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)
Emmett
Jake
Observations
10
9
8
8
7
averages 8.4
Deviations
10 - 8.4 = 1.6
9 – 8.4 = 0.6
8 – 8.4 = -0.4
8 – 8.4 = -0.4
7 – 8.4 = -1.4
0.0
871089
Variability
Deviation = distance between observations and the mean (or average)
Emmett
Jake
Observations
7
7
7
6
6
averages 6.6
Deviations
7 – 6.6 = 0.4
7 – 6.6 = 0.4
7 – 6.6 = 0.4
6 – 6.6 = -0.6
6 – 6.6 = -0.6
0.0
76776
Variability
Variance = average distance between observations and the mean squared
Emmett
Jake
Observations
10
9
8
8
7
averages 8.4
Deviations
10 - 8.4 = 1.6
9 – 8.4 = 0.6
8 – 8.4 = -0.4
8 – 8.4 = -0.4
7 – 8.4 = -1.4
0.0
871089
Squared Deviations
2.56
0.36
0.16
0.16
1.96
1.0 Variance
Variability
Variance = average distance between observations and the mean squared
Emmett
Jake
Observations
7
7
7
6
6
averages
Deviations Squared Deviations 76776
Variability
Variance = average distance between observations and the mean squared
Emmett
Jake
Observations
7
7
7
6
6
averages 6.6
Deviations
7 - 6.6 = 0.4
7 - 6.6 = 0.4
7 - 6.6 = 0.4
6 – 6.6 = -0.6
6 – 6.6 = -0.6
0.0
Squared Deviations
0.16
0.16
0.16
0.36
0.36
0.24
76776
Variance
Variability
Standard deviation = square root of variance Emmett
Jake
Variance Standard Deviation
Emmett 1.0 1.0
Jake 0.24 0.4898979
But what good is a standard deviation
Variability
The world tends to be bell-shaped
Most outcomes
occur in the middle
Fewer in the “tails”
(lower)
Fewer in the “tails” (upper)
Even very rare outcomes are
possible(probability > 0)
Even very rare outcomes are
possible(probability > 0)
Variability
Add up the dots on the dice
0
0.05
0.1
0.15
0.2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Sum of dots
Pro
ba
bili
ty 1 die
2 dice
3 dice
Here is why: Even outcomes that are equally likely (like dice), when you add them up, become bell shaped
“Normal” bell shaped curve
Add up about 30 of most things and you start to be “normal”
Normal distributions are divide upinto 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 XXXX XX X XX1 2 3 4 5 6 7 8
Causes of Variability
Common Causes: Random variation (usual) No pattern Inherent in process adjusting the process increases its variation
Special Causes Non-random variation (unusual) May exhibit a pattern 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 Limits = target ± tolerance Define acceptable & unacceptable
Process vs. control limits
Variance of averages < variance of individual items
Distribution of averages
Control limits
Process limits
Distribution of individuals
Specification limits
Usual v. Unusual, Acceptable v. Defective
μ Target
A B C D E
More about limits
Good quality: defects are rare (Cpk>1)
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.
μtarget
μtarget
Process capabilityGood quality: defects are rare (Cpk>1)Poor quality: defects are common (Cpk<1)
Cpk = min
USL – x3σ
=
x - LSL3σ
=
3σ = (UPL – x, or x – LPL) = =
14 20 26 15 24
24 – 203(2)
= =.667
20 – 153(2)
= =.833
Going out of control
When an observation is unusual, what can we conclude?
μ2
The mean has changed
X
μ1
Going out of control
When an observation is unusual, what can we conclude?
The standard deviationhas changed
σ2
X
σ1
Setting up control charts:
Calculating the limits1. Sample n items (often 4 or 5)
2. Find the mean of the sample (x-bar)
3. Find the range of the sample R
4. Plot on the chart
5. Plot the R on an R chart
6. Repeat steps 1-5 thirty times
7. Average the ’s to create (x-bar-bar)
8. Average the R’s to create (R-bar)
x
x
x
xxR
Setting up control charts:
Calculating the limits9. Find A2 on table (A2 times R estimates 3σ)
10. Use formula to find limits for x-bar chart:
11. Use formulas to find limits for R chart:
RAX 2
RDLCL 3 RDUCL 4
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 R chart
UCL
Centerline
LCL
Let’s try a small problem
smpl 1 smpl 2 smpl 3 smpl 4 smpl 5 smpl 6 Avg.
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 8.0556
R 1 3 6 1 5 5 3.5
X-bar chart R chart
UCL 11.6361 9.0125
Centerline 8.0556 3.5
LCL 4.4751 0
X-bar chart
0.0000
2.0000
4.0000
6.0000
8.0000
10.0000
12.0000
14.0000
1 2 3 4 5 6
11.6361
8.0556
4.4751
R chart
0
2
4
6
8
10
1 2 3 4 5 6
9.0125
3.5
0
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-of-control”
Random causes will on rare occasions indicate the process is probably “out-of-control” when it actually is not
Interpreting charts
In the excel spreadsheet, look for these shifts:
A B
C D
Show real time examples of charts here
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
“V” shaped or Curved control limits (calculate them by hiring a statistician)
n
ppp
)1(3
cc 3
n
uu 3
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)