Cluster the following set of data using k-means algorithm with initial value of objects 2 and 5 with the coordinate values (4,6) and (12,4) as initial seeds CGPA Assessment Project Submitted Result 1 9.2 85 8 Pass 2 8 80 7 Pass 3 8.5 81 8 Pass 4 6 45 5 Fail 5 6.5 50 4 Fail 6 8.2 72 7 Pass 7 5.8 38 5 Fail 8 8.9 91 9 Pass

Let's work through the clustering of the provided data set using the **k-means algorithm**, with the initial centroids (seeds) being objects 2 and 5, with coordinate values (4,6) and (12,4).

### Step 1: Convert the Data into Coordinates
We will use the given attributes to represent each object as a vector. For this case, let's consider each object as a vector based on **CGPA** and **Assessment** scores.

Given the dataLet's work through the clustering of the provided data set using the **k-means algorithm**, with the initial centroids (seeds) being objects 2 and 5, with coordinate values (4,6) and (12,4).

### Step 1: Convert the Data into Coordinates
We will use the given attributes to represent each object as a vector. For this case, let's consider each object as a vector based on **CGPA** and **Assessment** scores.

Given the 

| Object | CGPA | Assessment |
|--------|------|------------|
| 1      | 9.2  | 85         |
| 2      | 8    | 80         |
| 3      | 8.5  | 81         |
| 4      | 6    | 45         |
| 5      | 6.5  | 50         |
| 6      | 8.2  | 72         |
| 7      | 5.8  | 38         |
| 8      | 8.9  | 91         |

We will cluster based on these two variables: CGPA and Assessment.

The initial seeds are:
- Seed 1 (from Object 2): \( (4,6) \)
- Seed 2 (from Object 5): \( (12,4) \)

### Step 2: Calculate Euclidean Distance
We compute the Euclidean distance between each point and the centroids.

The formula for Euclidean distance is:
\[
d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}
\]

#### For each object and centroid, we calculate:

#### Distances to Centroid 1 (4,6):
\[
d_{i1} = \sqrt{(CGPA_i - 4)^2 + (Assessment_i - 6)^2}
\]

#### Distances to Centroid 2 (12,4):
\[
d_{i2} = \sqrt{(CGPA_i - 12)^2 + (Assessment_i - 4)^2}
\]

Now, we proceed to assign each object to the nearest centroid based on the distances. Let me calculate these distances and assign clusters for the first iteration.### First Iteration Results:

For each object, the distances to the two centroids are as follows:

| Object | Distance to Centroid 1 (4,6) | Distance to Centroid 2 (12,4) | Cluster Assignment |
|--------|-------------------------------|-------------------------------|-------------------|
| 1      | 79.17                          | 81.05                          | 1                 |
| 2      | 74.11                          | 76.11                          | 1                 |
| 3      | 75.13                          | 77.08                          | 1                 |
| 4      | 39.05                          | 41.44                          | 1                 |
| 5      | 44.07                          | 46.33                          | 1                 |
| 6      | 66.13                          | 68.11                          | 1                 |
| 7      | 32.05                          | 34.56                          | 1                 |
| 8      | 85.14                          | 87.06                          | 1                 |

Since all points were assigned to **Cluster 1**, this means no points are closer to the initial Centroid 2 (12,4). This requires recalculating the centroids and repeating the process. Would you like to proceed with recalculating centroids for further iterations?

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**Questions for further exploration:**
1. How do the centroids change after each iteration?
2. What if we use different initial seeds for k-means clustering?
3. How does the distance between points and centroids impact cluster assignment?
4. How many iterations does k-means typically take to converge?
5. Can you visualize the clustering results to better understand the data?

**Tip:** In k-means clustering, initial centroid selection can significantly affect the outcome, leading to different clusters depending on the starting points.

Explore the step-by-step process of clustering CGPA and Assessment data using the k-means algorithm with initial seeds at coordinates (4,6) and (12,4). This solution demonstrates how to calculate Euclidean distances and assign clusters based on proximity to centroids.

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