Download Cp vs Cpk Pp vs Ppk Practical Use of Statistics Point Estimates vs

9/20/2013
Practical Use of Statistics
Dorian Shanin was a great Quality Professional I had
met at the start of my career as a Quality Professional
He preached to always search for the “Red X”
Point Estimates vs. Confidence Intervals
The “Red X” being the critical characteristic of
the product
Cp vs Cpk
He said “Don’t ask your customer because they
haven’t identified it
Pp vs Ppk
Cp & Cpk vs Pp & Ppk
Normal vs Platikurtic vs Leptokurtic etc.
I determined through the years that he was right
In the immortal words of Vinnie Barbarino (a young John Travolta)
1
Relationships S
l
Samples
n
2
Defining a Process’ Identity
Every process is identified by:
Population
(N)
(µ , σ, and σ²)Parameters
“I’m so confused!”
Its measures of central tendency
(X1 , X
X2 , etc.)
etc )
(n1 , n2 , etc)
Statistics
Its measure of dispersion.
Processes produce Populations
3
Defining a Processes Identity
4
Defining a Processes Identity
The Central Tendency of a Distribution
indicates at what value a distribution appears to
be focused.
The measure of central tendency we will be
discussing this evening is the mean (average)
The Dispersion of a Distribution indicates
at how much a distribution varies around
the mean of the distribution
The measure of dispersion we will be
discussing is the standard deviation.
5
6
1
9/20/2013
Sample Mean
CENTRAL TENDENCY of a DISTRIBUTION
The Mean (Average) of a distribution is the
sum of all the values in the distribution divided
by the number of values in the distribution:
• Sample summary measures are called statistics
• The sample mean is the sum of the values in the sample divided by the sample size, n
n
X=
Mu is defined as the Average or Mean of the
Population and as such it is a Parameter
∑X
i =1
n
i
=
X1 + X 2 + L + Xn
n
X Bar is a Statistic is our estimate of µ
7
Measure of Dispersion
8
Sample Standard Deviation
The Standard Deviation is the most commonly used
measure of Dispersion
N
σ=
∑(X − μ)
2
i
i 1
i=
N
n
S =
Sigma is defined as the
Standard Deviation of the
Population and is a
Parameter
∑ (X
i=1
i
− X )2
n -1
S is used to designate the Sample Standard
Deviation. It is our estimate of Sigma
and is a Statistic
Shows variation about the mean
Has the same units as the original data
9
10
The Rules of Thumb
Concerning Normal Distribution
• If the data distribution is normal, then the interval:
• μ ± 1σ contains about 68% of the values in the
population or the sample
The Normal Distribution
Also called Gaussian or Bell-shaped
p
68%
μ
μ ± 1σ
11
12
2
9/20/2013
The Rule of Thumb cont.
•
μ ± 2σ contains about 95% of the values in
•
μ ± 3σ contains about 99.7% of the values
10
Average
-3σ
+3σ
9
9
9
the population or the sample
8
7
7
7
7
7
in the population or the sample
Frequency
6
5
4
4
4
4
4
3
2
2
2
2
2
1
95%
1
1
1
1
99.7%
0
0
0
0
0
0
1
2
3
4
5
6
0
0
0
0
0
0
0
μ ± 2σ
μ ± 3σ
Remember, These Rules of Thumb apply only with
a Normal Distribution
Average
‐3σ
10
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0.9
Expected
Low value
13
30.1
Bin
Expected
High value
Based on the
Normal Curve
14
+3σ
9
10
Average
8
9
‐3σ
7
+3σ
8
Frequency
6
7
Frequency
5
4
3
6
5
4
2
3
1
2
0
14
15
16
17
18
19
20
14.7
Expected
Low value
21
22
23
24
25
26
1
25.3
Expected
High value
Bin
0
4
5
6
7
8
9
5.7
Expected
Low value
10
11
12
13
14
15
16
17
16.3
Expected
High value
Bin
15
The Quality Professional should understand why a Process is not
Normally Distributed and attempt with the rest of the Process Team to
Normalize it. This falls under the umbrella of Continuous Improvement.
10
Average
-3σ
+3σ
9
16
The reason for the disparity is due to the calculation
for the Standard Deviation:
N
9
∑(X − μ)
9
8
7
7
7
σ=
7
7
i=1
N
6
5
4
4
4
4
4
3
1
2
2
1
1
2
Frequency
2
2
1
1
0
0
0
0
0
0
1
2
3
4
5
6
0
0
0
0
0
0
0
7
8
0.9
5.7
Expected Low Expected
Low value
value
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
14.7 Bin16.3
Expected Expected
Low value High value
25.3
Expected
High value
30.1
Expected High
value
10
9
8
7
6
5
4
3
2
1
0
10
9
8
7
6
5
4
4
4
4
4
3
2
2
2
2
2
1
1 1
1
0 0 0 0 0 0
0 0 0 0 0 0 1
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
9
7
9
7
7
Bin
17
7
Frequency
Freque
ency
2
i
4
5
6
7
8
9 10 11 12 13 14 15 16 17
Bin
The Quality Professional is unjustly Penalizing the Process if the
18
Shape of the Distribution is not taken into consideration
3
9/20/2013
Average vs. Confidence Interval of the Mean
The Calculation of the Average as I have so far discussed is a Point Estimate
There is Error associated with this point estimate when using it to
estimate µ. The magnitude of this Error depends on the Sample Size
and the Standard Deviation
X Bar is 18,
n
X=
Confidence Interval of the Mean
The average of 25 samples is 18 and these samples are
taken from a population with a known standard deviation of 10. Calculate
the 95% Confidence interval of the mean
∑X
i =1
n
i
=
σx = 10
n = 25,
z = 1.96
X1 + X 2 + L + Xn
n
14.08 ≤ µ ≤ 21.92
There is a 95% Confidence that the interval
between14.08 to 21.92 contains µ!
19
Cp /Cpk vs Pp/Ppk
Note this calculation our Risk (5%) and
Error Interval (14.08 to 21.92
20
Cp /Cpk vs Pp/Ppk
Pp = Process Capability = (USL-LSL) / 6σi
Process Capability Indicies Cp and Cpk
Cp = Process Capability = (USL-LSL) / 6σR
σi = σx
Note Pp doesn’t account for Measure of Central
Tendency
Note
N t Cp
C doesn’t
d
’t accountt for
f Measure
M
off Central
C t l
Tendency
Cpk is the minimum between Cpku and Cpkl
Cpkl = (µ - LSL)/3σR
Cpku = (USL-µ)/3σR
USL = upper specification limit
LSL = lower specification limit
σ = standard deviation
µ = mean
Ppk is the minimum between Ppku and Ppkl
Ppku = (USL-µ)/3σi
Ppkl = (µ - LSL)/3σi
USL = upper specification limit
LSL = lower specification limit
σ = standard deviation
µ = mean
21
Cp /Cpk vs Pp/Ppk
22
Rules of Thumb for Index Interpretation
Because Cpk uses the Average Range (R Bar) to estimate
sigma (Sigma hat) the Quality Professional must be sure
that the Process is in control and Normally distributed
otherwise Sigma Hat will be in Error
I always use Ppk as a Process index because
it is less likely to be in error
23
Value of Cpk or Ppk
Capability
Less than 1
Incapable
Action
Between 1 and 3
Capable
Do nothing or some
process improvement.
Dependent on sample
size.
Greater than 3
Very capable
Do nothing or reduce
specification limits. No
inspection necessary.
Improve by reducing
common causes of
variation in process
variables. Use 100%
inspection.
24
4
9/20/2013
X Bar and R Charts
Rational Subgroups
• Key to Shewart Control Charts • The subgroup should be selected so as to minimize piece to piece variation within the subgroup
• Selection of the next subgroup should be done to maximize the variation from subgroup to subgroup
Where n is the size of each Subgroup and Sx = Sp = Si (Std Dev of Individuals)25
Control Chart Interpretation
• Very important to ensure subgroup sequence is maintained along with timing of when the subgroup 26
was selected
Process Variability
• Common Causes
• Have predictable effects on processes
• It is present in all processes
To decrease it typically requires
• To decrease it typically requires an inherent change to the process
• This type of change is by and large “management controllable”
• Special Cause • Have unpredictable effects on processes
• Is present in most processes
• To decrease it requires To decrease it requires
identification of the “culprit”
• This type of change is best done by a Corrective Action Team
Common Cause Variation is also called Chance Variation
27
Special Cause Variation is also called Assignable
Cause Variation
28
X Bar and R Charts
Control Chart Interpretation
WESTERN ELECTRIC RULES
The typical X Bar and R Chart has “3 Sigma Control Limits”
In the past some quality professionals advocated “2 Sigma
Control Limits” for better Control of the Process
This is totally false.
3 Sigma Control Limits will tell us the Process is out of
control 0.3% of the time when it is actually in control
2 Sigma Control Limits will tell us the Process is out of
control 5.0% of the time when it is actually in control
29
30
5
9/20/2013
SQC vs. SPC
Statistical Quality Control (SQC) connotates
“Product Characteristic Control”
Measuring critical dimensions of the product and determining
the products acceptability
This is many times required by customers but it is a
reactive!
Statistical Process Control (SPC) connotates
“Process Characteristic Control”
Identifying the “critical process nodes” (bath concentration,
line speed, shut height, etc) and monitoring them
This is proactive
31
6