Find Spearman’s rank correlation coefficient for the following data.
X	75	88	95	70	60	80	81	50
Y	120	134	150	115	130	140	142	100

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

### 1. Rank the Data
Assign ranks to the \( X \) and \( Y \) values.

- For \( X \): 
  \[
  X: \ 75, \ 88, \ 95, \ 70, \ 60, \ 80, \ 81, \ 50 
  \]
  **Ranks**: (Rank 1 = smallest)
  \[
  \text{Ranks of } X: \ 4, \ 6, \ 8, \ 3, \ 2, \ 5, \ 7, \ 1
  \]

- For \( Y \):
  \[
  Y: \ 120, \ 134, \ 150, \ 115, \ 130, \ 140, \ 142, \ 100
  \]
  **Ranks**:
  \[
  \text{Ranks of } Y: \ 3, \ 5, \ 8, \ 2, \ 4, \ 6, \ 7, \ 1
  \]

### 2. Calculate the Differences in Ranks
Calculate the difference \( d_i \) between the ranks of each pair and then square these differences.

\[
\begin{array}{|c|c|c|c|c|}
\hline
\textbf{X} & \textbf{Rank of X} & \textbf{Y} & \textbf{Rank of Y} & d_i = \text{Rank of X} - \text{Rank of Y} & d_i^2 \\
\hline
75 & 4 & 120 & 3 & 1 & 1 \\
88 & 6 & 134 & 5 & 1 & 1 \\
95 & 8 & 150 & 8 & 0 & 0 \\
70 & 3 & 115 & 2 & 1 & 1 \\
60 & 2 & 130 & 4 & -2 & 4 \\
80 & 5 & 140 & 6 & -1 & 1 \\
81 & 7 & 142 & 7 & 0 & 0 \\
50 & 1 & 100 & 1 & 0 & 0 \\
\hline
\end{array}
\]

Sum of squared differences \( \sum d_i^2 \):
\[
\sum d_i^2 = 1 + 1 + 0 + 1 + 4 + 1 + 0 + 0 = 8
\]

### 3. Calculate Spearman's Rank Correlation Coefficient
Use the formula:
\[
\rho = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}
\]
where \( n \) is the number of pairs (in this case, \( n = 8 \)).

Substitute the values:
\[
\rho = 1 - \frac{6 \times 8}{8(8^2 - 1)} = 1 - \frac{48}{8 \times 63} = 1 - \frac{48}{504} = 1 - 0.0952 = 0.9048
\]

### Final Answer:
Spearman’s rank correlation coefficient \( \rho \) is approximately **0.9048**.

Would you like more details or have any questions?

---

Here are five related questions:
1. How does Spearman's rank correlation differ from Pearson's correlation coefficient?
2. What is the significance of a Spearman's rank correlation coefficient close to +1 or -1?
3. Can Spearman's rank correlation be used for data with tied ranks? How?
4. How do outliers impact Spearman's rank correlation coefficient compared to Pearson's?
5. How can Spearman's rank correlation coefficient be used in non-parametric statistics?

**Tip:** Spearman's rank correlation is particularly useful when your data doesn't meet the assumptions of Pearson's correlation, such as normality.

Learn how to find Spearman’s rank correlation coefficient for the given data sets of X and Y values. Follow step-by-step instructions including ranking the data, calculating differences in ranks, and applying the formula. Understand the significance of Spearman’s rank correlation in statistics. Suitable for advanced high school and college 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