Normally I expect most of the variance or at least a quarter of the variance to be explained in the first 2 dimensions of the PCA. However, recently I came across a PCA where the first was only about 12% and the 2 about 6%. What does this mean exactly? Does this mean that the PCA in question is not good enough to explain most of the variance?
The PCA is not good or bad. The values you have, where two principal components do not explain a big part of variance, mean that the data is far from being near a 2 dimensional subspace.
I can understand your data have many dimensions, as the variance of the second component only explains 6% and rest of components must explain even less each.
Here is an example. I create 1000 points in 5 dimensions with center (0,0,0,0,0) and different covariance matrices. In the first case, they are all distributed equally along each of the 5 dimensions. In the second case, points are also in 5 dimensions, but mostly near a plane in 2d. You can see the explained variance of each component.
data_5d = mvrnorm(1000, c(0,0,0,0,0), diag(5)) summary(prcomp(data_5d)) # Importance of components: # PC1 PC2 PC3 PC4 PC5 # Standard deviation 1.027 1.0064 0.9928 0.9604 0.9511 # Proportion of Variance 0.216 0.2076 0.2020 0.1890 0.1854 # Cumulative Proportion 0.216 0.4236 0.6256 0.8146 1.0000 data_almost_2d = mvrnorm(1000, c(0,0,0,0,0), diag(c(1,1,0.1,0.1,0.1))) summary(prcomp(data_almost_2d)) # Importance of components: # PC1 PC2 PC3 PC4 PC5 # Standard deviation 1.0074 0.9724 0.32140 0.30981 0.30016 # Proportion of Variance 0.4511 0.4203 0.04591 0.04266 0.04005 # Cumulative Proportion 0.4511 0.8714 0.91729 0.95995 1.00000