45 70 65 30 90 40 50 75 85 60
and 
35 90 70 40 95 40 60 80 80 50

calculate spearman's coefficient

To calculate Spearman's rank correlation coefficient, we follow these steps:

### 1. Rank the data
We need to assign ranks to both sets of data. For ties, assign the average of the ranks for the tied values.

#### First Dataset: 45, 70, 65, 30, 90, 40, 50, 75, 85, 60

- 30 → Rank 1
- 40 → Rank 2
- 45 → Rank 3
- 50 → Rank 4
- 60 → Rank 5
- 65 → Rank 6
- 70 → Rank 7
- 75 → Rank 8
- 85 → Rank 9
- 90 → Rank 10

#### Second Dataset: 35, 90, 70, 40, 95, 40, 60, 80, 80, 50

- 35 → Rank 1
- 40 → (Average of 2nd and 3rd rank) = 2.5
- 40 → (Tied with the other 40) = 2.5
- 50 → Rank 4
- 60 → Rank 5
- 70 → Rank 6
- 80 → (Average of 7th and 8th rank) = 7.5
- 80 → (Tied with the other 80) = 7.5
- 90 → Rank 9
- 95 → Rank 10

### 2. Calculate \(d_i\) (difference between ranks)

Let's calculate the difference in ranks for each pair of data points.

| Original Data 1 | Rank 1 | Original Data 2 | Rank 2 | \(d_i = R1 - R2\) | \(d_i^2\) |
|-----------------|--------|-----------------|--------|-------------------|-----------|
| 45              | 3      | 35              | 1      | 2                 | 4         |
| 70              | 7      | 90              | 9      | -2                | 4         |
| 65              | 6      | 70              | 6      | 0                 | 0         |
| 30              | 1      | 40              | 2.5    | -1.5              | 2.25      |
| 90              | 10     | 95              | 10     | 0                 | 0         |
| 40              | 2      | 40              | 2.5    | -0.5              | 0.25      |
| 50              | 4      | 60              | 5      | -1                | 1         |
| 75              | 8      | 80              | 7.5    | 0.5               | 0.25      |
| 85              | 9      | 80              | 7.5    | 1.5               | 2.25      |
| 60              | 5      | 50              | 4      | 1                 | 1         |

### 3. Apply the Spearman's rank correlation formula

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

Where:
- \( \sum d_i^2 \) is the sum of the squared differences
- \( n \) is the number of data points (in this case, \( n = 10 \))

#### Sum of \(d_i^2\):

\[
\sum d_i^2 = 4 + 4 + 0 + 2.25 + 0 + 0.25 + 1 + 0.25 + 2.25 + 1 = 15
\]

#### Substituting values into the formula:

\[
r_s = 1 - \frac{6 \times 15}{10(10^2 - 1)}
\]

\[
r_s = 1 - \frac{90}{990}
\]

\[
r_s = 1 - 0.0909 = 0.9091
\]

### Spearman's rank correlation coefficient (\(r_s\)) = **0.909**

This suggests a strong positive correlation between the two sets of data.

---

Would you like further details on this, or have any other questions?

Here are 5 related questions to expand on this:

1. What is the difference between Spearman's and Pearson's correlation?
2. How does ranking affect the correlation when data contains ties?
3. Can Spearman's correlation be applied to non-numerical data?
4. How is the significance of Spearman's coefficient tested?
5. What are some real-world applications of Spearman's correlation?

**Tip:** Spearman's rank correlation is particularly useful when data doesn't meet the assumptions of linearity, making it robust for non-parametric analysis.

The query submitted by the user: 45 70 65 30 90 40 50 75 85 60 and 35 90 70 40 95 40 60 80 80 50. Calculate Spearman's coefficient.

Learn how to calculate Spearman's rank correlation coefficient for the data sets [45, 70, 65, 30, 90, 40, 50, 75, 85, 60] and [35, 90, 70, 40, 95, 40, 60, 80, 80, 50]. This example demonstrates ranking, dealing with ties, and applying the formula to find a strong positive correlation (r_s = 0.909). Suitable for advanced high school and undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation