C & u charts

81© 2001 Jay Arthur Six Sigma Simplified

Stability A stable process produces predictable results consistently. Stabilitycan be easily determined from control charts. The upper control limit(UCL) and lower control limit (LCL) are calculated from the data.

How long does it take you to commute to work each morning?

Stabilize the ProcessUnderstanding Stability

Example

Stable=

Predictable

A process does not have to be stable to be capable of meeting thecustomer's requirements. Similarly, a stable process is not necessarilycapable. A managed process must be both stable and capable.Interpreting stability with control charts and capability with histograms willbe discussed in more detail on the following pages.

22 min.

29 min.

15 min.

Daily Commute (minutes)

Your Requirements1. Get to work in 30minutes or less.2. Get to work safely(no faster than 15minutes).

Stability andCapability

Stable

Unstable Trend

24 min.

32 min.

18 min.

Daily Commute (minutes)Snow Storm

UCL

LCL

22 min.

29 min.

15 min.


Point Unstable

USLLSL

3015T

rips

To

Wor

k

Daily Commute Time

Capable

USLLSL

3015 Minutes

Trip

s T

o W

ork

Daily Commute Time

Capable

USLLSL

3015 Minutes

Trip

s T

o W

ork

Daily Commute Time

NotCapable


Check StabilityInterpreting The Indicators

Purpose

Variation

You cannot steptwice into the sameriver. Heraclitus

Verify that the process system is stable andcan predictably meet customer requirements

A stable process produces predictable results. Understandingvariation helps us learn how to predict the performance of anyprocess. To ensure that the process is stable (i.e., predictable)we need to develop "run" or "control" charts of our indicators.

How can you tell if a process is stable? Processes are neverperfect. Common and special causes of variation make theprocess perform differently in different situations. Getting fromyour home to school or work takes varying amounts of timebecause of traffic or transportation delays. These are commoncauses of variation; they exist every day. A blizzard, a trafficaccident, a chemical spill, or other freak occurrence that causesmajor delays would be a special cause of variation.

In the 1920s, Dr. Shewhart, at Bell Labs, developed ways toevaluate whether the data on a line graph is common cause orspecial cause variation. Using 20-30 data points, you candetermine how stable and predictable the process is. Usingsimple equations, you can calculate the average (center line),and the upper and lower "control limits" from the data. 99% ofall expected (i.e., common cause variation) should lie betweenthese two limits. Control limits are not to be confused withspecification limits. Specification limits are defined by the cus-tomer. Control limits show what the process can deliver.

1 5 10

Center Line (average)

15 20 25 30

Upper Control Limit (UCL)

Lower Control Limit (LCL)

99.7% of all data points68

.3%

95.5

%

Your Requirements:1. Get to work fast!2. Get to work safely.

22 min.

29 min.

15 min.


Example

Stable


Check StabilityInterpreting The Indicators

SpecialCauseVariation

Processes that are "out of control" need to be stabilized beforethey can be improved using the problem-solving process.Special causes, require immediate cause-effect analysis toeliminate the special cause of variation.

The following diagram will help you evaluate stability in anycontrol chart. Unstable conditions can be any of the following:

Any point outside the upper or lower control limits is a clearexample of a special cause. The other forms of special causevariation are called "runs." Trends, cycling up and down, or"hugging" the center line or limits are special forms of a run.

EvaluatingStability

Points andRuns

1 5 10 15 20 25 30

Any point above UCL

Any point below LCL

CL

UCL

LCL

2 of 3 points in this area

4 of 5 points in this area or above

8 points in a row in this area or above

2 of 3 points in this area

8 points in a row in this area or below

4 of 5 points in this area or below

22 min.

29 min.

15 min.

Daily Commute (minutes)Snow Storm

Point Unstable

Unstable Trend

22 min.

29 min.

15 min.


Any point below LCL

UCL

LCL

CL

Trend

8 above CL

4 below B

2 above A

Point outside UCL

6 ascendingor descending

B

A

B

A


Step 4 - Check Stabilityc and u charts

The c and u charts will help you evaluate process stability whenthere can be more than one defect per unit. Examples mightinclude: the number of defective elements on a circuit board, thenumber of defects in a dining experience–order wrong, food toocold, check wrong, or the number of defects in bank statement,invoice, or bill. This chart is especially useful when you want toknow how many defects there are not just how many defectiveitems there are. It's one thing to know how many defective circuitboards, meals, statements, invoices, or bills there are; it isanother thing to know how many defects were found in thesedefective items.

The c chart is useful when it's easy to count the number ofdefects and the sample size is always the same. The u chart isused when the sample size varies: the number of circuit boards,meals, or bills delivered each day varies. The c chart belowshows the number of defects per day in a uniform sample.

Given this information, we would want to investigate whyFebruary 11th was "out of control." We would also want tounderstand why we were able to keep the defects so far belowaverage in the other circled areas. What did we do here that wasso successful?

A fully capable process delivers zero defects.

Stability

Capability

Number Defects Per Day

Num

ber

of D

efec

ts

0

1

2

3

4

5

6

7

1-F

eb

2-F

eb

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eb

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eb

5-F

eb

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eb

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eb

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9-F

eb

10-F

eb

11-F

eb

12-F

eb

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eb

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eb

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eb

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eb

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19-F

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eb

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eb

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eb

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eb

27-F

eb

28-F

eb

n=28

Point Outside Limits

Run Below CL

Approach to LimitsApproach to Limits

UCL

CL

LCL

X XX

Defects

c and uCharts(Attribute data)

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C Chart U ChartUCL: c + 3*sqrt(c) u + 3*sqrt(u/n )CL: c = ∑ci/n u = ∑ui/∑ni

LCL: c - 3*sqrt(c) u - 3*sqrt(u/n )

i

i

X XX

= More Than One Defect

Step 4 - Check Stabilityc and u charts

c

u

Title

Number or Percent of Defects

Measurement or Sample

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1Defects (c)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1Defects (u)

Sample Size (n)

Percent

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

UCL

LCL

Technology

C & u charts