Compute Spearman’s rho for the following data : Data 1 : 86, 80, 82, 79, 76, 63, 74, 65, 70, 69 Data 2 : 70, 79, 74, 65, 80, 90, 82, 85, 89, 84.

1. Introduction

Spearman's rank correlation coefficient, denoted by ( \rho ) (rho), is a non-parametric measure of correlation that assesses the strength and direction of the monotonic relationship between two variables. Unlike Pearson's correlation coefficient, Spearman's rho does not require the assumption of linearity and is suitable for ordinal or ranked data.

2. Given Data

Let's denote Data 1 as ( X ) and Data 2 as ( Y ):

Data 1: 86, 80, 82, 79, 76, 63, 74, 65, 70, 69

Data 2: 70, 79, 74, 65, 80, 90, 82, 85, 89, 84

3. Rank the Data

To compute Spearman's rho, we first rank the data in each dataset from lowest to highest, assigning ranks based on their positions. Tied ranks are assigned the average of the ranks they would occupy if not tied.

Data 1 ranks:
63 -> 1
65 -> 2.5
69 -> 4
70 -> 5
74 -> 6.5
76 -> 7
79 -> 8.5
80 -> 10
82 -> 11
86 -> 12

Data 2 ranks:
65 -> 1
70 -> 2
74 -> 3
79 -> 4
80 -> 5
82 -> 6
84 -> 7
85 -> 8
89 -> 9
90 -> 10

4. Calculate the Differences in Ranks

Next, we calculate the differences (( d )) between the ranks of corresponding values in Data 1 and Data 2.

( d_i = Rank(X_i) – Rank(Y_i) )

( d_1 = 1 – 2 = -1 )
( d_2 = 2.5 – 4 = -1.5 )
( d_3 = 4 – 3 = 1 )
( d_4 = 5 – 1 = 4 )
( d_5 = 6.5 – 5 = 1.5 )
( d_6 = 7 – 6 = 1 )
( d_7 = 8.5 – 7 = 1.5 )
( d_8 = 10 – 8 = 2 )
( d9 = 11 – 9 = 2 )
( d{10} = 12 – 10 = 2 )

5. Calculate Spearman's Rho

Spearman's rho (( \rho )) is calculated using the formula:

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

Where ( n ) is the number of pairs of observations.

Substituting the values:

[ \rho = 1 – \frac{6( (-1)^2 + (-1.5)^2 + 1^2 + 4^2 + 1.5^2 + 1^2 + 1.5^2 + 2^2 + 2^2 + 2^2 )}{10(10^2 – 1)} ]

[ \rho = 1 – \frac{6(1 + 2.25 + 1 + 16 + 2.25 + 1 + 2.25 + 4 + 4 + 4)}{10(100 – 1)} ]

[ \rho = 1 – \frac{6(34.75)}{990} ]

[ \rho = 1 – \frac{208.5}{990} ]

[ \rho ≈ 1 – 0.21 ]

[ \rho ≈ 0.79 ]

6. Interpretation

The calculated value of Spearman's rho (( \rho )) is approximately 0.79. This indicates a strong positive monotonic relationship between Data 1 and Data 2. As the values in Data 1 increase, the corresponding values in Data 2 tend to increase as well, and vice versa.

Conclusion

Spearman's rank correlation coefficient provides a robust measure of the monotonic relationship between two variables, making it suitable for analyzing data with ordinal or ranked values. In this example, Spearman's rho value of approximately 0.79 indicates a strong positive relationship between Data 1 and Data 2.