So far, we've performed hierarchical clustering in two dimensions. We used the course grades dataset, selecting only the scores in English and algebra, and visualize them on a scatter plot. There, we developed the concepts of instance distances and distances between clusters. As a refresher, to measure the distance between, say, Catherine and Jenna, we summed up the square differences between the grades in algebra and English. We referred to this as the Euclidean distance and noted that it comes from the Pythagorean theorem. Now what happens if we had another subject, say history? The data would then be in three dimensions. So we would need to extend our Euclidean distance and add the square distance of grades in history. And as you can probably guess, if we had another subject like biology, the distance equation gets extended with the square differences of those grades as well. In this way, we can use Euclidean distance in a space with an arbitrary number of dimensions, although that is not entirely true, as there is something called the curse of dimensionality. But we'll skip that for now and assume we are done with the distances between data items. So what about distances between clusters? That is, linkages. Well, they stay the same as defined in our previous video, depending only on the data item distances, which we now know how to compute. It's now time to try clustering our entire dataset. We eliminate the select columns and scatter plot and first check the data and data table. Here we remind ourselves that this data reports on the grades of seven different subjects. Therefore, each student lives in a seven-dimensional space. I hope they feel fine there. I sometimes feel dizzy in just three dimensions. Like in our previous video, we first need to get the distances between all the pairs of data instances, so we can perform our hierarchical clustering. Just to make sure, I again used Euclidean distance, and perhaps I will normalize the features this time. Let's not forget, I need to do this when variables have different ranges and domains. Now, taking a look at the dendagrams, I see I have three clusters. I would now like to get some intuition on what these clusters actually represent. I could in principle use scatter plot to try to understand what happened, but now I have seven-dimensional data, and just peaking at two dimensions at a time won't really tell me much. But let's take a quick look anyway. Here's the data projected onto the algebra English plane. The students with similar grades in these two subjects are indeed in the same cluster. However, we also see Bill and Ian close together, but in different clusters. Okay, so it seems I'm going to need a different tool to explain the clusters. And just as a hint, we have already used the widget that we will use for cluster explanation in one of our previous videos on data distributions. But before I jump into explaining clusters, I will in our next video show you the clustering of countries using human development index data and use geomaps to visualize the results.