Question
Download Solution PDFFor the two variables X and Y, the following observations are tabulated.
| X: | 3 | 4 | 4 |
| Y: | 10 | 10 | 9 |
The Spearman's correlation coefficient is:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFThe correct answer is -0.125.
Key Points
- Spearman's correlation coefficient is a measure of the monotonic association between two variables.
- It measures how well the rank order of one variable corresponds to the rank order of the other variable.
- A positive correlation means that as the rank of one variable increases, the rank of the other variable also increases.
- A negative correlation means that as the rank of one variable increases, the rank of the other variable decreases.
- To calculate Spearman's correlation coefficient, we first need to calculate the ranks of the data in X and Y.
- The ranks are assigned based on the order of the data, with the smallest value receiving the rank of 1 and the largest value receiving the rank of the number of data points.
- If there are ties, the average rank is assigned.
- The ranks for X are: 1, 2, 2
- The ranks for Y are: 3, 3, 1
- Next, we calculate the difference between the ranks for each data point, square the differences, and sum them.
The formula is as follows:
\(ρ = {1 - {{{6 \times \sum d^2}} \over {N \times (N^2 - 1)}}}\)
where N is the number of data points and d is the difference between the ranks of X and Y for each data point.
Putting the value of ds and n, we have ρ = -0.125
- The result of -0.125 means that there is a weak negative correlation between the two variables.
Additional Information
- Spearman's correlation coefficient measures the monotonic association between two variables.
- It measures the correspondence between the rank orders of the variables.
- A positive correlation means that as the rank of one variable increases, the rank of the other variable also increases.
- A negative correlation means that as the rank of one variable increases, the rank of the other variable decreases.
Important Points
- Spearman's correlation coefficient is a non-parametric measure, meaning that it does not make assumptions about the underlying distribution of the data.
- It can be used with both continuous and ordinal data.
- Spearman's correlation coefficient is only appropriate to use when there is a monotonic relationship between the variables.
- If the relationship is not monotonic, other measures, such as Pearson's correlation coefficient, may be more appropriate.
- The strength of the correlation is determined by the magnitude of the coefficient.
- A value close to +1 or -1 indicates a strong correlation, while a value close to 0 indicates a weak correlation.
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