring
value
Statistics
Median and mode
100 91 85 84 75 72 72 69 65
Mode
Median
Mean = 79.22
Statistics
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
Statistics
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
n
observations
2
 
2
 ( xi   
i 1
n
Statistics
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
2
i
i 1

 (x   
n
Statistics
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.
Statistics
Chebyshev’s 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.
1
P ( x    k  
k
Statistics
2
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
Statistics
Other measures of
dispersion: skewness


When a distribution lacks symmetry, it is
considered skewed.
<0 left
0 = symmetrical >0 right
n
a3 
 f i ( xi  x) / n
3
i 1
s
3
Statistics
Other measures of
dispersion: kurtosis

suggests “peakedness” of the data
•
“a” can be used to compare distributions
n
a4 
 f i ( xi  X) / n
4
i 1
s
4
Statistics
The normal frequency
distribution
1  ( x   ) / 2
f ( x) 
e
2
2
2
Statistics
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
Statistics
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%
Statistics
One standard deviation


68.3%
Statistics
Two standard deviations
2 2
95.5%
Statistics
Three standard deviations
3
3
99.73%
Statistics
The standardized normal
=0
=1
x scale -3
z scale
-3
-2
-2
- 
-1
0
Statistics
+
+2
+3
+1
+2
+3