Calculate Spearman’s rank correlation coefficient between the series A and B depicted in table:
Series 
A 
57 59 62 63 64 65 55 58 57 
Series 
B 
113 117 126 126 130 129 111 116 112

To calculate Spearman's rank correlation coefficient between Series A and Series B, we need to follow these steps:

### Step 1: Assign ranks to the data points in each series.
Assign ranks to each value in Series A and Series B, based on their values. If two or more data points are tied, assign the average rank for those tied values.

**Series A:**

| Values in A | Rank of A |
|-------------|-----------|
| 57          | 2.5       |
| 59          | 5         |
| 62          | 6         |
| 63          | 7         |
| 64          | 8         |
| 65          | 9         |
| 55          | 1         |
| 58          | 4         |
| 57          | 2.5       |

**Series B:**

| Values in B | Rank of B |
|-------------|-----------|
| 113         | 3         |
| 117         | 6         |
| 126         | 8.5       |
| 126         | 8.5       |
| 130         | 9         |
| 129         | 7         |
| 111         | 1         |
| 116         | 5         |
| 112         | 2         |

### Step 2: Calculate the difference in ranks \((d_i)\) and square the differences \((d_i^2)\).

| \(A_i\) | Rank of A | \(B_i\) | Rank of B | \(d_i = \text{Rank of A} - \text{Rank of B}\) | \(d_i^2\) |
|---------|-----------|---------|-----------|---------------------------------------------|-----------|
| 57      | 2.5       | 113     | 3         | -0.5                                        | 0.25      |
| 59      | 5         | 117     | 6         | -1                                          | 1         |
| 62      | 6         | 126     | 8.5       | -2.5                                        | 6.25      |
| 63      | 7         | 126     | 8.5       | -1.5                                        | 2.25      |
| 64      | 8         | 130     | 9         | -1                                          | 1         |
| 65      | 9         | 129     | 7         | 2                                           | 4         |
| 55      | 1         | 111     | 1         | 0                                           | 0         |
| 58      | 4         | 116     | 5         | -1                                          | 1         |
| 57      | 2.5       | 112     | 2         | 0.5                                         | 0.25      |

### Step 3: Calculate the sum of \(d_i^2\).
\[
\sum d_i^2 = 0.25 + 1 + 6.25 + 2.25 + 1 + 4 + 0 + 1 + 0.25 = 16
\]

### Step 4: Apply the formula for Spearman's rank correlation coefficient:
\[
\rho = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}
\]
where \(n\) is the number of data points.

For our case, \(n = 9\) (since there are 9 pairs of data points), so:
\[
\rho = 1 - \frac{6 \times 16}{9(9^2 - 1)} = 1 - \frac{96}{9(81 - 1)} = 1 - \frac{96}{9 \times 80} = 1 - \frac{96}{720} = 1 - 0.1333 = 0.8667
\]

### Conclusion:
The Spearman's rank correlation coefficient is approximately **0.867**.

Would you like more details on any step or have any questions?

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Here are five related questions:
1. How is the Pearson correlation coefficient different from the Spearman rank correlation coefficient?
2. What happens when there are ties in the ranks in Spearman’s correlation?
3. Can Spearman’s rank correlation coefficient be used with ordinal data?
4. What are the advantages of using rank-based correlation methods like Spearman’s over Pearson’s?
5. How does the Spearman correlation coefficient handle non-linear relationships?

**Tip:** Spearman's correlation is ideal for data that is not normally distributed or when dealing with ordinal variables.

Calculate Spearman’s rank correlation coefficient between the series A and B depicted in table:
Series A: 57, 59, 62, 63, 64, 65, 55, 58, 57
Series B: 113, 117, 126, 126, 130, 129, 111, 116, 112

Learn how to calculate Spearman’s rank correlation coefficient between two data series. Using ranks of Series A (57, 59, 62, etc.) and Series B (113, 117, 126, etc.), follow the step-by-step calculation with a detailed solution and formula application. Suitable for students in Grades 10-12.

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