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)