AREAS IN UPPER TAIL OF NORMAL DISTRIBUTION

FACTORS FOR CONTROL CHART

Table: Different control charts for variables (Source: Donna C., 2007))

Type Central Line Central Limits Comments

X and s X 3

3

X

X

UCL X A s

LCL X A s

Use when more sensitivity

is desired than R; when

n>10; and when data are

collected automatically. s 4

3

s

s

UCL B s

LCL B s

Moving Average,

M X and moving

range, MR

X 2

2

X

X

UCL X A R

LCL X A R

Use when only one

observation is possible at a

time. Data does not need to

be normal. R 4

3

R

R

UCL D R

LCL D R

X and moving R X 2.660

2.660

x

x

UCL X R

LCL X R

Use when only one

observation is possible at a

time and the data are

normal. Equations are based

on a moving range of two. R 3.276

(0)

R

R

UCL R

LCL R

Median and

Range

MDmd 5

5

MD MD MD

MD MD MD

UCL MD A R

LCL MD A R

Use when process is in a

maintenance mode. Benefits

are less arithmetic and

simplicity. Rmd 6

5

R MD

R MD

UCL D R

LCL D R

Examine the process: interpretation of control chart

Interpretation of Control Charts

Control chart (CC) is a device/tool to describe in a precise manner what is

meant by statistical control. CCS are used in a variety of ways, such as on-

line process surveillance

If sample points are falling within control limits and no systematic pattern is

exhibited, the process is al-right.

Surely, CC is importantly used to improve the process, because

1. Most process do not operate in a state of statistical control

2. Regular and attentive use of CCs will help identification of assignable

causes. CCs cannot direct any action for improvement

3. Engineering and other actions are necessary to eliminate assignable

causes.

So, need to identify root causes, apply long-term improvement solutions for

an effective SPC application.

Out-of-Control-Action Plan (OCAP)

It’s a flow-chart or text based description of the order of activities that must take

place to activate an event. Two elements of OCAP are:

Receive out of control signals from CCs and determine checkpoints

(potential assignable causes)

Terminators – action taken to resolve

OCAP should be living document so that it can be modified over time with the aid

of new knowledge and understanding. So, an initial OCAP is required when a CC

is introduced.

In figure below on out-of-control plan, there are two controllable variables,

pressure and time. When R-chart exhibits any out-of-control signal, then need to

contact process engineering immediately. If X-bar chart exhibits an out-of-control

signal, operators are directed to check process settings and calibration and then

make adjustments to pressure to bring back the process into a state of control. If

such adjustments are not successful, then need to contact process engineering

immediately.

Process

Measurement system

Input Output

Verify and

follow up

Detect

assignable

cause

Implement corrective

action

Identify root cause of

problem

Figure: out-of-control plan for a process

Out-of-control signal on X -bar

and R charts for certain

characteristic

Is data entered

correctly?

Which test/action

failed?

Are setting of time

and pressure correct?

Is an

equipment/instrumen

t in calibration?

Is the calibrated

item adjusted?

Edit data to correct entry

Contact process engineering

Reset pressure and/or time and

enter new data

Contact process engineering

Contact process engineering

Adjust pressure per specification

table. Reset and enter new data

Enter comments in log book for

describing actions

Yes

No

Average

Range

Yes

Yes

No

No

No

Yes

Correct interpretation of control charts is essential to managing a process. Understanding the

sources and potential causes of variation is critical to good management decisions.

The control chart is interpreted based in its patterns. There will be cases where the trend is so

unpredictable that totally not belong to any categories when the points are fluctuating randomly.

As had been proposed by Amjed Al-Ghanin and Jay Jordan in the article “Automated process

monitoring using statistical pattern recognition techniques on X-bar control charts”, pattern

recognition methodology has been pursued to identify unnatural behaviour on quality control

charts. This approach provides the ability to utilize patterning information of the chart and to

track back the root causes of process deviation, thus facilitating process diagnosis and

maintenance.

In controlling process in statistical charts, there is some checklist that provides a set of rule

that examining a process to determine if it is under control. These checklists are as follow:

No points beyond the control limits

The same number of points exist between the upper and lower limits of the center line

The points above or below the center line are falling randomly

Only few of the center points are close to control limits

The first assumption for these rules is the distribution of the sample must be normal. The

distribution of sample size increases regardless of the original distribution from the central limit

theorem of statistics. The distribution of original data must be normal for a small size of

distribution data sets. The upper and lower control limits are calculated as three standard

deviations for the overall mean. Thus, the probability that the sample mean beyond the control

limits is expected to be small. This is the rule one for the probability. The same distribution

makes the upper and lower line in balance. This characteristic makes the normal distribution is

the median. From statistic, about 68% of the normal distribution falls within one standard

deviation of the meant, thus most of the points should be close to the center line. During the date

were collected, the meant and variance of the original data should be kept constant and

unchanged. This process is stable and sometimes unusual patterns arise in control charts, which

will be viewed as typical case here.

Several types of unusual patterns which can be seen from the control charts were

reviewed in the following section with an indication of the typical causes of such patterns.

One points outside control limits

A single point outside the control limits was shown in Figure 2.15. This single point is

produced by special case where the R-chart can provide the similar indication. These points are a

normal part of process and sometimes occur simply by chance. The reason that contributes to the

single point is an error in calculation of x or R for the sample. Whenever these kinds of error

occur, re-check to verifying data should be done to process further. Other possibilities are a

sudden power surge, a broken tool, measurement error, or an incomplete or limited operation in

process.

Figure: Single point outside control limits (Source: James, R. and Wiliam, M., 1999)

Sudden shift in the process average

An unusual number of consecutive points falling on one side of the center line were shown in

Figure 2.16. This is an unusual number of consecutive points falling on one side of the center

line and is usually an indication that the process average has suddenly shifted. This is the result

of an external influence that has affected the process and can be considered as special case. The

possible cause might be a new inspector, a new operator, a new machine setting or a change in

setup and method. The process becomes less uniform when it is shifted in R-chart. Typical

causes are the carelessness of the operator, poor maintenance or fixture in need of repair. If it is

shifted down in R-chart, then the uniformity of the process can be improved. This might be

improved through the better use of machine or materials and improved workmanship. Every

efforts should be mentioned for future maintenance or improvement. There are three rules for

early detection of process shifts, which are as follow:

If eight executive points fall below one side of center line, then the mean has shifted

The region between the center line is divided into three equal parts

If two points fall in the outer one-third region between the center line and one of the control

(or if four or five consecutive points fall within the outer two-third region), then the process

is out of control (refer Figure 2.17)

Figure 2.16 Shift in process average (Source: James, R. and Wiliam, M., 1999)

Figure: Examples of out-of-control indicators (Source: James, R. and Wiliam, M., 1999)

Cycle

Cycles are short and repeated in the chart with alternating high peaks and low valleys (refer

Figure 2.18). These patterns are the result that comes at a regular basic. This is caused by the

operation rotation or fatigue at the end of a shift and different gauges used by different

inspectors, seasonal effects. For example, temperature or humidity or differences between day

and night shift. In the R-chart, cycles are rotation of fixtures or gauges differences between

shifts, operator fatigue and maintenance schedules.

Figure: Cycles (Source: James, R. and Wiliam, M., 1999)

Trends

A trend is the result of gradually affects the quality of characteristics of the product and

causes the points on a control chart to gradually move up or down from the center line as shown

in Figure 2.19. A trend may occur due to experience of operator increases over time and

improvement of equipments over time. In the x-chart, trend happens when improving operator

skills, dirt or chip build up in fixtures, tool wear, change in humidity or temperature, aging of

equipment and others as well. on the other hands, trend might be due to gradual decline in

material quality, gradual loosening of fixture or a tool, operator fatigue or dulling of a tool as

shown in R-chart. A decreasing trend often is the result of improved operator skill or work

methods, better purchased materials, or improved or frequent maintenance.

Figure: Gradual trend (Source: James, R. and Wiliam, M., 1999)

Hugging the center line

Hugging the center line occurs when nearly all the points are fall close to the center line as

shown in Figure 2.20. In the control chart, it appears that the control charts are too wide. A

common cause is that the sample includes one item systematically taken from each of several

machines, spindles, operators, and so on. When though a large variation will occur in the part

taken as whole, the sample averages will not reflect this variation. In this case, the control chart

should be constructed for each machine, spindle or operators. Another causes for this pattern is

miscalculation of the control limits, wrong factor from the table or misplacing the decimal point

in computations.

Figure: Hugging the center line (Source: James, R. and Wiliam, M., 1999)

Hugging the control limits

When many points are near the control limits with very few in between, the pattern as shown

in Figure 2.26 will show up. It is also known as mixture and is a combination of two different

patterns on the same chart. This mixture can be spitted into two different patterns. This kind of

pattern result from different lots of material are used in one process or when parts are produced

by different machines but fed into a common inspection group.

Figure :Hugging the control limits (Source: James, R. and Wiliam, M., 1999)

Instability

This characteristic is caused by unnatural and erratic fluctuations on both sides of the chart

over a period of time. Points are often lying outside both the upper and lower control limits

without consistent pattern, as shown in Figure 2.22. Assignable causes are more difficult to

identify in his case when specific patterns are present. A frequent cause of instability is over

adjustment of a machine or the same reasons that cause the hugging the control limits.

Figure: Instability (Source: James, R. and Wiliam, M., 1999)

Apart from that, when a process is in control, there occurs a natural pattern of variation.

Natural pattern has:

About 34% of the plotted point in an imaginary band between 1s on both side CL.

About 13.5% in an imaginary band between 1s and 2s on both sides CL.

About 2.5% of the plotted point in an imaginary band between 2s and 3s on both side CL.

The percentage of cases under portions of the normal curve is shown in Figure2.23

Figure: Percentage of cases under portions of the normal curve

Western Electric rules can also be used to interpret the control chart. The control chart is

being split into three areas or zone above and below the centerline. The zones are sometimes

referred to as Zone A , Zone B and Zone C, and or sometime it is refer to one sigma zone, two

sigma zone, etc. Zone C is the region within 1 sigma of the centerline. Zone B is the region

between 1 and 2 sigma of the centerline. Zone A is the region from 2 to 3 sigma from the

centerline. There are 4 basic rules with respect to how the data is situated on a control chart to

indicate if it is not in statistical control (i.e. special causes of variation are present instead of

random sources). The rules are:

Rule 1: Any single data point falls outside the 3-sigma limit from the centerline (i.e., any

point falls outside Zone A, beyond either the upper or lower control limit);

Rule 2: Two out of three consecutive points fall beyond the 2-sigma limit (in zone A or

beyond), on the same side of the centerline;

Rule 3: Four out of five consecutive points fall beyond the 1-sigma limit (in Zone B or

beyond), on the same side of the centerline;

Rule 4: Nine consecutive points fall on the same side of the centerline (in Zone C or

beyond);

Western Electric Rules (Source: http://en.wikipedia.org/wiki/Western_Electric_rules,

2006)

(a i) Bottle Thread Outer Diameter Measurement (Mold Cavity No. 1) Date measurements taken: 2/2/2007 and 3/2/2007 Time: 1.00pm-7.00pm and 9.00am-5.00pm

Measured by: Jia Yunn Interval Time: 30 minutes

Dimension of Interest: Thread Outer Diameter Cavity 1 Specification: 35.2 +/- 0.2mm

Gage used: Vernier Caliper

Bottle Thread Outer Diameter Measurement (Mold Cavity No. 1) Subgroup

No.

Time

Taken

Measurement on each of five bottles per subgroup

X1 X2 X3 X4 X5

1 13.00 35.02 35.40 35.12 35.08 35.18 2 13.30 35.22 35.20 35.18 35.26 35.18 3 14.00 35.32 35.26 35.26 35.22 35.22 4 14.30 35.14 35.24 35.32 35.36 35.32 5 15.00 35.36 35.26 35.32 35.36 35.20 6 15.30 35.22 35.34 35.40 35.24 35.20 7 16.00 35.28 35.38 35.28 35.24 35.24 8 16.30 35.38 35.36 35.26 35.20 35.16 9 17.00 35.14 35.24 35.32 35.24 35.26 10 17.30 35.20 35.24 35.32 35.06 35.20 11 18.00 35.28 35.26 35.40 35.16 35.28 12 18.30 35.04 35.28 35.32 35.12 35.12 13 19.00 35.12 35.14 35.30 35.10 35.28 14 9.00 35.22 35.12 35.06 35.10 35.28 15 9.30 35.30 35.20 35.14 35.16 35.18 16 10.00 35.24 35.28 35.20 35.34 35.10 17 10.30 35.34 35.24 35.28 35.12 35.38 18 11.00 35.36 35.40 35.34 35.40 35.10 19 11.30 35.34 35.22 35.30 35.10 35.16 20 12.00 35.00 35.22 35.18 35.14 35.18 21 12.30 35.20 35.24 35.36 35.04 35.22 22 13.00 35.28 35.16 35.40 35.26 35.28 23 13.30 35.14 35.26 35.18 35.20 35.22 24 14.00 35.26 35.24 35.32 35.24 35.14 25 14.30 35.10 35.20 35.30 35.20 35.24 26 15.00 35.18 35.16 35.14 35.20 35.20 27 15.30 35.22 35.04 35.20 35.24 35.00 28 16.00 35.38 35.22 35.26 35.26 35.20 29 16.30 35.16 35.10 35.30 35.20 35.34 30 17.00 35.14 35.06 35.32 35.36 35.20

(b i) Bottle Thread Inner Diameter Measurement (Mold Cavity No. 1) Date measurements taken: 26/2/2007 and 27/2/2007 Time: 9.00am-5.00pm and 9.00am-4.00pm

Measured by: Jia Yunn Interval Time: 30 minutes

Dimension of Interest: Bottle Thread Inner Diameter 1 Specification: 28.4 +/- 0.2mm

Gage used: Vernier Caliper

Subgroup

No.

Time

Taken

Measurement on each of five bottles per subgroup

X1 X2 X3 X4 X5

1 9.00 28.32 28.25 28.32 28.38 28.26 2 9.30 28.28 28.42 28.40 28.24 28.53 3 10.00 28.40 28.42 28.38 28.36 28.32

4 10.30 28.22 28.45 28.20 28.42 28.24 5 11.00 28.40 28.32 28.40 28.22 28.30 6 11.30 28.40 28.22 28.30 28.28 28.44 7 12.00 28.40 28.44 28.26 28.20 28.38 8 12.30 28.38 28.34 28.32 28.40 28.40 9 13.00 28.38 28.28 28.20 28.36 28.24 10 13.30 28.40 28.22 28.34 28.36 28.32 11 14.00 28.30 28.50 28.56 28.40 28.20 12 14.30 28.24 28.40 28.42 28.32 28.26 13 15.00 28.28 28.26 28.32 28.24 28.26 14 15.30 28.42 28.30 28.32 28.20 28.24 15 16.00 28.40 28.44 28.50 28.58 28.44 16 16.30 28.38 28.36 28.40 28.44 28.24 17 9.00 28.34 28.46 28.32 28.32 28.50 18 9.30 28.26 28.38 28.40 28.36 28.34 19 10.00 28.28 28.30 28.34 28.40 28.48 20 10.30 28.38 28.56 28.44 28.26 28.42 21 11.00 28.36 28.42 28.26 28.60 28.24 22 11.30 28.20 28.38 28.56 28.50 28.30 23 12.00 28.53 28.40 28.40 28.42 28.28 24 12.30 28.24 28.36 28.20 28.28 28.38 25 13.00 28.38 28.44 28.40 28.36 28.38 26 13.30 28.44 28.58 28.50 28.44 28.40 27 14.00 28.40 28.60 28.26 28.42 28.36 28 14.30 28.32 28.36 28.38 28.42 28.40 29 15.00 28.48 28.40 28.34 28.30 28.28 30 15.30 28.32 28.36 28.34 28.28 28.40

Table: Histograms for the measured dimensions

No. Dimension Measured Histogram Generated

a.i Bottle thread outer diameter for

cavity 1

Mean=35.2272

Median=35.22

Mode= 35.2

Range=0.04

Standard deviation= 0.09316

0.40.05

1+3.322*log150i

0.41 =9

0.05h

a.ii Bottle thread outer diameter for

cavity 2

Mean=35.2516

Median= 35.25

Mode= 35.2

Range = 0.36

Standard Deviation=0.08021

0.36 = 0.04

1+3.322*log150i

0.36= 1 10

0.04h

35.000 35.100 35.200 35.300 35.400

Thread Outer Diameter, mm

0

10

20

30

Fre

qu

en

cy

of

occu

rren

ces

35.227

Bottle thread outer diameter for cavity No.1

35.100 35.200 35.300 35.400

Thread outer diameter, mm

0

10

20

30

Fre

qu

en

cy

of

occu

rren

ces

35.252

Bottle thread outer diameter for cavity No.2