chapter 3 - control chart for variables
TRANSCRIPT
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.
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.
Source of variation
Equipment Tool wear, machine vibration, …
Material Raw material quality
Environment Temperature, pressure, humadity
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.
PROCESSPROCESS
Input Output
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”
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
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 measurablemeasurable characteristics of a product or service.
Measurement data is taken and arrayed on charts.
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.
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
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
Control Chart ExamplesControl Chart Examples
Nominal
UCL
LCL
Sample number
Var
iati
on
s
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
m
i
i 1
XX
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
deviation standard
3XLCL
3XUCL
Determining an alternative value for the standard deviation
m
m
ii
1
RR
RAXUCL 2
RAXLCL 2
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 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
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.
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
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
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
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
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
Another Example Another Example of X-bar & R of X-bar & R
chart chart
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)
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 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.
X-bar ChartX-bar Chart
R ChartR Chart
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
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 ,
,
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
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
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 NormalThe NormalDistributionDistribution
-3 -2 -1 +1 +2 +3Mean
68.26%95.44%99.74%
= Standard deviation
LSL USL
-3 +3CL
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.
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
Common CausesCommon 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 CausesAssignable Causes
(a) Mean
Grams
Average
Assignable CausesAssignable Causes
(b) Spread
Grams
Average
Assignable CausesAssignable Causes
(c) Shape
Grams
Average
Control ChartsControl Charts
UCL
Nominal
LCL
Assignable causes likely
1 2 3Samples
Control Chart ExamplesControl Chart Examples
Nominal
UCL
LCL
Sample number
Var
iati
on
s
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
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
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
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
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 -1 +1 +2 +3Mean
68.26%95.44%99.74%
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.