data set is 
x : 25,30,20,35,28,40,22,33
y: 10,15, 12,18,14,22,11, 17 spearman rank 

To calculate the **Spearman's Rank Correlation Coefficient**, we follow these steps:

### Step-by-step Procedure:
1. **Rank the Data**: Assign ranks to both \(x\) and \(y\) values.
   - In case of tied ranks, use the average of the ranks.
2. **Calculate \(d_i = R(x_i) - R(y_i)\)**: For each pair, find the difference between the ranks.
3. **Square the Differences**: Compute \(d_i^2\).
4. **Use the Spearman Rank Formula**:
   \[
   r_s = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}
   \]
   Where \(n\) is the number of data points.

### Step 1: Rank the Data
#### For \(x\):
- 20: Rank 1
- 22: Rank 2
- 25: Rank 3
- 28: Rank 4
- 30: Rank 5
- 33: Rank 6
- 35: Rank 7
- 40: Rank 8

#### For \(y\):
- 10: Rank 1
- 11: Rank 2
- 12: Rank 3
- 14: Rank 4
- 15: Rank 5
- 17: Rank 6
- 18: Rank 7
- 22: Rank 8

### Step 2: Calculate \(d_i = R(x_i) - R(y_i)\)
| \(x\) | \(R(x)\) | \(y\) | \(R(y)\) | \(d_i = R(x_i) - R(y_i)\) | \(d_i^2\) |
|------|--------|-----|--------|-----------------------|-----------|
| 25   | 3      | 10  | 1      | 3 - 1 = 2             | 4         |
| 30   | 5      | 15  | 5      | 5 - 5 = 0             | 0         |
| 20   | 1      | 12  | 3      | 1 - 3 = -2            | 4         |
| 35   | 7      | 18  | 7      | 7 - 7 = 0             | 0         |
| 28   | 4      | 14  | 4      | 4 - 4 = 0             | 0         |
| 40   | 8      | 22  | 8      | 8 - 8 = 0             | 0         |
| 22   | 2      | 11  | 2      | 2 - 2 = 0             | 0         |
| 33   | 6      | 17  | 6      | 6 - 6 = 0             | 0         |

### Step 3: Sum of \(d_i^2\)
\[
\sum d_i^2 = 4 + 0 + 4 + 0 + 0 + 0 + 0 + 0 = 8
\]

### Step 4: Spearman Rank Correlation Coefficient Formula
\[
r_s = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}
\]
where \(n = 8\) (since there are 8 data points).

\[
r_s = 1 - \frac{6 \times 8}{8(8^2 - 1)} = 1 - \frac{48}{8(64 - 1)} = 1 - \frac{48}{8 \times 63} = 1 - \frac{48}{504}
\]
\[
r_s = 1 - 0.0952 = 0.9048
\]

### Final Answer:
The **Spearman's Rank Correlation Coefficient** is approximately \(r_s = 0.905\).

---

Would you like more details or have any questions about this calculation?

Here are some related questions to explore:
1. What is the difference between Spearman's and Pearson's correlation coefficients?
2. How are tied ranks handled in Spearman's correlation?
3. What are the limitations of Spearman's correlation?
4. How does Spearman's correlation deal with non-linear relationships?
5. Can Spearman's correlation be used with categorical data?

**Tip:** Spearman's rank correlation is best suited for data that does not meet the assumptions of normality and linearity, as it is non-parametric.

data set is x : 25,30,20,35,28,40,22,33 y: 10,15,12,18,14,22,11,17 spearman rank

Calculate the Spearman's Rank Correlation Coefficient for the dataset x: 25,30,20,35,28,40,22,33 and y: 10,15,12,18,14,22,11,17. This step-by-step guide covers ranking data, calculating rank differences, and applying the Spearman's rank formula. Suitable for students in Grades 10-12.

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