Find the coefficient of correlation using rank method: where x = 32,55,44,60,43,37,43,49,10,20 and y = 40,30,70,20,30,50,72,60,45,25

To find the coefficient of correlation using the rank method (Spearman's Rank Correlation Coefficient), we follow these steps:

1. **Rank the data for both \( x \) and \( y \)**.
2. **Calculate the difference between the ranks** for each pair (\( D = R_x - R_y \)).
3. **Square these differences** (\( D^2 \)).
4. Use the Spearman rank correlation formula:

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

Where:
- \( D \) is the difference in ranks between corresponding values of \( x \) and \( y \),
- \( n \) is the number of data points.

Let’s start by calculating the ranks for \( x \) and \( y \), and then we can compute the correlation coefficient.

### Step 1: Rank the \( x \) and \( y \) values
- \( x = [32, 55, 44, 60, 43, 37, 43, 49, 10, 20] \)
- \( y = [40, 30, 70, 20, 30, 50, 72, 60, 45, 25] \)

Now, I'll compute the ranks of \( x \) and \( y \) values and proceed with the calculation.Here are the ranks and differences for \( x \) and \( y \):

\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & y & R_x & R_y & D & D^2 \\
\hline
32 & 40 & 3.0 & 5.0 & -2.0 & 4.00 \\
55 & 30 & 9.0 & 3.5 & 5.5 & 30.25 \\
44 & 70 & 7.0 & 9.0 & -2.0 & 4.00 \\
60 & 20 & 10.0 & 1.0 & 9.0 & 81.00 \\
43 & 30 & 5.5 & 3.5 & 2.0 & 4.00 \\
37 & 50 & 4.0 & 7.0 & -3.0 & 9.00 \\
43 & 72 & 5.5 & 10.0 & -4.5 & 20.25 \\
49 & 60 & 8.0 & 8.0 & 0.0 & 0.00 \\
10 & 45 & 1.0 & 6.0 & -5.0 & 25.00 \\
20 & 25 & 2.0 & 2.0 & 0.0 & 0.00 \\
\hline
\end{array}
\]

The sum of squared differences \( \sum D^2 \) is \( 177.5 \). 

### Spearman's rank correlation coefficient is:

\[
r_s = 1 - \frac{6 \times 177.5}{10 \times (10^2 - 1)} = -0.076
\]

Thus, the Spearman rank correlation coefficient is approximately **-0.076**. This indicates a very weak negative correlation between \( x \) and \( y \).

Would you like any further details or clarifications?

### Related Questions:
1. How do you interpret the value of the Spearman rank correlation coefficient?
2. What is the difference between Spearman's rank correlation and Pearson's correlation?
3. How do you handle tied ranks in Spearman's correlation?
4. Can Spearman's correlation be used for non-linear relationships?
5. What are other methods to measure correlation between two variables?

### Tip:
Spearman's correlation is particularly useful when the relationship between variables is not linear or when the data contain outliers.

Learn how to find the Spearman rank correlation coefficient for the given data sets x = [32, 55, 44, 60, 43, 37, 43, 49, 10, 20] and y = [40, 30, 70, 20, 30, 50, 72, 60, 45, 25]. This step-by-step guide covers ranking, calculating differences, and applying the formula to determine the correlation. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation