We made a statistical analysisto determine that the likelihood of the apparent differences in our measurement results being true differencesand the comparisons being significantly different.
For drawing conclusions between differences seen for the seven selected wavelengths, we didan ANOVA followed by Tukey's comparison using the tool of Minitab.The main purpose of our tests was to find the differences between the UV and the longer wavelengths.
'P-clr' represents the degree of polarization recorded under clear sky (see equation 7);
'e-clr' represents the degree of polarization recorded under clear sky (see equation 8);
'P-cldy' represents the degree of polarization recorded under cloudy sky (see equation 7);
'e-cldy' represents the degree of polarization recorded under cloudy sky (see equation 8).
Our results are as follows:
1 ANOVA of the degree of polarization of seven selected wavelengths under clear sky
One-way ANOVA: P-clr(whole), P-clr(IR), P-clr(Red), P-clr(Blue), P-clr(UV400), P-clr(UV380), P-clr(UV360)
Source DF SS MS F P
Factor 6 3.378733 0.563122 1922.62 0.000
Error 504 0.147618 0.000293
Total 510 3.526351
S = 0.01711 R-Sq = 95.81% R-Sq(adj) = 95.76%
Individual 95% CIs For Mean Based on
Pooled StDev
Level N Mean StDev --+------+------+------+------
P-clr(whole) 73 0.39059 0.01778 (*)
P-clr(IR) 73 0.42830 0.01962 *)
P-clr(Red) 73 0.43214 0.01911 (*)
P-clr(Blue) 73 0.40178 0.01673 (*)
P-clr(UV400) 73 0.28467 0.01608 *)
P-clr(UV380) 73 0.24168 0.01501 *)
P-clr(UV360) 73 0.23484 0.01483 (*)
--+------+------+------+------
0.240 0.300 0.360 0.420
Pooled StDev = 0.01711
Grouping Information Using Tukey Method
N Mean Grouping
P-clr(Red) 73 0.43214 A
P-clr(IR) 73 0.42830 A
P-clr(Blue) 73 0.40178 B
P-clr(whole) 73 0.39059 C
P-clr(UV400) 73 0.28467 D
P-clr(UV380) 73 0.24168 E
P-clr(UV360) 73 0.23484 E
Means that do not share a letter are significantly different.
Tukey 95% Simultaneous Confidence Intervals
All Pairwise Comparisons
Individual confidence level = 99.67%
P-clr(whole) subtracted from:
Lower Center Upper
P-clr(IR) 0.02936 0.03771 0.04607
P-clr(Red) 0.03320 0.04155 0.04990
P-clr(Blue) 0.00284 0.01119 0.01954
P-clr(UV400) -0.11427 -0.10592 -0.09757
P-clr(UV380) -0.15726 -0.14890 -0.14055
P-clr(UV360) -0.16411 -0.15575 -0.14740
------+------+------+------+--
P-clr(IR) (*)
P-clr(Red) *)
P-clr(Blue) (*)
P-clr(UV400) (*)
P-clr(UV380) (*
P-clr(UV360) (*)
------+------+------+------+--
-0.12 0.00 0.12 0.24
P-clr(IR) subtracted from:
Lower Center Upper
P-clr(Red) -0.00452 0.00384 0.01219
P-clr(Blue) -0.03487 -0.02652 -0.01817
P-clr(UV400) -0.15198 -0.14363 -0.13528
P-clr(UV380) -0.19497 -0.18662 -0.17826
P-clr(UV360) -0.20182 -0.19347 -0.18511
------+------+------+------+--
P-clr(Red) *)
P-clr(Blue) (*
P-clr(UV400) (*)
P-clr(UV380) *)
P-clr(UV360) (*)
------+------+------+------+--
-0.12 0.00 0.12 0.24
P-clr(Red) subtracted from:
Lower Center Upper
P-clr(Blue) -0.03871 -0.03036 -0.02200
P-clr(UV400) -0.15582 -0.14747 -0.13911
P-clr(UV380) -0.19880 -0.19045 -0.18210
P-clr(UV360) -0.20565 -0.19730 -0.18895
------+------+------+------+--
P-clr(Blue) *)
P-clr(UV400) (*
P-clr(UV380) (*)
P-clr(UV360) (*
------+------+------+------+--
-0.12 0.00 0.12 0.24
P-clr(Blue) subtracted from:
Lower Center Upper
P-clr(UV400) -0.12546 -0.11711 -0.10876
P-clr(UV380) -0.16845 -0.16010 -0.15174
P-clr(UV360) -0.17530 -0.16695 -0.15859
------+------+------+------+--
P-clr(UV400) *)
P-clr(UV380) (*
P-clr(UV360) (*)
------+------+------+------+--
-0.12 0.00 0.12 0.24
P-clr(UV400) subtracted from:
Lower Center Upper
P-clr(UV380) -0.05134 -0.04299 -0.03463
P-clr(UV360) -0.05819 -0.04984 -0.04148
------+------+------+------+--
P-clr(UV380) *)
P-clr(UV360) (*)
------+------+------+------+--
-0.12 0.00 0.12 0.24
P-clr(UV380) subtracted from:
Lower Center Upper
P-clr(UV360) -0.01520 -0.00685 0.00150
------+------+------+------+--
P-clr(UV360) *)
------+------+------+------+--
-0.12 0.00 0.12 0.24
2ANOVA of the polarization angle measurement error of seven selected wavelengths under clear sky
One-way ANOVA: e-clr(whole), e-clr(IR), e-clr(Red), e-clr(Blue), e-clr(UV400), e-clr(UV380), e-clr(UV360)
Source DF SS MS F P
Factor 6 1.42453 0.23742 168.57 0.000
Error 497 0.70001 0.00141
Total 503 2.12455
S = 0.03753 R-Sq = 67.05% R-Sq(adj) = 66.65%
Individual 95% CIs For Mean Based on
Pooled StDev
Level N Mean StDev ---+------+------+------+------
e-clr(whole) 72 0.35207 0.04317 (*-)
e-clr(IR) 72 0.23533 0.04540 (-*-)
e-clr(Red) 72 0.20140 0.04150 (*-)
e-clr(Blue) 72 0.21204 0.03669 (*-)
e-clr(UV400) 72 0.19769 0.03216 (-*)
e-clr(UV380) 72 0.19326 0.03083 (-*)
e-clr(UV360) 72 0.19235 0.02968 (*-)
---+------+------+------+------
0.200 0.250 0.300 0.350
Pooled StDev = 0.03753
Grouping Information Using Tukey Method
N Mean Grouping
e-clr(whole) 72 0.35207 A
e-clr(IR) 72 0.23533 B
e-clr(Blue) 72 0.21204 C
e-clr(Red) 72 0.20140 C D
e-clr(UV400) 72 0.19769 C D
e-clr(UV380) 72 0.19326 D
e-clr(UV360) 72 0.19235 D
Means that do not share a letter are significantly different.
Tukey 95% Simultaneous Confidence Intervals
All Pairwise Comparisons
Individual confidence level = 99.67%
e-clr(whole) subtracted from:
Lower Center Upper
e-clr(IR) -0.13518 -0.11674 -0.09829
e-clr(Red) -0.16911 -0.15067 -0.13222
e-clr(Blue) -0.15847 -0.14003 -0.12158
e-clr(UV400) -0.17282 -0.15438 -0.13593
e-clr(UV380) -0.17725 -0.15881 -0.14036
e-clr(UV360) -0.17817 -0.15972 -0.14128
+------+------+------+------
e-clr(IR) (---*--)
e-clr(Red) (--*--)
e-clr(Blue) (--*--)
e-clr(UV400) (--*--)
e-clr(UV380) (---*--)
e-clr(UV360) (--*--)
+------+------+------+------
-0.180 -0.120 -0.060 0.000
e-clr(IR) subtracted from:
Lower Center Upper
e-clr(Red) -0.05237 -0.03393 -0.01549
e-clr(Blue) -0.04174 -0.02329 -0.00485
e-clr(UV400) -0.05608 -0.03764 -0.01920
e-clr(UV380) -0.06051 -0.04207 -0.02363
e-clr(UV360) -0.06143 -0.04299 -0.02454
+------+------+------+------
e-clr(Red) (--*--)
e-clr(Blue) (--*--)
e-clr(UV400) (--*--)
e-clr(UV380) (--*--)
e-clr(UV360) (--*--)
+------+------+------+------
-0.180 -0.120 -0.060 0.000
e-clr(Red) subtracted from:
Lower Center Upper
e-clr(Blue) -0.00780 0.01064 0.02908
e-clr(UV400) -0.02215 -0.00371 0.01474
e-clr(UV380) -0.02658 -0.00814 0.01030
e-clr(UV360) -0.02750 -0.00906 0.00939
+------+------+------+------
e-clr(Blue) (--*--)
e-clr(UV400) (--*--)
e-clr(UV380) (--*--)
e-clr(UV360) (--*---)
+------+------+------+------
-0.180 -0.120 -0.060 0.000
e-clr(Blue) subtracted from:
Lower Center Upper
e-clr(UV400) -0.03279 -0.01435 0.00410
e-clr(UV380) -0.03722 -0.01878 -0.00033
e-clr(UV360) -0.03814 -0.01969 -0.00125
+------+------+------+------
e-clr(UV400) (--*--)
e-clr(UV380) (--*--)
e-clr(UV360) (--*--)
+------+------+------+------
-0.180 -0.120 -0.060 0.000
e-clr(UV400) subtracted from:
Lower Center Upper
e-clr(UV380) -0.02287 -0.00443 0.01401
e-clr(UV360) -0.02379 -0.00535 0.01310
+------+------+------+------
e-clr(UV380) (--*--)
e-clr(UV360) (--*--)
+------+------+------+------
-0.180 -0.120 -0.060 0.000
e-clr(UV380) subtracted from:
Lower Center Upper
e-clr(UV360) -0.01936 -0.00092 0.01753
+------+------+------+------
e-clr(UV360) (--*--)
+------+------+------+------
-0.180 -0.120 -0.060 0.000
3ANOVA of the degree of polarization of seven selected wavelengths under cloudy sky
One-way ANOVA: P-cldy(whole), P-cldy(IR), P-cldy(Red), P-cldy(Blue), P-cldy(UV400), P-cldy(UV380), P-cldy(UV360)
Source DF SS MS F P
Factor 6 0.654722 0.109120 805.55 0.000
Error 504 0.068272 0.000135
Total 510 0.722995
S = 0.01164 R-Sq = 90.56% R-Sq(adj) = 90.44%
Individual 95% CIs For Mean Based on
Pooled StDev
Level N Mean StDev ----+------+------+------+-----
P-cldy(whole) 73 0.20062 0.01482 (*)
P-cldy(IR) 73 0.11119 0.01148 (*)
P-cldy(Red) 73 0.13244 0.00951 (*)
P-cldy(Blue) 73 0.17760 0.01194 (*)
P-cldy(UV400) 73 0.19441 0.01210 (*)
P-cldy(UV380) 73 0.20277 0.01112 (*
P-cldy(UV360) 73 0.21075 0.00969 (*)
----+------+------+------+-----
0.120 0.150 0.180 0.210
Pooled StDev = 0.01164
Grouping Information Using Tukey Method
N Mean Grouping
P-cldy(UV360) 73 0.21075 A
P-cldy(UV380) 73 0.20277 B
P-cldy(whole) 73 0.20062 B
P-cldy(UV400) 73 0.19441 C
P-cldy(Blue) 73 0.17760 D
P-cldy(Red) 73 0.13244 E
P-cldy(IR) 73 0.11119 F
Means that do not share a letter are significantly different.
Tukey 95% Simultaneous Confidence Intervals
All Pairwise Comparisons
Individual confidence level = 99.67%
P-cldy(whole) subtracted from:
Lower Center Upper
P-cldy(IR) -0.09511 -0.08942 -0.08374
P-cldy(Red) -0.07386 -0.06818 -0.06250
P-cldy(Blue) -0.02869 -0.02301 -0.01733
P-cldy(UV400) -0.01189 -0.00621 -0.00053
P-cldy(UV380) -0.00353 0.00215 0.00783
P-cldy(UV360) 0.00446 0.01014 0.01582
------+------+------+------+-
P-cldy(IR) (*)
P-cldy(Red) (*)
P-cldy(Blue) (*)
P-cldy(UV400) (*)
P-cldy(UV380) (*)
P-cldy(UV360) (*)
------+------+------+------+-
-0.060 0.000 0.060 0.120
P-cldy(IR) subtracted from:
Lower Center Upper
P-cldy(Red) 0.01557 0.02125 0.02693
P-cldy(Blue) 0.06073 0.06641 0.07209
P-cldy(UV400) 0.07754 0.08322 0.08890
P-cldy(UV380) 0.08589 0.09158 0.09726
P-cldy(UV360) 0.09388 0.09956 0.10524
------+------+------+------+-
P-cldy(Red) (*
P-cldy(Blue) (*)
P-cldy(UV400) (*)
P-cldy(UV380) (*)
P-cldy(UV360) (*)
------+------+------+------+-
-0.060 0.000 0.060 0.120
P-cldy(Red) subtracted from:
Lower Center Upper
P-cldy(Blue) 0.03948 0.04516 0.05084
P-cldy(UV400) 0.05629 0.06197 0.06765
P-cldy(UV380) 0.06465 0.07033 0.07601
P-cldy(UV360) 0.07263 0.07832 0.08400
------+------+------+------+-
P-cldy(Blue) (*
P-cldy(UV400) (*)
P-cldy(UV380) (*)
P-cldy(UV360) (*)
------+------+------+------+-
-0.060 0.000 0.060 0.120
P-cldy(Blue) subtracted from:
Lower Center Upper
P-cldy(UV400) 0.01113 0.01681 0.02249
P-cldy(UV380) 0.01948 0.02516 0.03084
P-cldy(UV360) 0.02747 0.03315 0.03883
------+------+------+------+-
P-cldy(UV400) (*)
P-cldy(UV380) (*)
P-cldy(UV360) (*
------+------+------+------+-
-0.060 0.000 0.060 0.120
P-cldy(UV400) subtracted from:
Lower Center Upper
P-cldy(UV380) 0.00268 0.00836 0.01404
P-cldy(UV360) 0.01066 0.01634 0.02202
------+------+------+------+-
P-cldy(UV380) (*)
P-cldy(UV360) (*)
------+------+------+------+-
-0.060 0.000 0.060 0.120
P-cldy(UV380) subtracted from:
Lower Center Upper
P-cldy(UV360) 0.00231 0.00799 0.01367
------+------+------+------+-
P-cldy(UV360) (*)
------+------+------+------+-
-0.060 0.000 0.060 0.120
4ANOVA of the polarization angle measurement error of seven selected wavelengths under cloudy sky
One-way ANOVA: e-cldy(whole), e-cldy(IR), e-cldy(Red), e-cldy(Blue), e-cldy(UV400), e-cldy(UV380), e-cldy(UV360)
Source DF SS MS F P
Factor 6 6.00633 1.00106 610.30 0.000
Error 497 0.81522 0.00164
Total 503 6.82155
S = 0.04050 R-Sq = 88.05% R-Sq(adj) = 87.91%
Individual 95% CIs For Mean Based on
Pooled StDev
Level N Mean StDev ---+------+------+------+------
e-cldy(whole) 72 0.48207 0.04383 (*)
e-cldy(IR) 72 0.53113 0.04841 (*)
e-cldy(Red) 72 0.52211 0.04510 (-*)
e-cldy(Blue) 72 0.37640 0.03896 (-*)
e-cldy(UV400) 72 0.27065 0.03572 (-*)
e-cldy(UV380) 72 0.28190 0.03493 (*-)
e-cldy(UV360) 72 0.28275 0.03419 (*-)
---+------+------+------+------
0.280 0.350 0.420 0.490
Pooled StDev = 0.04050
Grouping Information Using Tukey Method
N Mean Grouping
e-cldy(IR) 72 0.53113 A
e-cldy(Red) 72 0.52211 A
e-cldy(whole) 72 0.48207 B
e-cldy(Blue) 72 0.37640 C
e-cldy(UV360) 72 0.28275 D
e-cldy(UV380) 72 0.28190 D
e-cldy(UV400) 72 0.27065 D
Means that do not share a letter are significantly different.
Tukey 95% Simultaneous Confidence Intervals
All Pairwise Comparisons
Individual confidence level = 99.67%
e-cldy(whole) subtracted from:
Lower Center Upper
e-cldy(IR) 0.02915 0.04906 0.06896
e-cldy(Red) 0.02014 0.04004 0.05995
e-cldy(Blue) -0.12557 -0.10567 -0.08576
e-cldy(UV400) -0.23132 -0.21142 -0.19151
e-cldy(UV380) -0.22007 -0.20017 -0.18026
e-cldy(UV360) -0.21922 -0.19932 -0.17942
------+------+------+------+
e-cldy(IR) (*-)
e-cldy(Red) (-*)
e-cldy(Blue) (*)
e-cldy(UV400) (*)
e-cldy(UV380) (-*)
e-cldy(UV360) (-*)
------+------+------+------+
-0.15 0.00 0.15 0.30
e-cldy(IR) subtracted from:
Lower Center Upper
e-cldy(Red) -0.02892 -0.00901 0.01089
e-cldy(Blue) -0.17463 -0.15472 -0.13482
e-cldy(UV400) -0.28038 -0.26047 -0.24057
e-cldy(UV380) -0.26913 -0.24922 -0.22932
e-cldy(UV360) -0.26828 -0.24838 -0.22847
------+------+------+------+
e-cldy(Red) (*-)
e-cldy(Blue) (-*)
e-cldy(UV400) (-*)
e-cldy(UV380) (*-)
e-cldy(UV360) (*-)
------+------+------+------+
-0.15 0.00 0.15 0.30
e-cldy(Red) subtracted from:
Lower Center Upper
e-cldy(Blue) -0.16561 -0.14571 -0.12580
e-cldy(UV400) -0.27136 -0.25146 -0.23155
e-cldy(UV380) -0.26011 -0.24021 -0.22030
e-cldy(UV360) -0.25926 -0.23936 -0.21946
------+------+------+------+
e-cldy(Blue) (*-)
e-cldy(UV400) (*-)
e-cldy(UV380) (*)
e-cldy(UV360) (*)
------+------+------+------+
-0.15 0.00 0.15 0.30
e-cldy(Blue) subtracted from:
Lower Center Upper
e-cldy(UV400) -0.12565 -0.10575 -0.08585
e-cldy(UV380) -0.11440 -0.09450 -0.07460
e-cldy(UV360) -0.11356 -0.09365 -0.07375
------+------+------+------+
e-cldy(UV400) (*)
e-cldy(UV380) (-*)
e-cldy(UV360) (-*)
------+------+------+------+
-0.15 0.00 0.15 0.30
e-cldy(UV400) subtracted from:
Lower Center Upper
e-cldy(UV380) -0.00865 0.01125 0.03115
e-cldy(UV360) -0.00781 0.01210 0.03200
------+------+------+------+
e-cldy(UV380) (-*)
e-cldy(UV360) (-*)
------+------+------+------+
-0.15 0.00 0.15 0.30
e-cldy(UV380) subtracted from:
Lower Center Upper
e-cldy(UV360) -0.01906 0.00085 0.02075
------+------+------+------+
e-cldy(UV360) (*)
------+------+------+------+
-0.15 0.00 0.15 0.30