The question is asking to cluster the given data set using the k-means algorithm, with two initial seeds: (4, 6) and (12, 4). The k-means algorithm works by first assigning the points to the nearest centroid, then updating the centroid based on the mean of the points assigned to each cluster, and repeating this process until convergence.

Steps to Solve:

1. Data Points: The data points need to be represented as coordinates. We can use the CGPA and Assessment columns as the two-dimensional coordinates for the k-means clustering. Therefore, each row can be represented as a point:

   • (9.2, 85)
   • (8, 80)
   • (8.5, 81)
   • (6, 45)
   • (6.5, 50)
   • (8.2, 72)
   • (5.8, 38)
   • (8.9, 91)

2. Initial Centroids (Given):

   • Centroid 1: (4, 6)
   • Centroid 2: (12, 4)

3. Distance Calculation (First Assignment Step): Compute the distance between each data point and the centroids using the Euclidean distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ where $x_1, y_1$ is the coordinate of the data point, and $x_2, y_2$ is the coordinate of the centroid.

4. Cluster Assignment: Assign each point to the nearest centroid.

5. Update Centroids: After assigning points, calculate the mean of the points in each cluster to get the new centroids.

Let me perform the calculations for the first iteration. I'll calculate the distance of each data point to both centroids and assign them to the closest one.After calculating the Euclidean distances between the data points and the two initial centroids $(4, 6)$ and $(12, 4)$:

• Distances to Centroid 1 (4, 6):

  • Point (9.2, 85): 79.17
  • Point (8, 80): 74.11
  • Point (8.5, 81): 75.13
  • Point (6, 45): 39.05
  • Point (6.5, 50): 44.07
  • Point (8.2, 72): 66.13
  • Point (5.8, 38): 32.05
  • Point (8.9, 91): 85.14

• Distances to Centroid 2 (12, 4):

  • Point (9.2, 85): 81.05
  • Point (8, 80): 76.11
  • Point (8.5, 81): 77.08
  • Point (6, 45): 41.44
  • Point (6.5, 50): 46.33
  • Point (8.2, 72): 68.11
  • Point (5.8, 38): 34.56
  • Point (8.9, 91): 87.06

Cluster Assignment:

After comparing the distances, all points are currently assigned to Cluster 1 because they are closer to the centroid at $(4, 6)$.

Next, we need to update the centroids based on the mean of the points in each cluster. Since all points are in Cluster 1, the new centroid for Cluster 1 will be the mean of all points, while Cluster 2 does not have any points assigned. I'll compute the updated centroids for the next iteration.After the first iteration, the new centroids are:

• New Centroid 1 (mean of all points assigned to Cluster 1): $(7.64, 67.75)$
• Centroid 2 remains unchanged at $(12, 4)$ since no points were assigned to Cluster 2.

In the next step, we would repeat the process of calculating distances from the data points to these updated centroids and reassign points to clusters.

Would you like to continue with the next iteration or have any questions so far?

Here are 5 questions related to the topic:

1. How does the k-means algorithm handle situations where no points are assigned to a centroid?
2. What is the significance of initializing centroids in k-means clustering?
3. Can the k-means algorithm always guarantee convergence to the optimal solution?
4. How does the choice of the number of clusters (k) impact the outcome of k-means clustering?
5. What are some common methods for initializing centroids in k-means?

Tip: In k-means, it is important to standardize the data when the features have different scales.