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Control Charts for Variables

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Control charts help us learn more about processes. Separate common and special causes of variation ... Length, size, weight, height, time, velocity. Attribute data ... – PowerPoint PPT presentation

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Title: Control Charts for Variables


1
Control Charts for Variables
  • EBB 341 Quality Control

2
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
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.

4
Source of variation
  • Equipment
  • Tool wear, machine vibration,
  • Material
  • Raw material quality
  • Environment
  • Temperature, pressure, humadity
  • Operator
  • Operator performs- physical emotional

5
Control Chart Viewpoint
  • Variation due to
  • Common or chance causes
  • Assignable causes
  • Control chart may be used to discover assignable
    causes

6
Some Terms
  • Run chart - without any upper/lower limits
  • Specification/tolerance limits - not statistical
  • Control limits - statistical

7
Control chart functions
  • Control charts are powerful aids to understanding
    the performance of a process over time.

Output
PROCESS
Input
Whats causing variability?
8
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

9
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

10
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)

11
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

12
Control chart for variables
  • Variables are the measurable characteristics of a
    product or service.
  • Measurement data is taken and arrayed on charts.

13
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.

14
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

15
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

16
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 (USL - LSL)/(6s) where s Rmean /d2

17
Control Chart Examples
UCL
Nominal
Variations
LCL
Sample number
18
How to develop a control chart?
19
Define the problem
  • Use other quality tools to help determine the
    general problem thats occurring and the process
    thats 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.

20
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.

21
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.

22
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.

23
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?.

24
UCL LCL calculation
25
Determining an alternative value for the standard
deviation
26
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)

27
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

28
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.
29
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
30
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
31
X-bar Chart
32
R Chart
33
Run Chart
34
  • Another Example of X-bar R chart

35
Given Data (Table 5.2)
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
36
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

37
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.

38
X-bar Chart
39
R Chart
40
Revised CL Control Limits
  • 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 mm

41
New Control Limits
  • New value
  • Using standard value, CL 3? control limit
    obtained using formula

42
  • From Table B
  • A 1.500 for a subgroup size of 4,
  • d2 2.059, D1 0, and D2 4.698
  • Calculation results

43
Trial Control Limits Revised Control Limit
Revised control limits
UCL 6.46
CL 6.40
LCL 6.34
UCL 0.18
CL 0.08
LCL 0
44
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.

45
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

46
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 1s on both side CL.
  • About 13.5 in an imaginary band between 1s and
    2s on both side CL.
  • About 2.5 of the plotted point in an imaginary
    band between 2s and 3s on both side CL.

47
The Normal Distribution
? Standard deviation
48
  • 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.

49
Normal Distribution Review
  • 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

50
Common Causes
51
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.

52
Assignable Causes
(a) Mean
Average
Grams
53
Assignable Causes
Average
(b) Spread
Grams
54
Assignable Causes
Average
(c) Shape
Grams
55
Control Charts
Assignable causes likely
UCL
Nominal
LCL
1 2
3 Samples
56
Control Chart Examples
UCL
Nominal
Variations
LCL
Sample number
57
Control Limits and Errors
Type I error Probability of searching for a
cause when none exists
(a) Three-sigma limits
UCL
Process average
LCL
58
Control Limits and Errors
Type I error Probability of searching for a
cause when none exists
(b) Two-sigma limits
UCL
Process average
LCL
59
Control Limits and Errors
Type II error Probability of concluding that
nothing has changed
(a) Three-sigma limits
UCL
Shift in process average
Process average
LCL
60
Control Limits and Errors
Type II error Probability of concluding that
nothing has changed
(b) Two-sigma limits
UCL
Shift in process average
Process average
LCL
61
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!!

62
Improvement
63
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.

64
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

65
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.

66
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.
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