1820. The distribution of features $X, Y$, and $Z$ among members of a certain statistical population is given in the following table:

| $X$ | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 5 | 5 | 6 | 7 | 7 | 8 |
| :---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: |
| $Y$ | 5 | 3 | 2 | 4 | 5 | 5 | 6 | 7 | 8 | 8 | 8 | 7 | 8 | 10 |
| $\rightarrow ⿱ 一 一$ | -2 | -4 | -1 | -3 | -3 | -4 | -4 | -6 | -7 | -6 | -9 | -8 | -8 | -10 |
| $m$ | 1 | 2 | 3 | 3 | 2 | 4 | 3 | 2 | 3 | 1 | 2 | 1 | 2 | 1 |

Find the combined linear correlation coefficient $R_{Z, X Y}$ and the partial correlation coefficients $r_{Y}(Z, X)$ and $r_{X}(Z, Y)$.
Solution. Since $R_{Z, X Y}, r_{X}(Y, Z)$, and $r_{Y}(X, Z)$ are defined through the coefficients of linear correlation between the features $X$ and $Y$, $X$ and $Z$, and $Y$ and $Z$, we first find these. All necessary calculations are visible from the table (see table on page 414):

$$
\begin{gathered}
\overline{X^{\prime}}=\frac{-9}{30}=-0.3 ; \quad \overline{Y^{\prime}}=\frac{22}{30}=-0.733 ; \quad \overline{Z^{\prime}}=\frac{3}{30}=0.1 \\
\overline{\left(X^{\prime}\right)^{2}}=\frac{115}{30}=3.833 ; \quad \overline{\left(Y^{\prime}\right)^{2}}=\frac{150}{30}=5 ; \quad \overline{\left(Z^{\prime}\right)^{2}}=\frac{185}{30}=6.167 \\
\sigma_{X^{\prime}}^{2}=3.833-(-0.3)^{2}=3.743 ; \sigma_{Y}^{2}=5-(0.733)^{2}=4.463 \\
\sigma_{Z^{\prime}}^{2}=6.167-(0.1)^{2}=6.157 ; \sigma_{X}=1.934 ; \sigma_{Y^{\prime}}=2.112 \\
\overline{\sigma_{Z^{\prime}}}=2.481 ; \\
\overline{X^{\prime} Y^{\prime}}=\frac{101}{30}=3.367 ; \overline{X^{\prime} Z^{\prime}}=\frac{-132}{30}=-4.4 ; \overline{Y^{\prime} Z^{\prime}}= \\
=\frac{-141}{30}=-4.7 ; \quad r(X, Y)=\frac{3.367-(-0.3) \cdot 0.733}{1.934 \cdot 2.112}=0.879
\end{gathered}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_1b9a1c1720bae4af5bc5g-414.jpg?height=1736&width=1079&top_left_y=57&top_left_x=51)

$$
\begin{aligned}
& r(X, Z)=\frac{-4.4-(-0.3)(0.1)}{1.934 \cdot 2.481}=-0.910 ; r(Y, Z)= \\
& \quad=\frac{-4.7-0.73 \cdot 0.1}{2.112 \cdot 2.481}=-0.901 ; r_{X}(Y, Z)= \\
& \quad=-0.163 ; r_{Y}(X, Z)=-0.570 ; R_{Z, X Y}=0.935
\end{aligned}
$$