## Control Charts for Variables CHAPTER-3

Control Charts for Variables CHAPTER-3 Variation    There is no two natural items in any category are the same. Variation may be quite large ...
Author: April Taylor
Control Charts for Variables

CHAPTER-3

Variation 

There is no two natural items in any category are the same. Variation may be quite large or very small. If variation very small, it may appear that items are identical, but precision instruments will show differences.

3 Categories of variation 

Within-piece variation 

Apiece-to-piece variation 

One portion of surface is rougher than another portion. Variation among pieces produced at the same time.

Time-to-time variation 

Service given early would be different from that given later in the day.

Source of variation 

Equipment 

Material 

Raw material quality

Environment 

Tool wear, machine vibration, …

Operator 

Operator performs- physical & emotional

Control Chart Viewpoint 

Variation due to Common or chance causes  Assignable causes 

Control chart may be used to discover “assignable causes”

Some Terms 

Run chart - without any upper/lower limits Specification/tolerance limits - not statistical Control limits - statistical

Control chart functions 

Control charts are powerful aids to understanding the performance of a process over time. Output

Input

PROCESS What’s causing variability?

Control charts identify variation 

Chance causes - “common cause” 

inherent to the process or random and not controllable if only common cause present, the process is considered stable or “in control”

Assignable causes - “special cause”  

variation due to outside influences if present, the process is “out of control”

Separate common and special causes of variation Determine whether a process is in a state of statistical control or out-of-control Estimate the process parameters (mean, variation) and assess the performance of a process or its capability

Control charts to monitor processes 

To monitor output, we use a control chart 

we check things like the mean, range, standard deviation

To monitor a process, we typically use two control charts  

mean (or some other central tendency measure) variation (typically using range or standard deviation)

Types of Data 

Variable data 

Product characteristic that can be measured 

Length, size, weight, height, time, velocity

Attribute data  Product

characteristic evaluated with a discrete

choice • Good/bad, yes/no

Control chart for variables Variables are the measurable characteristics of a product or service.  Measurement data is taken and arrayed on charts. 

Control charts for variables 

X-bar chart 

R chart 

In this chart, the sample ranges are plotted in order to control the variability of a variable.

S chart 

In this chart the sample means are plotted in order to control the mean value of a variable (e.g., size of piston rings, strength of materials, etc.).

In this chart, the sample standard deviations are plotted in order to control the variability of a variable.

S2 chart 

In this chart, the sample variances are plotted in order to control the variability of a variable.

X-bar and R charts 

The X- bar chart is developed from the average of each subgroup data. 

used to detect changes in the mean between subgroups.

The R- chart is developed from the ranges of each subgroup data 

used to detect changes in variation within subgroups

Control chart components 

Centerline 

shows where the process average is centered or the central tendency of the data

Upper control limit (UCL) and Lower control limit (LCL) 

describes the process spread

The Control Chart Method X bar Control Chart:

UCL = XDmean + A2 x Rmean LCL = XDmean - A2 x Rmean CL = XDmean R Control Chart: UCL = D4 x Rmean LCL = D3 x Rmean CL = Rmean

Capability Study: PCR = (UCL - LCL)/(6S); where S = Rmean /d2

Control Chart Examples Variations

UCL Nominal LCL

Sample number

How to develop a control chart?

Define the problem 

Use other quality tools to help determine the general problem that’s occurring and the process that’s suspected of causing it.

Select a quality characteristic to be measured 

Identify a characteristic to study - for example, part length or any other variable affecting performance.

Choose a subgroup size to be sampled 

Choose homogeneous subgroups 

Homogeneous subgroups are produced under the same conditions, by the same machine, the same operator, the same mold, at approximately the same time.

Try to maximize chance to detect differences between subgroups, while minimizing chance for difference with a group.

Collect the data 

Generally, collect 20-25 subgroups (100 total samples) before calculating the control limits. Each time a subgroup of sample size n is taken, an average is calculated for the subgroup and plotted on the control chart.

Determine trial centerline 

The centerline should be the population mean, Since it is unknown, we use X Double bar, or the grand average of the subgroup averages. m

Xi X

i 1

m

Determine trial control limits - Xbar

chart 

The normal curve displays the distribution of the sample averages. A control chart is a time-dependent pictorial representation of a normal curve. Processes that are considered under control will have 99.73% of their graphed averages fall within 6 .

UCL & LCL calculation

UCL

X 3

LCL X 3 standard deviation

Determining an alternative value for the standard deviation m

Ri R

i

1

m

UCL

X

A2 R

LCL

X

A2 R

Determine trial control limits - R chart 

The range chart shows the spread or dispersion of the individual samples within the subgroup. 

If the product shows a wide spread, then the individuals within the subgroup are not similar to each other. Equal averages can be deceiving.

Calculated similar to x-bar charts; 

Use D3 and D4 (appendix 2)

Example: Control Charts for Variable Data Sample 1 2 3 4 5 6 7 8 9 10

Slip Ring Diameter (cm) 1 2 3 4 5 5.02 5.01 4.94 4.99 4.96 5.01 5.03 5.07 4.95 4.96 4.99 5.00 4.93 4.92 4.99 5.03 4.91 5.01 4.98 4.89 4.95 4.92 5.03 5.05 5.01 4.97 5.06 5.06 4.96 5.03 5.05 5.01 5.10 4.96 4.99 5.09 5.10 5.00 4.99 5.08 5.14 5.10 4.99 5.08 5.09 5.01 4.98 5.08 5.07 4.99

X 4.98 5.00 4.97 4.96 4.99 5.01 5.02 5.05 5.08 5.03 50.09

R 0.08 0.12 0.08 0.14 0.13 0.10 0.14 0.11 0.15 0.10 1.15

Calculation From Table above:  Sigma X-bar = 50.09  Sigma R = 1.15  m = 10 Thus;  X-Double bar = 50.09/10 = 5.009 cm  R-bar = 1.15/10 = 0.115 cm Note: The control limits are only preliminary with 10 samples. It is desirable to have at least 25 samples.

Trial control limit 

UCLx-bar = X-D bar + A2 R-bar = 5.009 + (0.577)(0.115) = 5.075 cm LCLx-bar = X-D bar - A2 R-bar = 5.009 (0.577)(0.115) = 4.943 cm

UCLR = D4R-bar = (2.114)(0.115) = 0.243 cm  LCLR = D3R-bar = (0)(0.115) = 0 cm For A2, D3, D4: see Table B, Appendix n = 5 

3-Sigma Control Chart Factors Sample size n 2 3 4 5 6 7 8

X-chart A2 1.88 1.02 0.73 0.58 0.48 0.42 0.37

R-chart D3 0 0 0 0 0 0.08 0.14

D4 3.27 2.57 2.28 2.11 2.00 1.92 1.86

X-bar Chart 5.10 UCL

5.08

X bar

5.06 5.04 5.02 5.00

CL

4.98 4.96

LCL

4.94 0

1

2

3

4

5

6

Subgroup

7

8

9

10

11

R Chart UCL

0.25

Range

0.20 0.15

CL

0.10 0.05 LCL

0.00 0

1

2

3

4

5

6

Subgroup

7

8

9

10

11

Run Chart

6.70 6.65

Mean, X-bar

6.60 6.55 6.50 6.45 6.40 6.35 6.30 0

5

10

15

Subgroup number 0.35 0.3

Range, R

0.25 0.2 0.15 0.1 0.05 0 0

5

10

15

Subgroup number

20

25

20

25

Another Example of X-bar & R chart

Given Data (Table 5.2) Subgro up

X1

X2

X3

X4

X-bar

UCL-X-bar

X-Dbar

LCL-X-bar

R

UCL-R

R-bar

LCL-R

1

6.35

6.4

6.32

6.37

6.36

6.47

6.41

6.35

0.08

0.20

0.0876

0

2

6.46

6.37

6.36

6.41

6.4

6.47

6.41

6.35

0.1

0.20

0.0876

0

3

6.34

6.4

6.34

6.36

6.36

6.47

6.41

6.35

0.06

0.20

0.0876

0

4

6.69

6.64

6.68

6.59

6.65

6.47

6.41

6.35

0.1

0.20

0.0876

0

5

6.38

6.34

6.44

6.4

6.39

6.47

6.41

6.35

0.1

0.20

0.0876

0

6

6.42

6.41

6.43

6.34

6.4

6.47

6.41

6.35

0.09

0.20

0.0876

0

7

6.44

6.41

6.41

6.46

6.43

6.47

6.41

6.35

0.05

0.20

0.0876

0

8

6.33

6.41

6.38

6.36

6.37

6.47

6.41

6.35

0.08

0.20

0.0876

0

9

6.48

6.44

6.47

6.45

6.46

6.47

6.41

6.35

0.04

0.20

0.0876

0

10

6.47

6.43

6.36

6.42

6.42

6.47

6.41

6.35

0.11

0.20

0.0876

0

11

6.38

6.41

6.39

6.38

6.39

6.47

6.41

6.35

0.03

0.20

0.0876

0

12

6.37

6.37

6.41

6.37

6.38

6.47

6.41

6.35

0.04

0.20

0.0876

0

13

6.4

6.38

6.47

6.35

6.4

6.47

6.41

6.35

0.12

0.20

0.0876

0

14

6.38

6.39

6.45

6.42

6.41

6.47

6.41

6.35

0.07

0.20

0.0876

0

15

6.5

6.42

6.43

6.45

6.45

6.47

6.41

6.35

0.08

0.20

0.0876

0

16

6.33

6.35

6.29

6.39

6.34

6.47

6.41

6.35

0.1

0.20

0.0876

0

17

6.41

6.4

6.29

6.34

6.36

6.47

6.41

6.35

0.12

0.20

0.0876

0

18

6.38

6.44

6.28

6.58

6.42

6.47

6.41

6.35

0.3

0.20

0.0876

0

19

6.35

6.41

6.37

6.38

6.38

6.47

6.41

6.35

0.06

0.20

0.0876

0

20

6.56

6.55

6.45

6.48

6.51

6.47

6.41

6.35

0.11

0.20

0.0876

0

21

6.38

6.4

6.45

6.37

6.4

6.47

6.41

6.35

0.08

0.20

0.0876

0

22

6.39

6.42

6.35

6.4

6.39

6.47

6.41

6.35

0.07

0.20

0.0876

0

23

6.42

6.39

6.39

6.36

6.39

6.47

6.41

6.35

0.06

0.20

0.0876

0

24

6.43

6.36

6.35

6.38

6.38

6.47

6.41

6.35

0.08

0.20

0.0876

0

25

6.39

6.38

6.43

6.44

6.41

6.47

6.41

6.35

0.06

0.20

0.0876

0

Calculation From Table 5.2:  Sigma X-bar = 160.25  Sigma R = 2.19  m = 25 Thus;  X-double bar = 160.25/29 = 6.41 mm  R-bar = 2.19/25 = 0.0876 mm

Trial control limit 

UCLx-bar = X-double bar + A2R-bar = 6.41 + (0.729)(0.0876) = 6.47 mm LCLx-bar = X-double bar - A2R-bar = 6.41 – (0.729)(0.0876) = 6.35 mm UCLR = D4R-bar = (2.282)(0.0876) = 0.20 mm LCLR = D3R-bar = (0)(0.0876) = 0 mm

For A2, D3, D4: see Table B Appendix, n = 4.

X-bar Chart

R Chart

Revised CL & Control Limits 

Calculation based on discarding subgroup 4 & 20 (Xbar chart) and subgroup 18 for R chart:

X new

X X d = (160.25 - 6.65 - 6.51)/(25-2) m md = 6.40 mm

Rnew

R Rd = (2.19 - 0.30)/25 - 1 = 0.079 = 0.08 mm m md

New Control Limits New value:

X o X new, 

Ro Rnew,

o

RO d2

Using standard value, CL & 3 control limit obtained using formula:

UCLX

Xo

UCLR D2

A o

,

o

,

LCLX

Xo

LCLR D1

A o

o

From Table B:  A = 1.500 for a subgroup size of 4,  d2 = 2.059, D1 = 0, and D2 = 4.698 Calculation results: Xo

X new 6.40mm

Ro Rnew 0.079,

o

Ro d2

0.079 2.059

0.038 mm

UCLX

Xo

A

o

6.40 (1.500)(0.038)

6.46 mm

LCL X

Xo

A

o

6.40 (1.500)(0.038)

6.34 mm

UCLR

D2

LCLR

D1

o

o

(4.698)(0.038)

(0)(0.038)

0.18 mm

0 mm

Trial Control Limits & Revised Control Limit 6.65

Revised control limits

Mean, X-bar

6.60 6.55

UCL = 6.46

6.50 6.45

CL = 6.40

6.40 6.35

LCL = 6.34

6.30 0

2

4

6

8

Subgroup

UCL = 0.18

Range, R

0.20 0.15

CL = 0.08

0.10 0.05 0.00 0

2

4 Subgroup

6

8

LCL = 0

Revise the charts 

In certain cases, control limits are revised because:  out-of-control points were included in the calculation of the control limits.  the process is in-control but the within subgroup variation significantly improves.

Revising the charts    

Interpret the original charts Isolate the causes Take corrective action Revise the chart 

Only remove points for which you can determine an assignable cause

Process in Control 

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 1 on both side CL. About 13.5% in an imaginary band between 1 and 2 on both side CL. About 2.5% of the plotted point in an imaginary band between 2 and 3 on both side CL.

The Normal Distribution = Standard deviation -3 LSL

-3

CL

+3

Mean -2 -1 +1 68.26% 95.44% USL 99.74%

+2

+3

    

34.13% of data lie between and 1 above the mean ( ). 34.13% between and 1 below the mean. Approximately two-thirds (68.28 %) within 1 of the mean. 13.59% of the data lie between one and two standard deviations Finally, almost all of the data (99.74%) are within 3 of the mean.

Normal Distribution Review 

Define the 3-sigma limits for sample means as follows:

Upper Limit Lower Limit 

3 n 3 n

3(0.05) 5.01 5 3(0.05) 5.01 5

5.077 4.943

What is the probability that the sample means will lie outside 3-sigma limits? Note that the 3-sigma limits for sample means are different from natural tolerances which are at

3

Common Causes

Process Out of Control 

The term out of control is a change in the process due to an assignable cause. When a point (subgroup value) falls outside its control limits, the process is out of control.

Assignable Causes Average

Grams

(a) Mean

Assignable Causes Average

Grams

Assignable Causes Average

Grams

(c) Shape

Control Charts

Assignable causes likely UCL

Nominal

LCL

1

2 Samples

3

Control Chart Examples Variations

UCL Nominal LCL

Sample number

Control Limits and Errors Type I error: Probability of searching for a cause when none exists

(a) Three-sigma limits

UCL

Process average

LCL

Control Limits and Errors Type I error: Probability of searching for a cause when none exists

(b) Two-sigma limits

UCL

Process average

LCL

Control Limits and Errors (a) Three-sigma limits

Type II error: Probability of concluding that nothing has changed

UCL

Shift in process average Process average

LCL

Control Limits and Errors (b) Two-sigma limits

Type II error: Probability of concluding that nothing has changed

UCL

Shift in process average Process average

LCL

Achieve the purpose 

Our goal is to decrease the variation inherent in a process over time. As we improve the process, the spread of the data will continue to decrease. Quality improves!!

Improvement

Examine the process 

A process is considered to be stable and in a state of control, or under control, when the performance of the process falls within the statistically calculated control limits and exhibits only chance, or common causes.

Consequences of misinterpreting the process 

 

Blaming people for problems that they cannot control Spending time and money looking for problems that do not exist Spending time and money on unnecessary process adjustments Taking action where no action is warranted Asking for worker-related improvements when process improvements are needed first

Process variation 

When a system is subject to only chance causes of variation, 99.74% of the measurements will fall within 6 standard deviations  If 1000 subgroups are measured, 997 will fall within the six sigma limits. -3

-2

Mean -1 +1 68.26% 95.44% 99.74%

+2

+3

Chart zones 

Based on our knowledge of the normal curve, a control chart exhibits a state of control when: ♥ Two thirds of all points are near the center value. ♥ The points appear to float back and forth across the centerline. ♥ The points are balanced on both sides of the centerline. ♥ No points beyond the control limits. ♥ No patterns or trends.