control charts for variables ebb 341 quality control

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Control Charts for Variables EBB 341 Quality Control

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Page 1: Control Charts for Variables EBB 341 Quality Control

Control Charts for Variables

EBB 341 Quality Control

Page 2: Control Charts for Variables EBB 341 Quality Control

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.

Page 3: Control Charts for Variables EBB 341 Quality Control

3 Categories of variation

Within-piece variation One portion of surface is rougher than another

portion. Apiece-to-piece variation

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.

Page 4: Control Charts for Variables EBB 341 Quality Control

Source of variation

Equipment Tool wear, machine vibration, …

Material Raw material quality

Environment Temperature, pressure, humadity

Operator Operator performs- physical & emotional

Page 5: Control Charts for Variables EBB 341 Quality Control

Control Chart Viewpoint

Variation due to Common or chance causes Assignable causes

Control chart may be used to discover “assignable causes”

Page 6: Control Charts for Variables EBB 341 Quality Control

Some Terms

Run chart - without any upper/lower limits

Specification/tolerance limits - not statistical

Control limits - statistical

Page 7: Control Charts for Variables EBB 341 Quality Control

Control chart functions

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

PROCESSPROCESS

Input Output

What’s causing variability?

Page 8: Control Charts for Variables EBB 341 Quality Control

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”

Page 9: Control Charts for Variables EBB 341 Quality Control

Control charts help us learn more about processes

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

Page 10: Control Charts for Variables EBB 341 Quality Control

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)

Page 11: Control Charts for Variables EBB 341 Quality Control

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

Page 12: Control Charts for Variables EBB 341 Quality Control

Control chart for variables

Variables are the measurablemeasurable characteristics of a product or service.

Measurement data is taken and arrayed on charts.

Page 13: Control Charts for Variables EBB 341 Quality Control

Control charts for variables X-bar 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.).

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 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.

Page 14: Control Charts for Variables EBB 341 Quality Control

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

Page 15: Control Charts for Variables EBB 341 Quality Control

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

Page 16: Control Charts for Variables EBB 341 Quality Control

The Control Chart Method

R Control Chart: UCL = D4 x Rmean LCL = D3 x Rmean CL = Rmean Capability Study:PCR = (USL - LSL)/(6s); where s = Rmean /d2

X bar Control Chart: UCL = XDmean + A2 x Rmean LCL = XDmean - A2 x Rmean CL = XDmean 

Page 17: Control Charts for Variables EBB 341 Quality Control

Control Chart ExamplesControl Chart Examples

Nominal

UCL

LCL

Sample number

Var

iati

on

s

Page 18: Control Charts for Variables EBB 341 Quality Control

How to develop a control chart?

Page 19: Control Charts for Variables EBB 341 Quality Control

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.

Page 20: Control Charts for Variables EBB 341 Quality Control

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.

Page 21: Control Charts for Variables EBB 341 Quality Control

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.

Page 22: Control Charts for Variables EBB 341 Quality Control

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

m

i

i 1

XX

Page 23: Control Charts for Variables EBB 341 Quality Control

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.

Page 24: Control Charts for Variables EBB 341 Quality Control

UCL & LCL calculation

deviation standard

3XLCL

3XUCL

Page 25: Control Charts for Variables EBB 341 Quality Control

Determining an alternative value for the standard deviation

m

m

ii

1

RR

RAXUCL 2

RAXLCL 2

Page 26: Control Charts for Variables EBB 341 Quality Control

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)

Page 27: Control Charts for Variables EBB 341 Quality Control

Example: Control Charts for Variable Data Slip Ring Diameter (cm)

Sample 1 2 3 4 5 X R

1 5.02 5.01 4.94 4.99 4.96 4.98 0.08

2 5.01 5.03 5.07 4.95 4.96 5.00 0.12

3 4.99 5.00 4.93 4.92 4.99 4.97 0.08

4 5.03 4.91 5.01 4.98 4.89 4.96 0.14

5 4.95 4.92 5.03 5.05 5.01 4.99 0.13

6 4.97 5.06 5.06 4.96 5.03 5.01 0.10

7 5.05 5.01 5.10 4.96 4.99 5.02 0.14

8 5.09 5.10 5.00 4.99 5.08 5.05 0.11

9 5.14 5.10 4.99 5.08 5.09 5.08 0.15

10 5.01 4.98 5.08 5.07 4.99 5.03 0.10

50.09 1.15

Page 28: Control Charts for Variables EBB 341 Quality Control

CalculationFrom 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.

Page 29: Control Charts for Variables EBB 341 Quality Control

Trial control limitTrial 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, Appendixn = 5

Page 30: Control Charts for Variables EBB 341 Quality Control

3-Sigma Control Chart Factors

Sample size X-chart R-chart

n A2 D3 D4

2 1.88 0 3.27

3 1.02 0 2.57

4 0.73 0 2.28

5 0.58 0 2.11

6 0.48 0 2.00

7 0.42 0.08 1.92

8 0.37 0.14 1.86

Page 31: Control Charts for Variables EBB 341 Quality Control

X-bar Chart

4.94

4.96

4.98

5.00

5.02

5.04

5.06

5.08

5.10

0 1 2 3 4 5 6 7 8 9 10 11

Subgroup

X b

ar

LCL

CL

UCL

Page 32: Control Charts for Variables EBB 341 Quality Control

R Chart

0.00

0.05

0.10

0.15

0.20

0.25

0 1 2 3 4 5 6 7 8 9 10 11

Subgroup

Ran

ge

LCL

CL

UCL

Page 33: Control Charts for Variables EBB 341 Quality Control

Run Chart

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20 25

Subgroup number

Ra

ng

e, R

6.30

6.35

6.40

6.45

6.50

6.55

6.60

6.65

6.70

0 5 10 15 20 25

Subgroup numberM

ean

, X

-bar

Page 34: Control Charts for Variables EBB 341 Quality Control

Another Example Another Example of X-bar & R of X-bar & R

chart chart

Page 35: Control Charts for Variables EBB 341 Quality Control

Subgroup 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

Given Data (Table 5.2)

Page 36: Control Charts for Variables EBB 341 Quality Control

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

Page 37: Control Charts for Variables EBB 341 Quality Control

Trial control limitTrial 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.

Page 38: Control Charts for Variables EBB 341 Quality Control

X-bar ChartX-bar Chart

Page 39: Control Charts for Variables EBB 341 Quality Control

R ChartR Chart

Page 40: Control Charts for Variables EBB 341 Quality Control

Revised CL & Control Revised CL & Control LimitsLimits

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

= (160.25 - 6.65 - 6.51)/(25-2) = 6.40 mm

= (2.19 - 0.30)/25 - 1 = 0.079 = 0.08 mmd

dnew mm

RRR

d

dnew mm

XXX

Page 41: Control Charts for Variables EBB 341 Quality Control

New Control LimitsNew Control Limits

New value:

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

2

,,d

RRRXX O

onewonewo

oRoR

ooXooX

DLCLDUCL

AXLCLAXUCL

12 ,

,

Page 42: Control Charts for Variables EBB 341 Quality Control

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

Calculation results:

mmXX newo 40.6 mmd

RRR o

onewo 038.0059.2

079.0,079.0

2

mmAXUCL ooX46.6)038.0)(500.1(40.6

mmAXLCL ooX34.6)038.0)(500.1(40.6

mmDUCL oR 18.0)038.0)(698.4(2 mmDLCL oR 0)038.0)(0(1

Page 43: Control Charts for Variables EBB 341 Quality Control

Trial Control Limits & Revised Control Limit

6.30

6.35

6.40

6.45

6.50

6.55

6.60

6.65

0 2 4 6 8

Subgroup

Mea

n, X

-bar

0.00

0.05

0.10

0.15

0.20

0 2 4 6 8

Subgroup

Ran

ge,

R

UCL = 6.46

CL = 6.40

LCL = 6.34

LCL = 0

CL = 0.08

UCL = 0.18

Revised control limits

Page 44: Control Charts for Variables EBB 341 Quality Control

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.

Page 45: Control Charts for Variables EBB 341 Quality Control

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

Page 46: Control Charts for Variables EBB 341 Quality Control

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.

Page 47: Control Charts for Variables EBB 341 Quality Control

The NormalThe NormalDistributionDistribution

-3 -2 -1 +1 +2 +3Mean

68.26%95.44%99.74%

= Standard deviation

LSL USL

-3 +3CL

Page 48: Control Charts for Variables EBB 341 Quality Control

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.

Page 49: Control Charts for Variables EBB 341 Quality Control

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

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

Normal Distribution Review

94345

0503015

3

07755

0503015

3

.).(

. Limit Lower

.).(

. Limit Upper

n

n

3

Page 50: Control Charts for Variables EBB 341 Quality Control

Common CausesCommon Causes

Page 51: Control Charts for Variables EBB 341 Quality Control

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.

Page 52: Control Charts for Variables EBB 341 Quality Control

Assignable CausesAssignable Causes

(a) Mean

Grams

Average

Page 53: Control Charts for Variables EBB 341 Quality Control

Assignable CausesAssignable Causes

(b) Spread

Grams

Average

Page 54: Control Charts for Variables EBB 341 Quality Control

Assignable CausesAssignable Causes

(c) Shape

Grams

Average

Page 55: Control Charts for Variables EBB 341 Quality Control

Control ChartsControl Charts

UCL

Nominal

LCL

Assignable causes likely

1 2 3Samples

Page 56: Control Charts for Variables EBB 341 Quality Control

Control Chart ExamplesControl Chart Examples

Nominal

UCL

LCL

Sample number

Var

iati

on

s

Page 57: Control Charts for Variables EBB 341 Quality Control

Control Limits and Control Limits and ErrorsErrors

LCL

Processaverage

UCL

(a) Three-sigma limitsType I error:Probability of searching for a cause when none exists

Page 58: Control Charts for Variables EBB 341 Quality Control

Control Limits and Control Limits and ErrorsErrors

Type I error:Probability of searching for a cause when none exists

UCL

LCL

Processaverage

(b) Two-sigma limits

Page 59: Control Charts for Variables EBB 341 Quality Control

Type II error:Probability of concludingthat nothing has changed

Control Limits and Control Limits and ErrorsErrors

UCL

Shift in process average

LCL

Processaverage

(a) Three-sigma limits

Page 60: Control Charts for Variables EBB 341 Quality Control

Type II error:Probability of concludingthat nothing has changed

Control Limits and Control Limits and ErrorsErrors

UCL

Shift in process average

LCL

Processaverage

(b) Two-sigma limits

Page 61: Control Charts for Variables EBB 341 Quality Control

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!!

Page 62: Control Charts for Variables EBB 341 Quality Control

Improvement

Page 63: Control Charts for Variables EBB 341 Quality Control

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.

Page 64: Control Charts for Variables EBB 341 Quality Control

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

Page 65: Control Charts for Variables EBB 341 Quality Control

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 -1 +1 +2 +3Mean

68.26%95.44%99.74%

Page 66: Control Charts for Variables EBB 341 Quality Control

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.