Spearman's correlation

1. Spearman's correlation
Ms. Nigar K.Mujawar
Assistant Professor,
Shri.Balasaheb Mane Shikshan Prasarak Mandal Ambap
Womens College of Pharmacy, Peth-Vadgaon,
Kolhapur, M.S., INDIA.
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2. Spearman's correlation
Spearman's correlation coefficient (or Spearman's rho) is a non-
parametric measure of rank correlation between two variables. It
assesses how well the relationship between two variables can be
described using a monotonic function, regardless of the scale of
measurement. Here's a detailed explanation of Spearman's correlation
coefficient:
Calculation:
1.Ranking:
1.For each variable X and Y, rank the values separately. If there are
ties (i.e., identical values), assign them the average of the ranks they
would have received if they were distinct values.
2.Computing Differences:
1.For each pair of ranked values(xi​,yi​), calculate the difference in ranks
di​=rank of xi​−rank of yi.

3. 1.Applying the Formula:
1.Use the formula: = 1− 6∑di2 / n(n2−1)
2.6∑(2−1)ρ=1−​​ where n is the number of observations.
Interpretation:
The Spearman correlation coefficient ρ ranges between -1 and 1.
ρ=1 indicates a perfect positive monotonic relationship: as X increases,
Y tends to increase.
ρ=−1 indicates a perfect negative monotonic relationship: as X
increases, Y tends to decrease.
ρ=0 indicates no monotonic association between X and Y, but it doesn't
imply independence.

4. Advantages and Use Cases:
1. Non-parametric: Suitable for ordinal data or data that do not meet the
assumptions of normality required by Pearson correlation.
2. Robust: Less affected by outliers compared to Pearson correlation.
3. Interpretability: Describes the strength and direction of monotonic
relationships.
Example:
Suppose we have the following paired data representing ranks:
X:10,5,8,3,6
Y:7,4,9,2,1​
•Rank X values: Rank(10)=5,Rank(5)=3,Rank(8)=4,Rank(3)=2,Rank(6)=1
•Rank Y values: Rank(7)=4,Rank(4)=3,Rank(9)=5,Rank(2)=2,Rank(1)=1

5. •Compute differences:
• d1​=5−4=1
• d2​=3−3=0
• d3​=4−5=−1
• d4​=2−2=0
• d5​=1−1=0

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