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SPC Basic
Dr. Mohamed Riyazh Khan – DoMS
SNS. College of Engineering
Have you ever…
Shot a rifle?
 Played darts?
 Played basketball?
 Shot a round of golf?

What is the point of these sports?
What makes them hard?
Have you ever…
Shot a rifle?
 Played darts?
 Shot a round of golf?
 Played basketball?

Emmett
Jake
Who is the better shot?
Discussion
What do you measure in your process?
 Why do those measures matter?
 Are those measures consistently the
same?
 Why not?

Variability

Deviation = distance between
observations and the mean (or
average)
Observations
8
7
10
8
9
Emmett
Deviations
10 10 - 8.4 = 1.6
9
9 – 8.4 = 0.6
8 8 – 8.4 = -0.4
8 8 – 8.4 = -0.4
7 7 – 8.4 = -1.4
averages
8.4
0.0
Jake
Variability

Deviation = distance between
observations and the mean (or
average)
Observations
Deviations
7
7 – 6.6 = 0.4
7
7 – 6.6 = 0.4
7
7 – 6.6 = 0.4
6 6 – 6.6 = -0.6
6 6 – 6.6 = -0.6
averages
6.6
0.0
Emmett
7
6
7
7
6
Jake
Variability

8
7
10
8
9
Variance = average distance
between observations and the
mean squared
Deviations
Squared Deviations
10 10 - 8.4 = 1.6
2.56
9 – 8.4 = 0.6
0.36
8 8 – 8.4 = -0.4
0.16
8 8 – 8.4 = -0.4
0.16
7 7 – 8.4 = -1.4
1.96
0.0
1.0
Observations
9
averages
Emmett
8.4
Jake
Variance
Variability

Variance = average distance
between observations and the
mean squared
Observations
7
7
7
6
6
averages
Deviations
Squared Deviations
Emmett
7
6
7
7
6
Jake
Variability

Variance = average distance
between observations and the
mean squared
Deviations
Squared Deviations
7
7 - 6.6 = 0.4
0.16
7
7 - 6.6 = 0.4
0.16
7
7 - 6.6 = 0.4
0.16
6 6 – 6.6 = -0.6
0.36
6 6 – 6.6 = -0.6
0.36
0.0
0.24
Observations
averages
6.6
Emmett
7
6
7
7
6
Jake
Variance
Variability

Standard deviation = square root
of variance
Variance
Emmett
Jake
1.0
0.24
Standard
Deviation
1.0
0.4898979
But what good is a standard deviation
Emmett
Jake
Variability
The world tends to
be bell-shaped
Even very rare
outcomes are
possible
(probability > 0)
Fewer
in the
“tails”
(lower)
Most
outcomes
occur in the
middle
Fewer
in the
“tails”
(upper)
Even very rare
outcomes are
possible
(probability > 0)
Variability
Here is why:
Even outcomes that are equally
likely (like dice), when you add
them up, become bell shaped
Add up the dots on the dice
Probability
0.2
0.15
1 die
0.1
2 dice
0.05
3 dice
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Sum of dots
“Normal” bell shaped curve
Add up about 30 of most things
and you start to be “normal”
Normal distributions are divide up
into 3 standard deviations on
each side of the mean
Once your that, you
know a lot about
what is going on
And that is what a standard deviation
is good for
Usual or unusual?
1. One observation falls
outside 3 standard
deviations?
2. One observation falls in
zone A?
3. 2 out of 3 observations fall in
one zone A?
4. 2 out of 3 observations fall in
one zone B or beyond?
5. 4 out of 5 observations fall in
one zone B or beyond?
6. 8 consecutive points above
the mean, rising, or falling?
X
XX
X 34 56
X1X XX2
X
78
Causes of Variability

Common Causes:
 Random variation (usual)
 No pattern
 Inherent in process
 adjusting the process increases

its variation
Special Causes
 Non-random variation
 May exhibit a pattern
(unusual)
 Assignable, explainable, controllable
 adjusting the process decreases its variation
SPC uses samples to identify that special causes have occurred
Limits

Process and Control limits:
 Statistical
 Process
limits are used for individual items
 Control limits are used with averages
 Limits = μ ± 3σ
 Define usual (common causes) & unusual (special
causes)

Specification limits:
 Engineered
= target ± tolerance
 Define acceptable & unacceptable
 Limits
Process vs. control limits
Distribution of averages
Control limits
Specification limits
Variance of averages < variance of individual items
Distribution of individuals
Process limits
Usual v. Unusual,
Acceptable v. Defective
A
B
C
μ
Target
D
E
More about limits
Good quality:
defects are
rare (Cpk>1)
μ
target
μ
target
Poor quality:
defects are
common (Cpk<1)
Cpk measures “Process Capability”
If process limits and control limits are at the same location, Cpk = 1. Cpk ≥ 2 is exceptional.
Process capability
Good quality: defects are rare (Cpk>1)
Poor quality: defects are common (Cpk<1)
Cpk = min
=
USL – x
= 24 – 20 =.667
3σ
3(2)
=
x - LSL
= 20 – 15 =.833
3σ
3(2)
14
=
=
3σ = (UPL – x, or x – LPL)
15
20
24
26
Going out of control

When an observation is unusual, what can
we conclude?
The mean
has changed
X
μ1
μ2
Going out of control

When an observation is unusual, what can
we conclude?
σ1
The standard deviation
has changed
σ2
X
Setting up control charts:
Calculating the limits
1.
2.
3.
4.
5.
6.
7.
8.
Sample n items (often 4 or 5)
Find the mean of the sample x (x-bar)
Find the range of the sample R
Plot x on the x chart
Plot the R on an R chart
Repeat steps 1-5 thirty times
Average the x ’s to create x (x-bar-bar)
Average the R’s to create R (R-bar)
Setting up control charts:
Calculating the limits
9.
10.
Find A2 on table (A2 times R estimates 3σ)
Use formula to find limits for x-bar chart:
X  A2 R
11.
Use formulas to find limits for R chart:
LCL  D3 R
UCL  D4 R
Let’s try a small problem
smpl 1
smpl 2
smpl 3
smpl 4
smpl 5
smpl 6
observation 1
7
11
6
7
10
10
observation 2
7
8
10
8
5
5
observation 3
8
10
12
7
6
8
x-bar
R
X-bar chart
UCL
Centerline
LCL
R chart
Let’s try a small problem
smpl 1
smpl 2
smpl 3
smpl 4
smpl 5
smpl 6
observation 1
7
11
6
7
10
10
observation 2
7
8
10
8
5
5
observation 3
8
10
12
7
6
8
X-bar
7.3333
9.6667
9.3333
7.3333
7
7.6667
R
1
3
6
1
5
5
X-bar chart
8.0556
3.5
R chart
11.6361
9.0125
Centerline
8.0556
3.5
LCL
4.4751
0
UCL
Avg.
X-bar chart
14.0000
12.0000
10.0000
8.0000
11.6361
8.0556
6.0000
4.0000
2.0000
0.0000
4.4751
1
2
3
4
5
6
R chart
10
9.0125
8
6
4
3.5
2
0
0
1
2
3
4
5
6
Interpreting charts
Observations outside control limits indicate
the process is probably “out-of-control”
 Significant patterns in the observations
indicate the process is probably “out-ofcontrol”
 Random causes will on rare occasions
indicate the process is probably “out-ofcontrol” when it actually is not

Interpreting charts

In the excel spreadsheet, look for these
shifts:
A
B
C
Show real time examples of charts here
D
Lots of other charts exist
P chart
C charts
U charts
Cusum & EWMA
For yes-no
questions like
“is it defective?”
(binomial data)
For counting
number defects
where most items
have ≥1 defects
(eg. custom built
houses)
Average count
per unit (similar
to C chart)
Advanced charts
p(1  p)
p 3
n
c 3 c
u
u 3
n
“V” shaped or
Curved control
limits (calculate
them by hiring a
statistician)
Selecting rational samples



Chosen so that variation within the sample is
considered to be from common causes
Special causes should only occur between
samples
Special causes to avoid in sampling
 passage
of time
 workers
 shifts
 machines
 Locations
Chart advice







Larger samples are more accurate
Sample costs money, but so does being out-of-control
Don’t convert measurement data to “yes/no” binomial
data (X’s to P’s)
Not all out-of control points are bad
Don’t combine data (or mix product)
Have out-of-control procedures (what do I do now?)
Actual production volume matters (Average Run Length)