 Open Access
 Authors : Roniere Leite Soares , Walman Benicio De Castro
 Paper ID : IJERTV9IS030118
 Volume & Issue : Volume 09, Issue 03 (March 2020)
 Published (First Online): 11032020
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Application of Tukeys Test for Statistical Measurement of Percentage Indexes Derived from the Difference Between Predicted Crystallographic Data in Five Nitihf Alloys Obtained by Two Processes
Roniere Leite Soares
Federal University of Campina Grande UFCG CCT UAEP – AEG
Campina Grande ParaÃba – Brazil
Walman Benicio De Castro Federal University of Campina Grande UFCG CCT UAEM – LaMMEA
Campina Grande ParaÃba – Brazil
Abstract The present paper uses the Tukey test to measure the minimal significative difference (MSD) between crystallographic data predicted by nonlinear mathematical models generated from graphics published by Zarinejad (2008) and Potapov (1997) on 5 nominal compositions of Ni50Ti50XHfX
.at% substitutional alloys fused by arc melting and melt spinning processes. Results show that the calculated MSD do not represent a statistically significant difference. Therefore, it is concluded that the independent variable (Hafnium content) does not change the analysed dependent variables (lattice parameters, crystalline structure volume and monoclinic angle ) for any fusing processes.
Keywords ANOVA; Monoclinic; NiTiHf; Statistical test; Tukey.

PREDICTION OF THE B19 PHASE OF NI50TI50XHFX .AT% ALLOYS
Based on the graphs by Zarinejad (2008) [1], which measurements were obtained by XRD [3] refinement method, it was possible to calculate, through numerical proportions, the equivalence of the lattice parameters (a, b, c), volume of the monoclinic structure B19' and axial angle , as shown in table 1.

INTRODUCTION
The production of shape memory effect (SME) alloys depends sensibly on particular processes that, based on the variation of parameters involved in the equipment used (maximum melting temperature, vacuum magnitude, purge intensity, purity of the metallic charges, torch exposition time, etc), can or cannot generate different crystallographic features for samples which theoretically have the same chemical compositions. In this sense, it is important to compare the obtained results through predictive modelling functions to observe whether there are any significant statistical difference capable of indicating a great variability in crystallographic characteristics, such as: lattice parameters (a b c [Ã…]), cell volume (VOL [Ã…3]) and characteristic angle (beta [o]) of the monoclinic structure (space group P121/m1). In this work, they are studied as variables dependent on hafnium content, which is considered an independent variable in the nominal composition. The focus of the study is on room temperature B19 martensitic phase B19, which has a monoclinic structure (Pearson symbol mP8) and is considered by the literature as an unstable phase. In addition, the five substitutional compositions Ni50Ti50XHfX .at% considered in this instance are X = 8, 11, 14, 17 and 20 .at%, which were obtained by two different processes: Arc Melting (Zarinejad, 2008) [1] and Melt Spinning (Potapov, 1997) [2].
Hf
a
b
c
V
( at.% )
( Ã… )
( Ã… )
( Ã… )
( Ã…3 )
( o )
5
2.8228476
4.1605263
4.6628158
54.195804
98.068493
10
2.9049668
4.1210526
4.7220216
55.758741
98.49315
15
2.9980132
4.1092105
4.7707581
57.972027
99.189041
20
3.0814569
4.0503947
4.8754512
59.807692
100.04657
Hf
a
b
c
V
( at.% )
( Ã… )
( Ã… )
( Ã… )
( Ã…3 )
( o )
5
2.8228476
4.1605263
4.6628158
54.195804
98.068493
10
2.9049668
4.1210526
4.7220216
55.758741
98.49315
15
2.9980132
4.1092105
4.7707581
57.972027
99.189041
20
3.0814569
4.0503947
4.8754512
59.807692
100.04657
Table 1 – Values for the four compositions of Ni50Ti50XHfX .at% alloys (Zarinejad, 2008) obtained by conventional solidification (X = 5, 10, 15 and
20 at.%)
From the obtained results, the same variables were estimated for the five alloys of study (X = 8, 11, 14, 17 e 20 at.%). From the third order nonlinear regression (shown in Figure 1) it was adopted the polynomial function = 0 + 1X + 2X2 + 3X3 Â± for each parameter, which resulted in: a = 2,77219 + 0,00562x + 0,00104×2 2,73732E5×3 (R2=1); b = 4,30224 0,04354x + 0,00354×2 9,94737E5×3 (R2=1) and c = 4,52671 + 0,03934x 0,00287×2 + 8,85679E5×3
(R2=1). The determination coefficient R2 are equal to 1. This indicates that the proposed models are suitable to describe the phenomenon in which the parameters a and c increase, while the parameter b decreases. Equally, the models adopted for volume and angle for b19phase presented 100% fitting.
Figure 1 – Scatter diagrams & Cubic regression curves: with the values of lattice parameters (a, b, c) determined for the Ni50Ti50XHfX .at% alloys proposed by Zarinejad (2008)
All the values estimated for the five dependent variables are organized in table 2.
Ni50Ti50XHfX .at% Nirich alloys of study are plotted in figure 2:
Figure 2 – Scatter Plot with estimated values of the lattice parameters of the crystal structure B19' in Ni50Ti50XHfX .at% alloys (X = 8, 11, 14, 17, 20 at.%)
According to Figure 3(a), from the regressive
Hf (at.%)
a ( Ã… )
b ( Ã… )
c ( Ã… )
VOLreg ( Ã…3
)
( o )
8
2.897725
0
4.1295494
6
4.7030967
6
54.99855
98.2847324
6
11
2.996283
7
4.1192405
0
4.7300638
7
56.18361
98.6143636
3
14
3.129822
0
4.1135641
6
4.7579803
1
57.52923
99.0337747
7
17
3.302774
5
4.0964057
1
4.8011940
9
58.81347
99.5192931
4
20
3.519575
4.0516500
4.874053
59.81439
100.047246
td>
98.2847324
6
Hf (at.%)
a ( Ã… )
b ( Ã… )
c ( Ã… )
VOLreg ( Ã…3
)
( o )
8
2.897725
0
4.1295494
6
4.7030967
6
54.99855
11
2.996283
7
4.1192405
0
4.7300638
7
56.18361
98.6143636
3
14
3.129822
0
4.1135641
6
4.7579803
1
57.52923
99.0337747
7
17
3.302774
5
4.0964057
1
4.8011940
9
58.81347
99.5192931
4
20
3.519575
4.0516500
4.874053
59.81439
100.047246
function
VOLreg
= 54,31119 0,25944x + 0,05413×2
Table 2 – Values of the parameters predicted by the 3rd order polynomials: lattice parameters (a, b, c), monoclinic volume and , for Ni50Ti50XHfX .at% alloys (X = 8, 11, 14, 17 and 20 .at %)
A comparison between the values of table 1 and table 2, confirms a coherence in the direct proportion between the percentage of hafnium and the volume of the monoclinic structure (phase B19'). The lattice parameters for the five
0,00137×3 (R2 = 1), adopted for all four Zarinejad (2008) alloys, in which X = 5, 10, 15 e 20 at.%, it is possible to predict the volume (Ã…3) of the B19 monoclinic martensitic structure for the five compositions of the alloy Ni50Ti50XHfX at.% (X=8, 11, 14 17 and 20 at.%), shown in Figure 3(b).
Similarly, using the cubic polynomial function y = 98,02467 0,03662x + 0,00981×2 1,46128E4×3 (R2 = 1)
adopted for Zarinejad alloys (2008), as seen in Figure 4(a), the angles of the B19 phase for the five Ni50Ti50XHfX at.% (Hf = 8, 11, 14, 17 e 20 .at%) alloys of study were predicted and plotted in figure 4(b).
Figure 3 – Nonlinear regression (order 3) from which the volumes of the martensite phase (B19 ') of the five substitutive Ni50Ti50XHfX .at% (X = 8, 11, 14, 17 and 20 %.at) alloys were calculated
Figure 4 – Nonlinear cubic regression from which the angles of the B19martensitic phases were estimated for the five Ni50Ti50XHfX .at% (X = 8, 11, 14, 17 and 20 at.%) alloys rich in nickel
For the functions used as statistical models it was calculated the analysis of variance in each polynomial, according to table 3.
Variables
ANOVA
DF
SQ
MQ
F
Significance F
a
Regression
3
0.03776864
0.012589547
Residue
0
0
Total
3
0.03776864
b
Regression
3
0.006228132
0.002076044
Residue
0
0
Total
3
0.006228132
c
Regression
3
0.024311803
0.008103934
Residue
0
0
Total
3
0.024311803
Angle
Regression
3
2.245370993
0.748456998
Residue
0
0
Total
3
2.245370993
VolReg
Regression
3
18.21455606
6.071518687
Residue
0
0
Total
3
18.21455606
VolCalc
Regression
3
18.46596165
6.155320551
Residue
0
0
Total
3
18.46596165
Table 3 – ANOVAS calculated for each model = 0 + 1x + 2×2 + 3×3 referring to the 5 variables of the B19martensitic phase of the 5 Ni50Ti50 XHfX at.% alloys
To make explicit the coherence of the regressive model adopted for the volume, the calculated volume (Vol = a.b.c.sen) was added in Table 3, on top of which the model VOLcalc = 53,5806847 + 0,018979973x + 0,033586976×2 +
8,39734E04×3 (R2 = 1) was created.
All of the R2 values were equal to 1. This makes unnecessary the calculation of AdjustedR2, normally used to measure the real degree of modeling reliability. According to Table 3, since there was no standard error in the coefficients, the general error of the models is 0. For this reason, F value is not presented in table 3. Thus, it can be stated that the mathematical models adopted here are statistically significant as well as predictive for all five dependent variables observed.
Taking the four Zarinejad alloys (2008) as a reference, the residues found for each regression, in each variable of the monoclinic structure B19, are presented in table 4 and figure 5.
Hf (.at%)
a ( Ã… )
b ( Ã… )
c ( Ã… )
( o )
Volregressive
Volcalculated
5
1.78E15
8.88E16
1.78E15
2.84E14
7.11E15
2.84E14
10
8.88E16
1.78E15
2.66E15
1.42E14
1.42E14
0
15
8.88E16
8.88E16
1.78E15
1.42E14
1.42E14
0
20
8.88E16
0
2.66E15
1.42E14
1.42E14
0
Table 4 – Residues of each variable calculated for phase B19 'of the 4 substitutive Ni50Ti50XHfX .at% (X = 5, 10, 15 and 20) alloys from Zarinejad
(2008)
Figure 5 – Dispersive plot of residues in each variable of the B19monoclinic structure (a, b, c, Volume and Beta [, o]) for the 4 alloys of Zarinejad (2008)


PREDICTION OF THE PHASE B19ON NI50TI50XHFX
.AT% RIBBONS QUICKLY SOLIDIFIED BY MELT SPINNING
Based on the Potapov plots (1997) for the six ribbons of substitutional compositions Ni49.8Ti50.2XHfX .at% (X= 8, 9.5, 11, 15, 20 e 25 .at%) obtained by melt spinning, the calculation of lattice parameters (a, b, c), B19monoclinic crystal volume and characteristic angle are shown in table 5:
Hf (.at%)
a ( Ã… )
b ( Ã… )
c ( Ã… )
VOL ( Ã…3 )
( o )
8
2.985357
4.100357
4.725
56.961414
99.992857
9.5
2.989285
4.096428
4.755642
57.337329
100.07142
11
3.001071
4.094857
4.779214
57.769323
100.38571
15
.027785
4.0925
4.827142
58.499286
102.03571
20
3.06000
4.084642
4.89000
59.539886
103.05714
25
3.099285
4.072857
4.937142
60.25763
104.78571
Table 5 – Crystallographic characteristics of the martensitic phase obtained by XRD refinement for the six ribbons with Potapov (1997) nominal compositions Ni49.8Ti50.2xHfx .at% obtained by fast melt spinning solidification
Considering the similarity of the composition used by Potapov (1997) regarding the composition Ni50Ti50XHfX
.at%, adopted in this work (difference of 0,2 .at% in the contents of Ni e Ti), the values of table 5 were used to proportionally calculate the same variables having as reference an alloy rich in Ni of nominal composition Ni50Ti50 XHfX .at%, quickly solidified by meltspinning. The results are shown in Table 6:
Hf (.at%)
a (Ã…)
b (Ã…)
c (Ã…)
Vol (Ã…3)
( o )
8
2.99735
4.08402
4.74398
57.19017
100.39443
9.5
3.00129
4.08011
4.77474
57.5676
100.47331
11
3.01312
4.07854
4.79841
58.00133
100.78887
15
3.03994
4.0762
4.84653
58.73422
102.44549
20
3.07229
4.06837
4.90964
59.779
103.47102
25
3.11173
4.05663
4.95697
60.49963
105.20654
Table 6 – Crystallographic characteristics of the martensitic phase calculated proportionally from Potapov's ribbons Ni49.8Ti50.2xHfx .at% (1997) for the theoretical ribbons Ni50Ti50XHfX .at% (X = 8, 9.5, 11, 15, 20 and 25 .at%) obtained by fast melt spinning solidification
0 1
0 1
Based on the data in table 6, these five variables were calculated with the nonlinear cubic model, which is composed by the third order function = + +
2 + 3 + , ( = 1, 2, ). The values of R2 are,
For each of the cubic functions, the values of AdjustedR2 computed are, in the decrescent order, in table 8: 0.99643, 0.98445, 0.99682, 0.99658 e 0.97115.
2 3
for the dependent variables a, b, c, volume and , respectively, 0.9986, 0.9938, 0.9987, 0.9986 and 0.9885. This certifies a large percentage of correct answers. The estimated quantities for the five compositions are presented in table 7.
Hf (.at%)
a ( Ã… )
b ( Ã… )
c ( Ã… )
VOL ( Ã…3 )
( o )
8
2.995688
4.083342
4.746384
57.208961
100.237795
11
3.012892
4.07906
4.795981
57.93075
100.997707
14
3.031679
4.076048
4.838146
58.594993
101.835911
17
3.051887
4.07311
4.87463
59.199742
102.726977
20
3.073354
4.06905
4.907188
59.743049
103.645474
Table 7 – Estimative of the crystallographic characteristics of the B19phase in the five ribbons quickly solidified by melt spinning for the composition Ni50Ti50XHfX .at% (820 .at%)
Using the matrix adopted for monoclinic crystals [4], according to the Equation 01, the volumes were recalculated to show the coherence of the statistical model. The volumes found were, for the five compositions (Hf = 8, 11, 14, 17 and 20.at%), in this order: 57.13535181 Ã…3, 57.85901627 Ã…3,
58.51514303 Ã…3, 59.10615326 Ã…3 e 59.63533041 Ã…3.
2 0 . . 1/2
= [ 0 2 0 ]
. . 0 2
(1)
The model functions were determined based on the coefficients listed in table 8, accompanied by the respective errors.
Coeffici ents
0
0
1
1
2
2
3
3
Sum
Value
Erro
Â±
Value
Erro
Â±
Value
Erro Â±
Value
Erro Â±
a
2.958
78
0.034
34
0.003
71
0.007
16
1.21E
04
4.61E
04
– 1.00E
06
9.25E
06
b
4.110
06
0.015
79
– 0.005
38
0.003
29
3.14E
04
2.12E
04
– 7.38E
06
4.25E
06
c
4.564
45
0.058
9
0.028
21
0.012
29
– 7.70E
04
7.90E
04
1.08E
05
1.59E
05
Volume
55.01
768
0.964
61
0.297
08
0.201
2
– 0.0028
0.012
94
– 1.20E
05
2.60E
04
98.78
753
4.197
37
0.115
09
0.875
51
0.0095
3
0.056
3
– 1.57E
04
0.001
13
Table 8 – Coefficient values calculated for third order statistical models for Ni50Ti50XHfX .at%
Figure 6 Left column presents the graphs with estimated values for the six Potapov ribbons (1997), and right column shows the calculated values for the five theoretical ribbons of nominal compositions Ni50Ti50xHfx .at%
For a graphic visualization, figure 6 presents the values calculated for the six Potapov ribbons (1997) plotted in the left column (figure 6(a), figure 6(c) and figure 6(e)) in a way that they are compared with the values predicted for the five ribbons studied here, plotted in the right column (figure 6(b), figure 6(d) and figure 6(f)).
According to the obtained numbers , the values of angle, volume and "a" and "c" are directly proportional to the atomic percentage of hafnium from the alloy, except in the lattice parameter "b", which is inversely proportional to the content of Hf (.at%).

COMPARISON OF CRYSTALLOGRAPHIC CHARACTERISTICS OF THE MARTENSITIC PHASE IN
ZARINEJADS ALLOYS AND POTAPOVS RIBBONS
Due to the fact that these variables are different, i.e., onedimensional (lattice parameters) twodimensional ( angles) and threedimensional (monoclinic structures volumes) entities. To enable a dimensionless comparison between the predicted data from Zarinejad (2008) and Potapov (1997), it was necessary to consider the percentage indexess of the differences between the estimated values for each parameter (a, b, c, Vol e ) of the fie compositions Ni50Ti50XHfX .at%. This makes them arbitrary data. Therefore, it was necessary to organize the sample series from table 11 with the dimensionless format of table 9. Percentage indices have arbitrary units.
Using descriptive statistics, the measures of central tendency and dispersion were summarized in table 10.
The normal distribution (differences percentage indexes versus x indexes density) is represented by equation
2 and plotted in figure 7. It is composed of: (standard deviation), (Constante de EulerMascheroni: 2.718), (proportion) and (population mean).
Percentage indexes ranked in ascending order
0.0012
0.0068
0.0137
0.0236
0.0327
0.0043
0.0091
0.0151
0.0275
0.0347
0.0055
0.0091
0.0166
0.0302
0.0386
0.0057
0.0098
0.0182
0.0312
0.076
0.0065
0.0112
0.0195
0.0314
0.1268
Table 9 – List of 25 data organized sequentially (indexes of dimensional differences verified between both authors)
Mean (average)
0.0242
Standard deviation
0.0267
Interval
0.1256
Standard error
0.0053
Sample variance
0.0007
Min and Max
0.0012; 0.1268
Median
0.0166
Kurtosis
9.1143
Sum
0.605
Mode (fashion)
0.0091
Asymmetry
2.7719
Confidence level (95 %)
0.011025366
Table 10 – Descriptive summary of the main statistical measurements
1 1 )2
() = ( , (, )
2
2
(2)
It is noticeable, in figure 7, that there is a concentration of minor differences close to bigger densities. This certifies that the values from both authors are close.
Characteristics of phase B19Â´ (monoclinic)
Zarinejad
Potapov
Zarinejad
Potapov
Zarinejad
Potapov
Zarinejad
Potapov
Zarinejad
Potapov
Hf (.a%)
a
b
c
VOL
8
Estimated values
2.897.725
2.995.688
4.129.549
4.083.342
4.703.097
4.746.384
5.499.855
5.720.896
9.828.473
1.002.378
Absolute difference (Ã…)
0.097963
0.046207
0.043287
2.210.411
1.953.065
Proximity (%)
96.73%
98.88%
99.09%
96.14%
98.05%
Mean
29.467.065
41.064.455
47.247.405
561.037.555
992.612.625
Percentage difference
3.27%
1.12%
0.91%
3.86%
1.95%
11
Estimated values
2.996.284
3.012.892
4.119.241
407.906
4.730.064
4.795.981
5.618.361
5.793.075
9.861.436
1.009.977
Absolute difference (Ã…)
0.016608
0.040181
0.065917
174.714
2.383.347
Proximity (%)
99.45%
99.02%
98.63%
96.98%
97.64%
Mean
3.004.588
40.991.505
47.630.225
5.705.718
998.060.335
Percentage difference
0.55%
0.98%
1.37%
3.02%
2.36%
14
Estimated values
3.129.822
3.031.679
4.113.564
4.076.048
475.798
4.838.146
5.752.923
5.859.499
9.903.377
1.018.359
Absolute difference (Ã…)
0.098143
0.037516
0.080166
1.065.763
2.802.141
Proximity (%)
96.86%
99.09%
98.34%
98.18%
97.25%
Mean
30.807.505
4.094.806
4.798.063
580.621.115
1.004.348.405
Percentage difference
3.14%
0.91%
1.66%
1.82%
2.75%
17
Estimated values
3.302.775
3.051.887
4.096.406
407.311
4.801.194
487.463
5.881.347
5.919.974
9.951.929
102.727
Absolute difference (Ã…)
3.302.775
4.096.406
4.801.194
5.881.347
9.951.929
Proximity (%)
92.40%
99.43%
98.49%
99.35%
96.88%
Mean
3.177.331
4.084.758
4.837.912
59.006.606
1.011.231.335
Percentage difference
7.60%
0.57%
1.51%
0.65%
3.12%
20
Estimated values
3.519.576
3.073.354
405.165
406.905
4.874.053
4.907.188
5.981.439
5.974.305
1.000.472
1.036.455
Absolute difference (Ã…)
0.446222
0.0174
0.033135
0.071341
3.598.274
Proximity (%)
87.32%
99.57%
99.32%
99.88%
96.53%
Mean
3.296.465
406.035
48.906.205
597.787.195
101.846.337
Percentage difference
12.68%
0.43%
0.68%
0.12%
3.47%
Table 11 – Comparison between Ni50Ti50XHfX .at% alloys and ribbons based on the models by Zarinejad (2008) and Potapov (1997)
However, there is a clear comprehension that the crystallographic characteristics depend not only on the hafnium content, but they are also sensitive to the peculiarities of the two production processes as well as the typical variables of each one of them: electric arc melting (Arc Melting) and quick solidification Melt Spinning.
Figure 7 – Plot of the probability curve (100 points) of the difference indexes estimated by the statistical models adopted for Zarinejad alloys (2008) and Potapov ribbons (1997)
Depending on how the peculiarities involving each one of the processes are treated, one can directly influence the main properties of the samples such as: homogeneity, amount of residual stress and minimization of oxidation, among others. These physical and mechanic properties, among others are derived from atomic arrangementknown as crystallographic structures (or crystalline structures).
It can be mentioned as variables of these processes: the number of times that the bulk was melted, exposure time to the torch, type of material of the mold , efficiency of the applied vacuum, rotating speed of the copper flywheel in quickly solidified ribbons etc. Hence, all the possibilities of variation and instrumental errors generate conditions of solidification which interfere in the micro structure of the alloys and obtained ribbons.

CONCLUSIONS
The analysis of variance (ANOVA) is an unilateral test based on the FSnedecor table, which after calculating the Fvalue, it measures whether it is inside or outside of the acceptance area, according to the following hypothesis test:
{ H0: Potapov = Zarinejad
H1: there is at least one difference between the means
As it is an OneWay ANOVA, the only factor (independent variable) is the atomic percentage of Hf (.at%). As a rule, in terms of null hypothesis (H0), the calculated F value is inside the area of acceptance, i.e., minor than critical

Otherwise, if H0 is rejected, the alternative hypothesis (H1) will be accepted having necessarily, the Fvalue calculated outside the area of acceptance, i.e., bigger than the critical F.
In this case, we considered that F(5%); DF1 (DFW); DF2 (DFR) = 3,12 (tabulated), according to the standard table for Tukeys Test (=0.05) [5]. According to Table 12, as the calculated F = 3.13 is bigger than the critical tabulated value, we concluded that there is a difference between the averages of both treatment groups: Potapov and Zarinejad. This conclusion is also confirmed by the Pvalue, which is smaller than 0.05. The lower the Pvalue, lower is the possibility for H0 be true. It is important to note that DF1 [horizontal =
Variation source
Sum of sQrs
DF
Mean square
Fcalc
valorP (same)
critical Ftabulated
Between groups:
6739.09
2
3369.55
3.131
0.04968
3.123907449
Within groups:
77488
72
1076.22
Permutation p (n=99999)
Total:
84227.1
74
0.04909
Table FSnedecor – F(5%): 2.74 = 3.12
Table 12 – OneWay ANOVA calculated for two series of samples of the crystalline dimensions of the B19 'monoclinic phase in Ni50Ti50
XHfX .at% alloys and ribbons obtained by conventional and rapid solidifications (Zarinejad and Potapov, respectively)

Between groups (W) = Treatment = Between treatments = Between

Within groups (R) = Residue = Error = Within

DF = degree of freedom

Therefore, the null hypothesis is rejected (H0). However, to ascertain whether this difference is statistically significant, it is necessary to perform a parametric test to certify this significance. This decision was taken based on the normality test of AndersonDarling [6], which predicts as normal the set of 25 data organized in table 9, as shown in figure 8:
Figure 8 – AndersonDarling Normality Test for the percentage indexes
A comparative analysis was made between two different solidification methods (arc melting e melt spin), when applying the TUKEYs Test (through table 13) [7] to, in a complementary form to ANOVA, calculate the minimum significant difference (m. s. d.) there is between the averages in the martensitic crystallographic characteristics, according to the original dimensional data predicted by the adjustment models, shown in table 2 and table 7. It is concluded that the method to obtain the alloys doesnt interfere in the dimensions here evaluated, that is, there is no significant statistical difference. This is due to the fact that the
numerator] is the degree of freedom between groups (DFW)
and that DF2 [vertical = denominator] is the degree of
calculated difference (
) was of 0.75.
freedom of the residues (DFR).
According to equation 3, the m.s.d [] necessary for both groups (data series) to have a significant difference is 19.027.
() = 
(03) 
In this calculation, q [q(5%); k; DF of residues = 2.899943] is a tabulated value (FSnedecor table, =0.05), the QMR is 1076.22 (highlighted in table 12) and the number of repetitions is equal to 25. This count of 25 is contained in table 13, before the summary (resume).
Hf ( at.% ) 
Dimensions 
Zarinejad (2008) 
Potapov (1997) 
8 
a (Ã…) 
2.897725078 
2.995688 
b (Ã…) 
4.129549466 
4.083342 

c (Ã…) 
4.703096765 
4.746384 

Volume (Ã…3) 
54.99855 
57.208961 

( Âº ) 
98.28473246 
100.237795 

11 
a (Ã…) 
2.996283729 
3.012892 
b (Ã…) 
4.119240505 
4.07906 

c (Ã…) 
4.730063875 
4.795981 

Volume (Ã…3) 
56.18361 
57.93075 

( Âº ) 
98.61436363 
100.997707 

14 
a (Ã…) 
3.129822061 
3.031679 
b (Ã…) 
4.113564167 
4.076048 

c (Ã…) 
4.757980318 
4.838146 

Volume (Ã…3) 
57.52923 
58.59493 

( Âº ) 
99.03377477 
101.835911 

17 
a (Ã…) 
3.302774532 
3.051887 
b (Ã…) 
4.096405712 
4.07311 

c (Ã…) 
4.801194093 
4.87463 

Volume (Ã…3) 
58.81347 
59.199742 

( Âº ) 
99.51929314 
102.726977 

20 
a (Ã…) 
3.5195756 
3.073354 
b (Ã…) 
4.0516504 
4.06905 

c (Ã…) 
4.8740532 
4.907188 

Volume (Ã…3) 
59.81439 
59.743049 

( Âº ) 
100.047246 
103.645474 

Resume 
Sum 
843.0616395 
861.829798 
Mean (average) 
33.72246558 
34.47319192 

Standard deviation 
39.50325635 
40.6129109 

Variance 
1560.507263 
1649.408532 

Difference between means 
39.05 40.61 = 0.75072634 
Table 13 – Paired values of the crystallographic characteristics of the B19 martensitic phase in Ni50Ti50XHfX .at% alloys predicted by the fitting
REFERENCES

Zarinejad, M., Y. Liu, and T.J. White, The crystal chemistry of martensite in NiTiHf shape memory alloys. Intermetallics, 2008. 16(7): p. 876883.

Potapov, P., et al., Effect of Hf on the structure of NiTi martensitic alloys. Materials Letters, 1997. 32(4): p. 247250.

3. Rietveld, H., A profile refinement method for nuclear and magnetic structures. Journal of applied Crystallography, 1969. 2(2): p. 6571.

Koch, E., Twinning, in International Tables for Crystallography Volume C: Mathematical, physical and chemical tables. 2006, Springer. p. 1014.

Tukey, J.W., The future of data analysis. The annals of mathematical statistics, 1962. 33(1): p. 167.

Pettitt, A., Testing the normality of several independent samples using the AndersonDarling statistic. Journal of the Royal Statistical Society: Series C (Applied Statistics), 1977. 26(2): p. 156161.

Rojas, I., et al., Analysis of the functional block involved in the design of radial basis function networks. Neural Processing Letters, 2000. 12(1): p. 117.
ATTACHMENTS
DF 
k levels 

(n – k) 
2 
3 
4 
5 
6 
7 
8 
9 
10 
5 
3.64 
4.6 
5.22 
5.67 
6.03 
6.33 
6.58 
6.8 
6.99 
6 
3.46 
4.34 
4.9 
5.3 
5.63 
5.9 
6.12 
6.32 
6.49 
7 
3.34 
4.16 
4.68 
5.06 
5.36 
5.61 
5.82 
6 
6.16 
8 
3.26 
4.04 
4.53 
4.89 
5.17 
5.4 
5.6 
5.77 
5.92 
9 
3.2 
3.95 
4.41 
4.76 
5.02 
5.24 
5.43 
5.59 
5.74 
10 
3.15 
3.88 
4.33 
4.65 
4.91 
5.12 
5.3 
5.46 
5.6 
11 
3.11 
3.82 
4.26 
4.57 
4.82 
5.03 
5.2 
5.35 
5.49 
12 
3.08 
3.77 
4.2 
4.51 
4.75 
4.95 
5.12 
5.27 
5.39 
13 
3.06 
3.73 
4.15 
4.45 
4.69 
4.88 
5.05 
5.19 
5.32 
14 
3.03 
3.7 
4.11 
4.41 
4.64 
4.83 
4.99 
5.13 
5.25 
15 
3.01 
3.67 
4.08 
4.37 
4.59 
4.78 
4.94 
5.08 
5.2 
16 
3 
3.65 
4.05 
4.33 
4.56 
4.74 
4.9 
5.03 
5.15 
17 
2.98 
3.63 
4.02 
4.3 
4.52 
4.7 
4.86 
4.99 
5.11 
18 
2.97 
3.61 
4 
4.28 
4.49 
4.67 
4.82 
4.96 
5.07 
19 
2.96 
3.59 
3.98 
4.25 
4.47 
4.65 
4.79 
4.92 
5.04 
20 
2.95 
3.58 
3.96 
4.23 
4.45 
4.62 
4.77 
4.9 
5.01 
24 
2.92 
3.53 
3.9 
4.17 
4.37 
4.54 
4.68 
4.81 
4.92 
30 
2.89 
3.49 
3.85 
4.1 
4.3 
4.46 
4.6 
4.72 
4.82 
40 
2.86 
3.44 
3.79 
4.04 
4.23 
4.39 
4.52 
4.63 
4.73 
60 
2.83 
3.4 
3.74 
3.98 
4.16 
4.31 
4.44 
4.55 
4.65 
120 
2.8 
3.36 
3.68 
3.92 
4.1 
4.24 
4.36 
4.47 
4.56 
2.77 
3.31 
3.63 
3.86 
4.03 
4.17 
4.29 
4.39 
4.47 
Table 14 – Tukey test table ( = 0.05): degrees of freedom and levels [studentized]
199.5
Table VI 
Degrees of Freedom (DF) in the numerator 

DF in denominator 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
1 
161.45 
215.71 
224.58 
230.16 
233.99 
236.77 
238.88 
240.54 
241.88 

2 
18.51 
19 
19.16 
19.25 
19.3 
19.33 
19.35 
19.37 
19.38 
19.4 
3 
10.13 
9.55 
9.28 
9.12 
9.01 
8.94 
8.89 
8.85 
8.81 
8.79 
4 
7.71 
6.94 
6.59 
6.39 
6.26 
6.16 
6.09 
6.04 
6 
5.96 
5 
6.61 
5.79 
5.41 
5.19 
5.05 
4.95 
4.88 
4.82 
4.77 
4.74 
6 
5.99 
5.14 
4.76 
4.53 
4.39 
4.28 
4.21 
4.15 
4.1 
4.06 
7 
5.59 
4.74 
4.35 
4.12 
3.97 
3.87 
3.79 
3.73 
3.68 
3.64 
8 
5.32 
4.46 
4.07 
3.84 
3.69 
3.58 
3.5 
3.44 
3.39 
3.35 
9 
5.12 
4.26 
3.86 
3.63 
3.48 
3.37 
3.29 
3.23 
3.18 
3.14 
10 
4.96 
4.1 
3.71 
3.48 
3.33 
3.22 
3.14 
3.07 
3.02 
2.98 
11 
4.84 
3.98 
3.59 
3.36 
3.2 
3.09 
3.01 
2.95 
2.9 
2.85 
12 
4.75 
3.89 
3.49 
3.26 
3.11 
3 
2.91 
2.85 
2.8 
2.75 
13 
4.67 
3.81 
3.41 
3.18 
3.03 
2.92 
2.83 
2.77 
2.71 
2.67 
14 
4.6 
3.74 
3.34 
3.11 
2.96 
2.85 
2.76 
2.7 
2.65 
2.6 
15 
4.54 
3.68 
3.29 
3.06 
2.9 
2.79 
2.71 
2.64 
2.59 
2.54 
16 
4.49 
3.63 
3.24 
3.01 
2.85 
2.74 
2.66 
2.59 
2.54 
2.49 
17 
4.45 
3.59 
3.2 
2.96 
2.81 
2.7 
2.61 
2.55 
2.49 
2.45 
18 
4.41 
3.55 
3.16 
2.93 
2.77 
2.66 
2.58 
2.51 
2.46 
2.41 
19 
4.38 
3.52 
3.13 
2.9 
2.74 
2.63 
2.54 
2.48 
2.42 
2.38 
20 
4.35 
3.49 
3.1 
2.87 
2.71 
2.6 
2.51 
2.45 
2.39 
2.35 
21 
4.32 
3.47 
3.07 
2.84 
2.68 
2.57 
2.49 
2.42 
2.37 
2.32 
22 
4.3 
3.44 
3.05 
2.82 
2.66 
2.55 
2.46 
2.4 
2.34 
2.3 
23 
4.28 
3.42 
3.03 
2.8 
2.64 
2.53 
2.44 
2.37 
2.32 
2.27 
24 
4.26 
3.4 
3.01 
2.78 
2.62 
2.51 
2.42 
2.36 
2.3 
2.25 
25 
4.24 
3.39 
2.99 
2.76 
2.6 
2.49 
2.4 
2.34 
2.28 
2.24 
26 
4.23 
3.37 
2.98 
2.74 
2.59 
2.47 
2.39 
2.32 
2.27 
2.22 
27 
4.21 
3.35 
2.96 
2.73 
2.57 
2.46 
2.37 
2.31 
2.25 
2.2 
28 
4.2 
3.34 
2.95 
2.71 
2.56 
2.45 
2.36 
2.29 
2.24 
2.19 
29 
4.18 
3.33 
2.93 
2.7 
2.55 
2.43 
2.35 
2.28 
2.22 
2.18 
30 
4.17 
3.32 
2.92 
2.69 
2.53 
2.42 
2.33 
2.27 
2.21 
2.16 
35 
4.12 
3.27 
2.87 
2.64 
2.49 
2.37 
2.29 
2.22 
2.16 
2.11 
40 
4.08 
3.23 
2.84 
2.61 
2.45 
2.34 
2.25 
2.18 
2.12 
2.08 
45 
4.06 
3.2 
2.81 
2.58 
2.42 
2.31 
2.22 
2.15 
2.1 
2.05 
50 
4.03 
3.18 
2.79 
2.56 
2.4 
2.29 
2.2 
2.13 
2.07 
2.03 
100 
3.94 
3.09 
2.7 
2.46 
2.31 
2.19 
2.1 
2.03 
1.97 
1.93 
Table 15 – F distribution of Snedecor ( = 0.05)