#### 3.1.1. Univariate Analysis

For the univariate analysis, it was possible to obtain the results presented in

Table 5. This analysis, although simple to be made, allows to know the characteristics and tendencies of the thermal insulating mortars considered in the sample.

It was seen that

AC,

Ed, and

TK were influenced by the presence of extreme values, verified by the distance from the median to the average and confirmed by the difference of the 25 % and 75 % quartiles, when compared with the minimum and maximum values, showing a significant variability of values on these materials. Regarding mode, the

Cons,

AC,

w/p,

Ed,

fc, and

ft presented multiple modes, indicating the probable influence of the classifications T1 and T2 of EN 998-1 [

7] on the thermal performance that manufacturers declare.

The variance of

Cons (Consistency),

BDf (Bulk density fresh-state), and Λ was the lowest and thus less discriminant. Of all the variables, ten are characterised by low dispersion (coefficient of variation < 50%), the remaining showing more dispersion (

fu,

Ed,

fc,

ft, and

TK), with

TK having a high coefficient of variation, as expected [

98].

For the asymmetry, the variables Bar (Powder bulk density), Cons, BDf, BDh (Bulk density hardened-state), ft (flexural strength), and λ_{10°C,28days} presented negative values, meaning that current thermal insulating mortars tend to present a concentration towards higher values on those variables.

#### 3.1.2. Bivariate Analysis

The matrix with the linear correlation coefficients between the different pairs of variables is presented in

Table 6. The correlations considered significant (

p-value < 0.05) are presented in bold, where cells with thick black lines indicate the highest positive correlations (>0.80), and cells with dashed-lines indicate the highest negative correlations.

It should be noted that correlations marked in bold, due to

p-value < 0.05, are relevant since the higher than that threshold the

p-value is, the less the observed relationship between variables might be a reliable indicator of their relation in the population [

86,

90], limiting the relevance of the former.

In this analysis, it was also built a scatterplot matrix for all pairs of variables,

Figure 1 briefly presenting an extract of the full matrix. In that Figure, the correlations between variables are shown in the upper right portion, the values’ distribution for each variable is shown on its diagonal, and, in the lower-left portion, the values’ scatter, with a line of the best linear correlation (in

blue) and a best-fit line for identifying other types of possible regression (in

orange), is shown. This allowed us to visualise the correlations and the scatter of the values (e.g., outliers).

#### 3.1.3. Discussion of the Uni and Bivariate Analysis

From

Table 5, it is seen that the average value of

fc is ~ 1.33 MPa and of

fu is ~ 0.16 MPa, with both properties presenting a higher concentration of formulations (positive asymmetry) on lower values. This indicates the influence that a small number of mortars with higher values can present. It is also possible to see that, as already known [

25,

69], these mortars tend to present lower mechanical performance than current cement-based mortars, which seems associated with the inclusion of the nanomaterial silica aerogel [

99]. Something that is also noted is the need for an increased

w/p ratio when compared to conventional cement-based mortars [

100], which can also influence their physical and mechanical performance due to an increased porosity.

For the physical characteristics (

λ_{10°C,dry},

W and

Λ) it was found, as shown in

Table 5, that the average values are of 0.122 W·m

^{−1}·K

^{−1}; 0.479 kg·m

^{−2}·min

^{−0.5} and 2.0 × 10

^{−11} kg·m

^{−2}·s

^{−1}·Pa

^{−1}. These are results with lower

λ_{10°C,dry} and

Λ than conventional cement-based mortars, but with higher

W [

101,

102]. Both

λ_{10°C,dry} and

Λ present interesting performance, but

W can be further improved. All these characteristics present positive asymmetries, but, due to the high degree of magnitudes, the higher values influence the average performance. These results also show that for the average thermal insulation mortar, although presenting higher mechanical performance than required by the EN 998-1 [

7] (fc ≥ 0.40 MPa), the associated thermal conductivities are also high (classified as a T2, according to the same standard), even more so if

λ is compared with EPS (λ ~ 0.032 W·m

^{−1}·K

^{−1} [

30]).

From this analysis, it was verified that these mortars still have a high margin of development if the objective is to decrease their λ and W while maintaining high Λ and mechanical behaviour. However, the use of formulations incorporating nanomaterials significantly reduced their thermal conductivities, allowing us to obtain mortars with λ below classic thermal insulating materials.

In the bivariate analysis and using the matrices of linear correlation and of scatterplots, it was observed how the different variables relate to each other. The linear correlation coefficients ranged from ~ 0 (fu and Λ) to |1| (λ_{10°C,28days}, and λ_{10°C,dry}) and, in most cases, with p-value < 0.05, increasing the confidence in accepting the observed result as representative of all thermal insulating mortars, indicating that independently of the insulation aggregate, nanomaterial or not, the correlations were maintained.

The bivariate analysis showed interesting correlations:

BDh presents a high correlation (0.91) with

BDf, this being somehow expected, as it was also verified by Soares [

68]. However, some other significant correlations were also identified: for the

λ_{10°C,28days}, high correlations (≥ 0.80) were found with

BDf,

BDh,

Ed, and

ft, which is an interesting behaviour. Although, for the bulk density values, this was a known fact [

68,

103,

104], it is a new finding that

Ed and

ft also showed high correlations. This can be related to a less porous microstructure that better conducts the heat-flux [

105], increasing the thermal conductivity. This behaviour was also verified for other types of mortars by Badache et al. [

106] and is something to consider when designing new formulations.

For the dry state (λ_{10°C,dry}), the correlation with BDh and BDf was also high, but it was higher with λ_{10°C,28days}. So, despite the drying process in which the free water lost is unpredictable, a very high correlation was found between λ_{10°C,dry} and λ_{10°C,28days.}

The behaviour of the variable

w/p was expected (inverse with other variables). This fact agrees with what is known, since the increase of

w/p, to improve the workability, reduces

fc,

ft, and

BD_{H}, because of the increase in voids, capillaries, and pores in the microstructure, when the free water evaporates [

107,

108]. This would be expected to correlate well with a decrease in the thermal conductivity values (due to the low thermal conductivity of the air), but that was not verified. Some correlation between

W and

Λ was also expected, due to the presence of capillaries, as well as between

fc and

fu, as a result of the internal cohesion, but that was not verified. As for

fc, it was correlated with

Ed and

ft, as expected [

109], since the increment of one increases the others.

Figure 1 complemented the matrix of linear correlation, allowing us to visualise the values’ dispersion for all formulations. This way, it was possible to see that all pairs of variables did not show distribution patterns other than linear, and that outlier behaviours were not significant.

With this analysis, it was verified that the use of nanomaterials in thermal insulating mortars leads to a significant lowering of BDf, BDh, λ_{10°C,dry}, and λ_{10°C,28days}, however, it also lowered fc and fu. This seems related to the higher degree of kneading water needed to achieve an adequate Cons (associated with workability) that is related with an increased porosity and microstructural fragility.