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 
Variety Replication
1 2 3 4
A 6 4 8 6 
B 7 6 6 9 
C 8 5 10 9 
Series 
B 
113 117 126 126 130 129 111 116 112

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

### Step 1: Rank the data
First, we rank each series individually. If there are tied ranks, assign to each tied value the average of the ranks they would have received if they had been different.

#### Series A: 
\[ 57, 59, 62, 63, 64, 65, 55, 58, 57 \]

#### Series B: 
\[ 113, 117, 126, 126, 130, 129, 111, 116, 112 \]

### Step 2: Assign ranks
Assign ranks to each value in the series.

#### Ranks for Series A:
- For \( A = [57, 57, 58, 59, 62, 63, 64, 65, 55] \)
  - Rank: 1 (for 55)
  - Rank: 2.5 (for the two 57s)
  - Rank: 4 (for 58)
  - Rank: 5 (for 59)
  - Rank: 6 (for 62)
  - Rank: 7 (for 63)
  - Rank: 8 (for 64)
  - Rank: 9 (for 65)

Ranks: \[ 2.5, 5, 6, 7, 8, 9, 1, 4, 2.5 \]

#### Ranks for Series B:
- For \( B = [113, 117, 126, 126, 130, 129, 111, 116, 112] \)
  - Rank: 1 (for 111)
  - Rank: 2 (for 112)
  - Rank: 3 (for 113)
  - Rank: 4 (for 116)
  - Rank: 5 (for 117)
  - Rank: 6.5 (for the two 126s)
  - Rank: 8 (for 129)
  - Rank: 9 (for 130)

Ranks: \[ 3, 5, 6.5, 6.5, 9, 8, 1, 4, 2 \]

### Step 3: Calculate the difference in ranks (\(d_i\)) and \(d_i^2\)
Compute the difference between the ranks of each pair, and then square these differences.

| Rank A | Rank B | \( d_i = \text{Rank A} - \text{Rank B} \) | \( d_i^2 \) |
|--------|--------|------------------------------------------|-------------|
| 2.5    | 3      | -0.5                                     | 0.25        |
| 5      | 5      | 0                                        | 0           |
| 6      | 6.5    | -0.5                                     | 0.25        |
| 7      | 6.5    | 0.5                                      | 0.25        |
| 8      | 9      | -1                                       | 1           |
| 9      | 8      | 1                                        | 1           |
| 1      | 1      | 0                                        | 0           |
| 4      | 4      | 0                                        | 0           |
| 2.5    | 2      | 0.5                                      | 0.25        |

### Step 4: Calculate Spearman’s rank correlation coefficient (\(\rho\))
Spearman’s rank correlation coefficient is calculated using the formula:

\[
\rho = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}
\]

Where:
- \( n \) is the number of data points (here, \( n = 9 \))
- \( \sum d_i^2 = 3 \) (sum of the squared differences)

Now, substituting the values:

\[
\rho = 1 - \frac{6 \times 3}{9(9^2 - 1)} = 1 - \frac{18}{9 \times 80} = 1 - \frac{18}{720} = 1 - 0.025 = 0.975
\]

### Final Answer
The Spearman’s rank correlation coefficient between the two series A and B is \( \rho = 0.975 \).

Would you like more details, or do you have any questions?

Here are 5 related questions you might find interesting:
1. How does Spearman's rank correlation coefficient differ from Pearson's correlation coefficient?
2. What are some assumptions behind using Spearman’s rank correlation?
3. How do you interpret the Spearman’s rank correlation coefficient?
4. What happens to the Spearman coefficient if there are many tied ranks in the data?
5. Can Spearman’s rank correlation be used for non-linear relationships?

**Tip:** When dealing with tied ranks in Spearman’s rank correlation, it's important to adjust ranks by assigning the average of the ranks that would have been assigned if they were not tied.

Learn how to calculate Spearman’s rank correlation coefficient between Series A and B using a detailed step-by-step solution. This problem involves ranking data, calculating differences in ranks, and applying the Spearman’s rank correlation formula. Suitable for advanced high school students and college-level statistics.

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