= Solution
The three retained covariance components explain
$$
\frac{155.2+30.7+15.6}{218.4}=0.9226\ldots,
$$
leaving only $7.7\%$ of the raw total <variance>. \b[Three components are a reasonable covariance-based descriptive compression], if that amount of discarded variation is acceptable for the intended use. Even two explain about $85.1\%$, so three is a useful choice rather than a uniquely forced one.
For the correlation analysis,
$$
\frac{4.01+1.64+1.13}{9}=0.7533\ldots,
$$
so three retain $75.3\%$ and discard $24.7\%$ of standardized variation. \b[Three can provide an interpretable summary, but the supplied information does not justify treating them as sufficient for every purpose.] Tasks requiring high standardized reconstruction accuracy may need more.
Each of the three displayed correlation <eigenvalues> exceeds the average <eigenvalue> one, but that alone does not show all remaining <eigenvalues> are below one. Their sum is $9-6.78=2.22$, and a fourth <eigenvalue> just above one is compatible with both that sum and the ordering. A complete <eigenvalue> sequence, a plot of successive <eigenvalues>, and the stability and interpretation of the directions would help decide the cutoff. With only 25 countries, estimated directions also have sampling uncertainty. The retained covariance and correlation dimensions should therefore be judged against their own <variance> criterion and substantive aim, not chosen solely because three columns were printed.
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