ken black qa ch11
TRANSCRIPT
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Business Statistics, 5th ed.
by Ken Black
Chapter 11
Analysis ofVariance
& Design ofExperiments
Discrete Distributions
PowerPoint presentations prepared by Lloyd Jaisingh,Morehead State University
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Learning Objectives
Understand the differences between variousexperimental designs and when to use them.
Compute and interpret the results of a one-wayANOVA.
Compute and interpret the results of a randomblock design. Compute and interpret the results of a two-way
ANOVA. Understand and interpret interaction.
Know when and how to use multiple comparisontechniques.
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Introduction to Design
of Experiments
Experimental Design
- a plan and a structure to test hypotheses inwhich the researcher controls or manipulatesone or more variables.
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Introduction to Design of Experiments
Independent Variable Treatment variable is one that the experimenter
controls or modifies in the experiment.
Classification variable is a characteristic of theexperimental subjects that was present prior to theexperiment, and is not a result of theexperimenters manipulations or control.
Levels or Classifications are the subcategories of
the independent variable used by the researcher inthe experimental design.
Independent variables are also referred to asfactors.
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Introduction to Design
of Experiments
Dependent Variable
- the response to the different levels of the
independent variables. Analysis of Variance (ANOVA)a group
of statistical techniques used to analyzeexperimental designs.
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Three Types
of Experimental Designs
Completely Randomized Designsubjects areassigned randomly to treatments; single
independent variable. Randomized Block Designincludes a blocking
variable; single independent variable.
Factorial Experimentstwo or more independent
variables are explored at the same time; everylevel of each factor are studied under every levelof all other factors.
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Completely Randomized Design
Machine Operator
Valve Opening
Measurements
1
.
.
.
2
.
.
.
4
.
.
.
.
.
.
3
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Valve Openings by Operator
1 2 3 4
6.33 6.26 6.44 6.29
6.26 6.36 6.38 6.23
6.31 6.23 6.58 6.19
6.29 6.27 6.54 6.21
6.4 6.19 6.56
6.5 6.34
6.19 6.58
6.22
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Analysis of Variance: Assumptions
Observations are drawn from normallydistributed populations.
Observations represent random samples
from the populations. Variances of the populations are equal.
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One-Way ANOVA: Procedural
OverviewH
H
ok
a
:
:
1 2 3
At least one of the means is different from the others
FMSC
MSE
If F > , reject H .
If F , do not reject H .
co
co
F
F
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One-Way ANOVA:
Sums of Squares Definitions
valueindividual
levelorgrouptreatmentaofmean=
meangrand=X
leveltmentgiven treaainnsobservatioofnumber
levelstreatmentofnumber=
leveltreatmenta=
leveltreatmentaofmemberparticular:
nn
ij
SSE+SSC=SST
squaresofsumbetween+squaresofsumerror=squaresofsumtotal
X
X
n
X
ij
j
j
1 1
2
1
2
1=i 1j=
2 jj
C
j
iwhere
jijjj i
C
j
C
j
C
XXX
XnX
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Partitioning Total Sum
of Squares of Variation
SST(Total Sum of Squares)
SSC(Treatment Sum of Squares)
SSE(Error Sum of Squares)
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One-Way ANOVA:
Computational Formulas
MSE
MSCF
SSEMSE
SSCMSC
Nn
ijSST
CNn
jijSSE
Cj
SSC
df
df
dfXX
dfXX
dfXXn
E
C
Tj
C
i
Ei
C
j
C
C
jj
j
j
1
1
1 1
2
1 1
2
1
2
where
X
: i = a particular member of a treatment level
j = a treatment level
C = number of treatment levels
= number of observations in a given treatment level
X = grand mean
column mean
= individual value
j
j
ij
n
X
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One-Way ANOVA:
Preliminary Calculations
1 2 3 4
6.33 6.26 6.44 6.29
6.26 6.36 6.38 6.23
6.31 6.23 6.58 6.19
6.29 6.27 6.54 6.21
6.4 6.19 6.56
6.5 6.34
6.19 6.58
6.22
Tj T1 = 31.59 T2 = 50.22 T3 = 45.42 T4 = 24.92 T = 152.15
nj n1 = 5 n2 = 8 n3 = 7 n4 = 4 N = 24
Mean 6.318000 6.277500 6.488571 6.230000 6.339583
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15492.0
)230.619.6()230.622.6()2775.636.6()2775.626.6()318.64.6(
)318.629.6()318.631.6()318.626.6()318.633.6(
23658.0
)339583.623.6()339583.6488571.6(
)339583.62775.6()339583.6318.6(
22
222
2222
1 1
2
22
22
1
2
47
85[
n
jijSSE
jSSC
j
i
C
j
C
jj
XX
XXn
One-Way ANOVA:
Sum of Squares Calculations
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39150.0
)339583.619.6(
)339583.622.6()339583.631.6(
)339583.626.6()339583.633.6(
2
22
22
1 1
2
n
ijSST
j
i
C
j
XX
One-Way ANOVA:
Sum of Squares Calculations
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One-Way
ANOVA: Mean
Square
and F Calculations
18.10007746.078860.
007746.20
15492.
078860.3
23658.
231241
20424
3141
MSEMSCF
SSEMSE
SSC
MSC
N
CN
C
df
df
df
df
df
E
C
T
E
C
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Analysis of Variance
for Valve Openings
Source of Variance df SS MS F
Between 3 0.23658 0.078860 10.18
Error 20 0.15492 0.007746
Total 23 0.39150
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F 20,3,05.df1
df2
A Portion of the F Table for = 0.05
1 2 3 4 5 6 7 8 9
1 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54
18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.4619 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42
20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39
21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37
df2
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One-Way ANOVA:
Procedural Summary
.Hrejectdo,10.3F
.Hreject,10.3>F
oc
oc
F
F
If
If
Rejection Region
Critical Value10.3
11,9,05.F
Non rejection
Region
20
3
2
1
othersthefromdifferentis
meanstheofoneleastAt:H
:H
a
4321o
.Hreject,10.3>10.18=FSince ocF
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Excel Output
for the Valve Opening Example
Anova: Single Factor
SUMMARY
Groups Count Sum Average Variance
Operator 1 5 31.59 6.318 0.00277
Operator 2 8 50.22 6.2775 0.0110786
Operator 3 7 45.42 6.488571429 0.0101143
Operator 4 4 24.92 6.23 0.0018667
ANOVA
Source of Variation SS df MS F P-value F crit
Between Groups 0.236580119 3 0.07886004 10.181025 0.00028 3.09839
Within Groups 0.154915714 20 0.007745786
Total 0.391495833 23
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MINITAB Output
for the Valve Opening Example
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Multiple Comparison Tests
An analysis of variance (ANOVA) test is anoverall test of differences among groups.
Multiple Comparison techniques are used toidentify which pairs of means are
significantly differentgiven that theANOVA test reveals overall significance. Tukeys honestly significant difference
(HSD) test requires equal sample sizes
Tukey-Kramer Procedure is used whensample sizes are unequal.
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Tukeys Honestly Significant
Difference (HSD) Test
HSDMSE
n
,C,N-C
,C,N-C
q
q
where: MSE = mean square error
n = sample size
= critical value of the studentized range distribution from Table A.10
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q Values for = .01
Degrees ofFreedom
1
2
3
4
.
11
12
2 3 4 5
90 135 164 186
14 19 22.3 24.7
8.26 10.6 12.2 13.3
6.51 8.12 9.17 9.96
4.39 5.14 5.62 5.97
4.32 5.04 5.50 5.84
.
...
Number of Populations
. , ,.
01 3 12504q
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Tukeys HSD Test
for the Employee Age Data
HSDMSE
nC N Cq
X
X
X
, ,.
..
. . .
. . .
. . .
504163
52 88
28 2 32 0 38
28 2 24 8 34
32 0 24 8 7 2
2
3
3
1
1
2
X
X
X
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Tukeys HSD Test for the Employee
Age Data using MINITAB
Intervals
do not
contain 0,
so significant
differences
between the
means.
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Tukey-Kramer Procedure:
The Case of Unequal Sample Sizes
HSDMSE
r sn n
,C,N-C
r
th
s
th
,C,N-C
q
n r
n s
q
where: MSE = mean square error
= sample size for sample
= sample size for sample
= critical value of the studentized range distribution from Table A.10
2
1 1( )
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Freighter Example: Means and
Sample Sizes for the Four Operators
Operator Sample Size Mean
1 5 6.31802 8 6.2775
3 7 6.4886
4 4 6.2300
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Tukey-Kramer Results
for the Four Operators
Pair
Critical
Difference
|Actual
Differences|
1 and 2 .1405 .0405
1 and 3 .1443 .1706*
1 and 4 .1653 .0880
2 and 3 .1275 .2111*
2 and 4 .1509 .0475
3 and 4 .1545 .2586*
*denotes significant at .05
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Partitioning the Total Sum of Squares
in the Randomized Block Design
SST(Total Sum of Squares)
SSC(Treatment
Sum of Squares)
SSE(Error Sum of Squares)
SSR(Sum of Squares
Blocks)
SSE(Sum of Squares
Error)
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A Randomized Block Design
Individual
observations
.
.
.
.
.
.
.
.
.
.
.
.
Single Independent Variable
Blocking
Variable
.
.
.
.
.
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Randomized Block Design Treatment
Effects: Procedural Overview
othersthefromdifferentismeanstheofoneleastAt:H
:H
a
321o
k
FMSC
MSE
If F > , reject H .If F , do not reject H .
c o
co
FF
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Randomized Block Design:
Computational Formulas
SSC n j C
SSR C i
n
SSE ij i iC n N n C
SST ij N
MSCSSC
C
MSRSSR
n
MSESSE
N n C
MSC
MSE
MSR
MSE
X X df
X X df
X X X X df
X X df
F
F
j
C
C
i
n
R
i
n
j
n
E
i
n
j
n
E
treatments
blocks
2
1
2
1
2
11
2
11
1
1
1 1 1
1
1
1
1
( )
( )
( )
( )where: i = block group (row)
j = a treatment level (column)
C = number of treatment levels (columns)
n = number of observations in each treatment level (number of blocks - rows)
individual observation
treatment (column) mean
block (row) mean
X = grand mean
N = total number of observations
ij
j
i
X
X
X
SSC sum of squares columns (treatment)
SSR = sum of squares rows (blocking)
SSE = sum of squares error
SST = sum of squares total
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Randomized Block Design:Tread-Wear Example
Supplier
1
2
3
4
Slow Medium FastBlockMeans( )
3.7 4.5 3.1 3.77
3.4 3.9 2.8 3.37
3.5 4.1 3.0 3.53
3.2 3.5 2.6 3.10
5Treatment
Means( )
3.9 4.8 3.4 4.03
3.54 4.16 2.98 3.56
Speed
jX
iX
X
C = 3
n = 5
N = 15
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SSC n j
SSR C i
X X
X X
j
C
i
n
2
1
2 2 2
2
1
2 2 2 2 2
5
3
54 356 16 356 98 3563484
77 356 37 356 53 356 10 356 03 356
1549
( )
(3. . ) (4. . ) (2. . ).
( )
(3. . ) (3. . ) (3. . ) (3. . ) (4. . )
.
[
[ ]
Randomized Block Design:
Sum of Squares Calculations (Part 1)
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Randomized Block Design:
Sum of Squares Calculations (Part 2)
176.5
)56.34.3()56.36.2()56.34.3()56.37.3(
)(
143.0
)56.303.498.24.3()56.310.398.26.2(
)56.337.354.34.3()56.377.354.37.3(
)(
2222
1 1
2
22
22
1 1
2
n
i
C
j
n
i
C
j
XX
XXXX
ijSST
ijijSSE
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Randomized Block Design:
Mean Square Calculations
MSCSSC
C
MSRSSR
n
MSESSE
N n C
FMSC
MSE
1
3484
2
1742
1
1549
40 387
1
0143
80 018
1742
0 01896 78
..
..
..
.
..
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Analysis of Variance
for the Tread-Wear Example
Source of VarianceSS df MS F
Treatment 3.484 2 1.742 96.78Block 1.549 4 0.387 21.50
Error 0.143 8 0.018
Total 5.176 14
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Randomized Block Design Treatment
Effects: Procedural SummaryH
H
o
a
:
:
1 2 3
At least one of the means is different from the others
78.96018.0
742.1
MSE
MSCF
F = 96.78 > = 8.65, reject H ..01,2,8
oF
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Randomized Block Design Blocking
Effects: Procedural Overview
H
H
o
a
:
:
1 2 3 4 5
At least one of the blocking means is different from the others
5.21018.
387.
MSE
MSRF
F = 21.5 > = 7.01, reject H .F o. , ,01 4 8
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Excel Output for Tread-Wear
Example: Randomized Block DesignAnova: Two-Factor Without Replication
SUMMARY Count Sum Average Variance
Supplier 1 3 11.3 3.7666667 0.4933333
Supplier 2 3 10.1 3.3666667 0.3033333
Supplier 3 3 10.6 3.5333333 0.3033333
Supplier 4 3 9.3 3.1 0.21
Supplier 5 3 12.1 4.0333333 0.5033333
Slow 5 17.7 3.54 0.073
Medium 5 20.8 4.16 0.258
Fast 5 14.9 2.98 0.092
ANOVASource of Variation SS df MS F P-value F critRows 1.5493333 4 0.3873333 21.719626 0.0002357 7.0060651
Columns 3.484 2 1.742 97.682243 2.395E-06 8.6490672
Error 0.1426667 8 0.0178333
Total 5.176 14
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MINITAB Output for Tread-Wear
Example: Randomized Block Design
Blocking variable
Suppliers
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Two-Way Factorial Design
Cells
.
.
.
.
.
.
.
.
.
.
.
.
Column Treatment
Row
Treatment
.
.
.
.
.
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Formulas for Computing
a Two-Way ANOVA
SSR nC i
R
SSC nR j C
SSI n ij i j R C
SSE ijk ij RC n
SST ijk N
MSRSSR
R
MSR
MSE
MSC
X X df
X X df
X X X X df
X X df
X X df
F
i
R
R
j
C
C
j
C
i
R
I
k
n
j
C
i
R
E
a
n
r
R
c
C
T
R
2
1
2
1
2
11
2
111
2
111
1
1
1 1
1
1
1
( )
( )
( )
( )
( )
SSC
C
MSC
MSE
MSISSI
R C
MSI
MSE
MSESSE
RC n
where
C
I
FF
1
1 1
1
:
n = number of observations per cell
C = number of column treatments
R = number of row treatments
i = row treatment level
j = column treatment level
k = cell member
= individual observation
= cell mean
= row mean
= column mean
X = grand mean
ijk
ij
i
j
X
X
XX
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A 2 3 Factorial Designwith Interaction
Cell
Means
C1 C2 C3
Row effects
R1
R2
Column
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A 2 3 Factorial Designwith Some Interaction
Cell
Means
C1 C2 C3
Row effects
R1
R2
Column
A 2 3 i i
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A 2 3 Factorial Designwith No Interaction
Cell
Means
C1 C2 C3
Row effects
R1
R2
Column
A 2 3 Factorial Design: Data and
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A 2 3 Factorial Design: Data andMeasurements for CEO Dividend Example
N = 24
n = 4
X=2.7083
1.75 2.75 3.625
Location Where CompanyStock is Traded
How Stockholdersare Informed of
DividendsNYSE AMEX OTC
Annual/QuarterlyReports
2
121
2
332
4
343
2.5
Presentations to
Analysts
23
12
33
24
44
34 2.9167
Xj
Xi
X11=1.5
X23=3.75X22=3.0X21=2.0
X13=3.5X12=2.5
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A 2 3 Factorial Design: Calculationsfor the CEO Dividend Example (Part 1)
SSR X X
SSCX
X
SSI X X X X
nCi
nR
j
n ij i j
i
R
j
C
j
C
i
R
2
1
2 2
2
1
2 2 2
2
11
2
4 3 2 5 2 7083 2 9167 2 7083
4 2 175 2 7083 2 75 2 7083 3625 2 7083
4 15 2 5 175 2 7083
10418
140833
( )
.
( )
.
( )
( )( )[( . . ) ( . . ) ]
( )( )[( . . ) ( . . ) ( . . ) ]
[( . . . . ) ( . . . . )
( . . . . ) ( . . . . )
( . . . . ) ( . . . . ) ]
.
2 5 2 5 2 75 2 7083
35 2 5 3625 2 7083 2 0 2 9167 175 2 7083
30 2 9167 2 75 2 7083 375 2 9167 3625 2 7083
2
2 2
2 2
00833
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A 2 3 Factorial Design: Calculationsfor the CEO Dividend Example (Part 2)
SSE X X
SST X X
ijk ij
ijk
k
n
j
C
i
R
a
n
r
R
c
C
2
1112 2 2 2
2
111
2 2 2 2
2 15 1 15 3 375 4 375
77500
2 27083 1 27083 3 27083 4 27083
229583
( )
( . ) ( . ) ( . ) ( . )
.
( )
( . ) ( . ) ( . ) ( . )
.
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A 2 3 Factorial Design: Calculationsfor the CEO Dividend Example (Part 3)
MSRSSR
R
MSR
MSE
MSCSSC
C
MSC
MSE
MSISSI
R C
MSI
MSE
MSESSE
RC n
R
C
I
F
F
F
1
10418
110418
10418
043062 42
1
14 0833
2
7 04177 0417
04306
1635
1 1
00833
200417
00417
04306010
1
7 7500
1804306
..
.
..
..
.
.
.
..
.
..
..
A l i f V i
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Analysis of Variance
for the CEO Dividend Problem
Source of VarianceSS df MS F
Row 1.0418 1 1.0418 2.42
Column 14.0833 2 7.0417 16.35*
Interaction 0.0833 2 0.0417 0.10
Error 7.7500 18 0.4306
Total 22.9583 23
*Denotes significance at = .01.
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Excel
Outputfor the
CEO
DividendExample
(Part 1)
Anova: Two-Factor With Replication
SUMMARY NYSE ASE OTC Total
AQReportCount 4 4 4 12
Sum 6 10 14 30
Average 1.5 2.5 3.5 2.5
Variance 0.3333 0.3333 0.3333 1
Presentation
Count 4 4 4 12Sum 8 12 15 35
Average 2 3 3.75 2.9167
Variance 0.6667 0.6667 0.25 0.9924
Total
Count 8 8 8
Sum 14 22 29Average 1.75 2.75 3.625
Variance 0.5 0.5 0.2679
E l O t t f th
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Excel Output for the
CEO Dividend Example (Part 2)
ANOVA
Source of Variation SS df MS F P-value F critSample 1.0417 1 1.0417 2.4194 0.1373 4.4139
Columns 14.083 2 7.0417 16.355 9E-05 3.5546Interaction 0.0833 2 0.0417 0.0968 0.9082 3.5546
Within 7.75 18 0.4306
Total 22.958 23
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MINITAB Output for the
Demonstration Problem 11.4:
MINITAB Output for the
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MINITAB Output for the
Demonstration Problem 11.4:
Interaction Plots
321
4
3
2
1
4321
4
3
2
1
Warehouses
Length
1
2
34
Warehouses
1
2
3
Length
Interaction Plot (data means) for DaysAbsent
-
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