Which of the following statements regarding variance explained by principal components is true?

The statement that together, the first four principal components explain 100% of the variance is true. In the context of Principal Component Analysis (PCA), the principal components are linear combinations of the original variables aimed at capturing the maximum variance in the data.

The first principal component always explains the most variance, followed by the second, then the third, and so on. When you consider multiple principal components, while they collectively aim to account for the total variance present in the dataset, it is important to note that the total variance explained by all principal components will always sum up to the total variance of the original data. Therefore, as you accumulate more principal components, the percentage of variance explained approaches 100%.

In practice, specifically saying that the first four principal components explain 100% of the variance means that these four components account for all the information and variability in the data, without losing anything from the original set. This scenario typically occurs in cases where the original variables can be fully represented by the first four components, which can happen in low-dimensional datasets or datasets with certain characteristics.

Other statements fail to represent the nature of PCA accurately:

• The idea that only the first principal component explains all the variance is misleading, as multiple components together can