4.2.3. The Correlation Analysis

According to above analyses, it is noticed that the toughness and hardness of the weld beads were relative to the grain size. To analyze the relations between grain size, and toughness as well as the hardness, corresponding correlation analyses had been also conducted.

Firstly, the nine weld beads can be ranked according to their grain sizes as shown in Figure 4. The weld bead with the smallest grain size can be marked as "Excellent", while the weld bead with the largest grain size was marked as "Poor", the other weld beads can be considered as "Good" and "Medium". Also, because the ranking values, toughness and hardness values had different value ranges, the normalized data processing must be conducted, in other words, the grain sizes, elongations and HRB values in the tensile testing were processed in between 0 and 1. The final results can be seen in Figure 7.

**Figure 7.** The variation tendency of grain size, toughness and hardness.

The vertical axis was the normalized parameter value whose range was between 0 and 1, the horizontal axis denoted the weld bead of L1–L9. It can be seen that the toughness, hardness and grain size of the weld beads had the approximately the same variation tendency.

Then, the correlation coefficients of above relation can also be calculated following the Equation (4), and the larger value of correlation coefficient denotes a greater degree of correlation.

$$r(\mathbf{X}, \mathbf{Y}) = \frac{Cov(\mathbf{X}, \mathbf{Y})}{\sqrt{Var[\mathbf{X}]Var[\mathbf{Y}]}} \tag{4}$$

where *r*(*X*,*Y*) referred to the correlation between *X* and *Y*, *Cov*(*X*,*Y*) denoted that the Covariance of *X* and *Y*, and *Var*[*X*] and *Var*[*Y*], respectively, referred the variances of *X* and *Y*. The correlation coefficient between the ranking by grain sizes and the elongations of weld bead in the tensile test was calculated to be 77.56%, and the corresponding values between the ranking by grain sizes and hardness values was 84.35%, which can be considered as significant correlation. It can be concluded that a smaller grain size meant better toughness and higher hardness. Hence, in the following multi-parameter analysis, only considering the grain size should be enough.
