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Subgroup

A sample of units produced under a similar set

of conditions.

Process average (also, process center)

An estimate of the overall mean or average forthe process. An estimate of where the process

is located, or centered.

Process variation (also, process spread,

process variability)

An estimate of the variation or spread present

in the process.

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Two Types of Variation

Common Cause

Noise Predictable

Routine

No Assignable Cause

Expected

Special Cause

Signal Not Predictable

Exceptional

Assignable Cause

Unusual

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Stable vs Unstable Processes

A stable (or in control) process is one in

which the key process responses and

product properties show no signs of specialcauses.

An unstable (or out of control) process hasboth common and special causes present.

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5

Visualizing Short-Term and Long-Term

Variation

Time

Short-Term

Measuremen

t

Long-Term

Time

Short-Term

Measurement

Long-Term

Long-Term

Time

Short-Term

Measurement

Time

Short-Term

Measurement

Long-Term

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6

Visualizing Short-Term and Long-Term

Variation "Individual World"

Time

Short-Term

Measuremen

t

Long-Term

Time

Short-Term

Measurement

Long-Term

Time

Short-Term

Measurement

Long-Term

Time

Short-Term

Measurement

Long-Term

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7

Short-Term vs Long-Term Variation

Short-term (sST)

Represents the processcapability

Captures variation due tocommon causes

Measures variation within asubgroup or between

successive values

Used for calculating controlchart limits

Long-term (sLT)

Represents the total processvariation

Captures variation due tocommon and special causes

Measures variation in all data

Should not be used forcalculating control chart limits

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8

Calculating Short-Term and Long-

Term Standard Deviations

Long-term (sLT)

XiX2

i = 1

n

n1

sLT =

Short-term (sST)sST = R / d2

sST = s / c4

sST = MR / d2

STLT

STLT

UnstableIf

StableIf

ss

ss

,

,

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Rule of Thumb

< 1 Month Short-Term Data

1 to 3 Months Judgment Call

>3 Months Long-Term Data

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10

Stability Index

Stability Index =sSTsLT

For a stable process, you would expect index values near 1.

For an unstable process, you would expect index values greater than 1.

Rule of Thumb

< 1.33 Good Process Stability.

1.33 to 1.67 Marginal Process Stability.

> 1.67 Major Process Stability Issues.

Note: For use when n>75. If n

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11

Switching Gears to Process Capability

+3s-3s

USLLSL

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12

Specification Limits and Capable

Processes

Specification Limits - often called voice of the customer and used todetermine if the product meets a customer requirement. Usually stated as a LSLand USL but sometimes you may only have one of these.

A capable process is a stable process that demonstrates the ability to meetcustomer requirements. (A purist definition The simple fact is that no processis stable forever and ever and we still need to address the capability of ourprocesses in the presence of instability)

When we talk capability indices, we're now comparing the process variation(and sometimes average) to the specification limits.

Before when we were talking stability, we were comparing the process variationto the control limits.

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13

The Four Major Capability /

Performance Indices

Cp

and Cpk

address short-term capability

Pp and Ppk address long-term performance

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14

Process Capability Indices - Cp

+3s-3s

USLLSL

Spec Width (door*) USL - LSL

Cp =

Mfg Capability (car)

=

3s

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15

Process Capability Indices - Cp

Cp

=USL - LSL

6sST

Cp < 1 - not capable

Cp = 1 - marginally capable

Cp > 1 - capable

The average is not part of the formula.

A measure ofpotential "best case" process capability if stable and on-target.

Can be misleading if process is unstable or off target.

Must have both a LSL and USL to calculate.

LSL USL

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16

Visualizing Process Capability

43210- 1- 2- 3-4

0 .4

0 .3

0 .2

0 .1

0 .0

Lower

Spec. Limit

Upper

Spec.Limit

Cust. Tolerance

Cp

=1

Process Capability

86420- 2- 4-6- 8

0 .4

0 .3

0 .2

0 .1

0 .0

Lower

SpecLimit

Upper

Spec.Limit

Cust. Tolerance

Cp

=2

Process

Capability

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17

What if Process is Off-Target?

Cp = 1.33

Cpk = 1.33

5.334.02.671.33-1.33-2.67-4.0-5.33 0

0 .4

0 .3

0 .2

0 .1

0 .0

Lower

Spec.

Limit

Upper

Spec.

Limit

Cust. Tolerance

0

0 .4

0 .3

0 .2

0 .1

0 .0

5.334.02.671.33-1.33-2.67-4.0-5.33

LowerSpec.

Limit

UpperSpec.

Limit

Cust. Tolerance

Cp = 1.33

Cpk = 0.83

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18

Process Capability Indices - Cpk

Cpk

=Min (USL - avg, avg - LSL)

3sST

Cpk < 1 - not capable

Cpk = 1 - marginally capable

Cpk > 1 - capable

Cp = Cpk if process is on target.

Still a measure ofpotential capability if the process is stable.

Can be used for 1-sided specs. (Cpu, Cpl)

A negative Cpk is possible if the average is outside specifications.

LSL USL

avg

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19

Process Performance Indices - Pp

Pp

=USL - LSL

6sLT

Pp < 1 - not meeting specs

Pp = 1 - marginally meeting specs

Pp > 1 - meeting specs

Considered a Process Performance Index

If stable, Cp = PpCan be misleading if process is off target.

LSL USL

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20

Process Performance Indices - Ppk

Ppk

=Min (USL - avg, avg - LSL)

3sLT

Ppk < 1 - not meeting specs

Ppk = 1 - marginally meeting specs

Ppk > 1 - meeting specs

If stable, Cpk = Ppk

If on target, Pp = Ppk

If stable and on target, Cp = PpkCan be used for 1-sided specs. (Ppu, Ppl)

Best indicator of actual process performance.

A negative Ppk is possible if the average is outside specifications.

LSL USL

avg

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21

Summary of Indices

Cp is the best indicator ofpotential process capabilitybecause it assumes a stable and on-target process.

Cpk is an indicator of potential process capability if theprocess is stable. It does take into consideration if theprocess is off-target.

Pp is an indicator of actual process performance if theprocess is on-target. It does take into consideration the

long-term variability. Ppk is the best indicator ofactual process performance

because it considers the process average and long-termvariability.

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One more Index

Sometimes it is desirable to operate the process

at a target that is not midway between the

specification limits. Cpk can still be used to

estimate defect levels but does not reflectwhether the process is centered on the target.

Use Cpm for this.

22

Cpm =USL - LSL

6 sST2 + (avg-tgt)2

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A run chart is the simplest of charts.It is a single line plotting some value over time.

A run chart can help you spot upward and downward trends

it can show you a general picture of a process

RUN CHART

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Control charts are designed to prevent two common

mistakes:

1) Adjusting the process when it should be left alone;

2) Ignoring the process when it may need to be adjusted.

Control Charts

Control (also, in control)

The absence ofspecial-cause variation.

A process that is in control exhibits only random variation. In other words, the

process is stable or consistent.

Control charts assess statistical control by determining whether the process

output falls within statistically calculated control limits and exhibits only random

variation.

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EXAMPLE

Median Line

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A control chart is a statistical tool used todistinguish between variation in a process resulting

from common causes and variation resulting from

special causes.

What is a Control Chart?

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One goal of using a Control Chart is to achieve and

maintain process stability.

Process stability is defined as a state in which a processhas displayed a certain degree of consistency in the

past and is expected to continue to do so in the future.

This consistency is characterized by a stream of datafalling within control limits

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An automobile assembly plant collected camshaft length

measurements to assess the quality of the process.

Five camshafts were measured from each of four shiftsdaily for five days.

The five samples that comprise each subgroup were

selected within a short period of time to minimizevariation from one camshaft to the next.

Example

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Control limits represent the limits of variation that shouldbe expected from a process in a state of statistical control.

Control limits

Control limitsSpecification limits?

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Specification limits

Limits that are established for the process - they do not reflect

how the process is actually performing. Specification limits are

based on the customer requirements and detail how one wants

the process to behave.

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A stable process is one that is consistent over time with

respect to the center and the spread of the data.

X bar Chart - Center

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R bar Chart - Spread

A stable process is one that is consistent over time with

respect to the center and the spread of the data.

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Why Use Control Charts?

Monitor process variation over time

Differentiate between special causeand common cause variation

Assess effectiveness of changes

Communicate process performance

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What are the types of Control Charts?

There are two main categories of Control Charts,

1.Those that display attribute data.

2.Those that display variables data.

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Attribute Data:

This category of Control Chart displays data that result

from counting the number of occurrences or

items in a single category of similar

items or occurrences.

These count data may be expressed as pass/fail,

yes/no, or presence/absence of a defect.

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Variables Data:

This category of Control Chart displays values

resulting from the measurement of a continuous

variable.

Examples of variables data

elapsed time, temperature, and radiation dose.

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There are three types that will work for the

majority of the data analysis cases you will

encounter

X-Bar and R Chart

Individual X and Moving Range Chart for

Variables Data

Individual X and Moving Range Chart for

Attribute Data

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What are the elements of a Control Chart?

Each Control Chart actually consists of twographs an upper and lower

A Control Chart is made up of eight elements.

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1.Briefly describes the information which is

displayed.

2.Information on how and when the data were

collected.

3.The counts or measurements are recorded in

the data collection section of the Control Chart

prior to being graphed.

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Plotting Areas. A Control Chart has two areasan upper

graph and a lower graphwhere the data is plotted.

a. The upper graph plots either the individual values, in the

case of an Individual X and Moving Range chart, or the average(mean value) of the sample or subgroup in the case of an X-Bar

and R chart.

b. The lower graph plots the moving range for Individual X andMoving Range charts, or the range of values found in the

subgroups for X-Bar and R charts.

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6. Horizontal or X-Axis. This axis displays thechronological order in which the data were collected.

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l

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8. Centerline.

This line is drawn at the average or mean value of

all the plotted data. The upper and lower graphseach have a separate centerline.

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What are the steps for calculating and plotting an

X-Bar and R Control Chart for Variables Data?

The X-Bar (arithmetic mean) and R (range) Control Chart is used

with variables data when subgroup or sample size is between 2

and 15. The steps for constructing this type of Control Chart are:

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Step 1 - Determine the data to be collected.

Decide what questions about the process you plan to answer.

1.Determine what characteristic to be measured




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Step 2 - Collect and enter the data by subgroup. A subgroup is made up

ofvariables data that represent a characteristic of a product produced by

a process.

The sample size relates to how large the subgroups are. Enter the

individualsubgroup measurements in time sequence in the portion of

the data collection section of the Control Chart labeled MEASUREMENTS

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STEP 3 - Calculate and enter the average for each

subgroup.

Use the formula below to calculate the average (mean)for each subgroup and enter it on the line labeled

Average in the data collection section

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Step 4 - Calculate and enter the range for each

subgroup.

Use the following formula to calculate the range (R) for

each subgroup. Enter the range for each subgroup on the

line labeled Range in the data collection section

RANGE = (Largest Value in each Subgroup) -(Smallest

Value in each Subgroup)

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Range

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Step 5 - Calculate the grand mean of the subgroups

average.

The grand mean of the subgroups average (X-Bar)becomes the centerline for the upper

plot.

S 6 C l l h f h b

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Step 6 - Calculate the average of the subgroup

ranges.

The average of all subgroups becomes thecenterline for the lower plotting area.

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Step 7 - Calculate the upper control limit (UCL)

and lower control limit (LCL) for the averages of

the subgroups.

Control limits define the parameters for

determining whether a process is in statistical

control.

NOTE: Constants, based on the subgroup size (n), are

used in determining control limits for variables charts.

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Use the following constants (A2) in the computation

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Step 8 - Calculate the upper control limit for the ranges.

When the subgroup or sample size (n) is less than 7, thereis no lower control limit. To find the upper

control limit for the ranges, use the formula:

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Use the following constants (D4) in the computation

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Step 9 - Select the scales and plot the control limits,

centerline, and data points, in each plotting area.

The scales must be determined before the data

points and centerline can be plotted. Once the upper

and lower control limits have been computed, the

easiest way to select the scales is to have the currentdata take up approximately 60 percent of the vertical

(Y) axis.

The scales for both the upper and lower plotting areasshould allow for future high or low out-of control

data points.

Plot each subgroup average as an individual data point in

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Plot each subgroup average as an individual data point in

the upper plotting area. Plot individual range data points in

the lower plotting area

UCL

CENTRE LINE

LCL

i bl l h f i di id l

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Variables control charts for individuals are powerful and simple visualtools for determining whether a process is in or out of control.

Variables control charts for individuals

When to use individuals control charts

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When collecting samples to learn about a process, it is sometimes easier to combine the

samples into subgroups, if it makes sense to group the samples together. When grouping is

not appropriate, then a subgroup size of one provides a method for evaluating the process.

Samples that cannot logically be grouped together are good candidates for individuals (I) and

moving range (MR) charts.

Some conditions that make using subgroups unfeasible or undesirable include:

When each sample is uniquely identified with a specific period of time

When each sample represents one distinct batch

When there are long time intervals between each sample

When sampling or testing is destructive and/or expensive

When output is continuous and homogenous

When there is a long cycle time for production

When the measurements are not necessarily related in time (for example, business

data such as time to complete a contract or time to answer a question)

Example

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Example A liquid detergent company wants to assess whether

or not its process is in control.

The liquid detergent is made in batches by mixingtogether a number of ingredients.

The quality characteristic of interest is the pH value

for each batch. Measurements of the pH value for 25

consecutive batches were taken.

Because the liquid detergent is created in a batch

process and only one measurement is taken for each

batch, it is not appropriate to place the data insubgroups. Instead, the liquid detergent data should

be analyzed using an Individuals (I) control chart,

which considers each measurement to be an

independent observation.

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Step 2 - Collect and enter the individual measurements.

Enter the individual measurements in time sequence on the

line labeled Individual X in the data collection section of theControl Chart.

These measurements will

be plotted as individual data points in the upper plotting

area.

Step 1 - Determine the data to be collected.

Decide what questions about the process you plan to

answer.

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STEP 3 - Calculate and enter the moving ranges.

Use the following formula to calculate the moving ranges

between successive data entries. Enter them on theline labeled Moving R in the data collection section.

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Step 4 Calculate the overall average of the individual

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Step 4 - Calculate the overall average of the individual

data points.

The average of the Individual-X data becomes the

centerline for the upper plot.

Step 5 - Calculate the average of the moving ranges.

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Step 5 Calculate the average of the moving ranges.

The average of all moving ranges becomes the centerline

for the lower plotting area.

Step 6 - Calculate the upper and lower control limits for

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Step 6 Calculate the upper and lower control limits for

the individual X values.

The calculation will compute the upper and lower control

limits for the upper plotting area. To find these controllimits, use the formula:

Y h ld th t t l t 2 66 i b th

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You should use the constant equal to 2.66 in both

formulas when you compute the upper and lower

control limits for the Individual X values.

Step 7 - Calculate the upper control limit for the ranges.

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This calculation will compute the upper control limit for the

lower plotting area. There is no lowercontrol limit.

You should use the constant equal to 3.268 in the formula

when you compute the upper control limit for the moving

range data.

Step 8 - Select the scales and plot the data points and centerline

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in each plotting area.

Once the upper and lower control limits have been computed,

the easiest way to select the scales is to have the current

spread of the control limits take up approximately 60 percent

of the vertical (Y) axis.

Y Axis

X Axis

UCL

LCL

60 % of Y Axis

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Step 9 - Provide the appropriate documentation.

Each Control Chart should be labeled with who, what, when,

where, why, and how information to describe where the data

originated, when it was collected, who collected it, anyidentifiable equipment or work groups, sample size, and all the

other things necessary for understanding and interpreting it.

It is important that the legend include all of theinformation that clarifies what the data describe.

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If you are working with attribute data, continue through steps 10,

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11, 12a, and 12b

Step 10 - Check for inflated control limits.

When either of the following conditions exists, the control limits

are said to be inflated, and you must

recalculate them:

If any point is outside of the upper control limit for the

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moving range (UCLmR)

If two-thirds or more of the moving range values are

below the average of the moving ranges computed inStep 5.

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When there is an odd number of values the median is

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When there is an odd number of values, the median is

the middle value.

When there is an even number of values, average the

two middle values to obtain the median.

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Step 12a - Do not compute new limits if the product of

3.144 times the median moving range value is greater

than the product of 2.66 times the average of the moving

ranges.

Step 12b - Recompute all of the control limits and

centerlines for both the upper and lower plotting areas if

the product of 3.144 times the median moving range

value is less than the product of 2.66 times the average ofthe moving range.

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These new limits must be redrawn on the corresponding

charts before you look for signals of special causes. Theold control limits and centerlines are ignored in

any further assessment of the data.

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What do we need to know to interpret Control Charts?

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p

If a process is stable, the likelihood of a point falling

outside this band is so small that such an occurrence is

taken as a signal of a special cause of variation.

Even though all the points fall inside the control limits, special

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cause variation may be at work.

The presence of unusual patterns can be evidence that your

process is not in statistical control.

Such patterns are more likely to occur when one or more special

causes is present.

The three standard deviations are sometimes identified by

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zones.

Each zones dividing line is exactly one-third the distance

from the centerline to either the upper control limit or thelower control limit

What are the rules for interpreting X-Bar and R Charts?

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p g

Interpreting the tests for special causes

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Note that only tests 1-4 are available for R, S, and moving range charts.

One point more than 3-s from center line. Test 1 evaluates the patternof variation for stability. Test 1 provides the strongest evidence of lack of

control. If small shifts in the process are of concern, Tests 2, 5, and 6 can

be used to supplement Test 1 to produce a control chart with greater

sensitivity.

Test 1

Test 2

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Nine points in a row one the same side of the center line. Test 2

evaluates the pattern of variation for stability. If small shifts in the

process are of concern, Test 2 can be used to supplement Test 1 toproduce a control chart with greater sensitivity.

Test 3

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Six points in a row, all increasing or all decreasing.

Detects a trend or continuous movement up or down. This test looks

for long series of consecutive points without a change in direction.

Test 4

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Test 4

Fourteen points in a row, alternating up or down.Points tothe presence of a systematic variable. The pattern of

variation should be random, but when a point fails Test 4 it

means that the pattern of variation is predictable.

Test 5

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Two out of three points more than 2s from the centerline (same side).Test 5 evaluates the pattern of variation

for small shifts in the process.

Test 5

Test 6

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Four out of five points more than 1s from center line

(same side).Test 6 evaluates the pattern of variation forsmall shifts in the process.

Test 6

Test 7

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Fifteen points in a row within 1s of center line (either side).Test 7

identifies a pattern of variation that is sometimes mistaken as a

display of good control. This type of variation is called stratificationand is characterized by points that hug the center line too closely.

Test 7 identifies a pattern of variation that is sometimes mistaken as a

display of good control. This type of variation is called stratification and

is characterized by points that hug the center line too closely.

Test 8

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Eight points in a row more than 1s from center line (either side).Test 8

is referred to as a mixture. A mixture pattern occurs when the pointstend to avoid the center line and instead plot near the control limits.

What are the rules for interpreting XmR Control Charts?

To interpret XmR Control Charts you have to apply a set of rules

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To interpret XmR Control Charts, you have to apply a set of rules

for interpreting the X part of the chart, and a further single rule

for the mR part of the chart.

RULES FOR INTERPRETING THE X-PORTION of XmR Control

Charts:

Apply the four rules discussed above, EXCEPT apply them only to

the upper plotting area graph.

RULE FOR INTERPRETING THE mR PORTION of XmR Control

Charts for attribute data:

Rule 1 is the only rule used to assess signals of special causes in

the lower plotting area graph. Therefore, you dont need to

identify the zones on the moving range portion of an XmR chart.

When should we change the control limits?

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There are only three situations in which it is appropriate to change the control

limits:

When removing out-of-control data points.When a special cause has been identified and removed while you are working to achieve

process stability, you may want to delete the data points affected by special causes and use

the remaining data to compute new control limits.

When replacing trial limits.

When a process has just started up, or has changed, you may want to calculate control limits

using only the limited data available. These limits are usually called trial control limits. You

can calculate new limits every time you add new data. Once you have 20 or 30 groups of 4 or

5 measurements without a signal, you can use the limits to monitor future performance. You

dont need to recalculate the limits again unless fundamental changes are made to the

process.

When there are changes in the process.When there are indications that your process has changed, it is necessary to recompute the

control limits based on data collected since the change occurred. Some examples of such

changes are the application of new or modified procedures, the use of different machines,

the overhaul of existing machines, and the introduction of new suppliers of critical input

materials.

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EXERCISE 1: A team collected the variables data recorded in the table below.

Use these data to answer the following questions and plot a Control

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Use these data to answer the following questions and plot a Control

Chart:

1. What type of Control Chart would you use with these data?

2. Why?

3. What are the values of X-Bar for each subgroup?

4. What are the values of the ranges for each subgroup?

5. What is the grand mean for the X-Bar data?6. What is the average of the range values?

7. Compute the values for the upper and lower control limits for both

the upper and lower plotting areas.

8. Plot the Control Chart.

9. Are there any signals of special cause variation? If so, what rule did

you apply to identify the signal?

EXERCISE 1 ANSWER KEY:

1 X B d R

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1. X-Bar and R.

2. There is more than one measurement within each

subgroup.3. Refer to Fig. 24.

4. Refer to Fig. 24.

5. Grand Mean of X = 4.52.

6. Average of R = 4.38.

7. UCLX = 4.52 + (0.729) (4.38) = 7.71.

LCLX = 4.52 - (0.729) (4.38) = 1.33.

UCLR = (2.282) (4.38) = 10.0.

LCLR = 0.


9. No.

EXERCISE 2: A team collected the dated shown in the

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chart below.

Use these data to answer the following questions and plot aControl Chart:

f

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1. What are the values of the moving ranges?

2. What is the average for the individual X data?

3. What is the average of the moving range data?4. Compute the values for the upper and lower control limits for

both the upper and lower plotting areas.

5. Plot the Control Chart.

6. Are the control limits inflated? How did you determine your

answer?

7. If the control limits are inflated, what elements of the Control

Chart did you have to recompute?

8. If the original control limits were inflated, what are the new

values for the upper and lower control limits and centerlines?9. If the original limits were inflated, replot the Control Chart using

the new information.

10. After checking for inflated limits, are there any signals of special

cause variation? If so, what rule did you use to identify the signal?

EXERCISE 2 ANSWER KEY:

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2. 19.2.

3. 5.5.4. UCLX = 33.8. LCLX = 4.6. UCLmR =18.0. LCLmR = 0.


6. Yes; one point out of control, and 2/3 of all points below the

centerline.7. All control limits and the centerline for the lower chart. The

median value will be used in the re computation rather than the

average.

8. UCLX = 31.8. LCLX = 6.6. UCLmR = 15.5. LCLmR = 0.

CenterlinemR = 4 (median value).

9. Refer to Viewgraph 28.

10. Yes; the same point on the mR chart is out of control. Rule 1.

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