For Spearman's rank correlation, if the correlation coefficient is 0.7 and \(\rm \Sigma_{i=1}^n d_i^2=49.5\) then the value of sample size 'n' is:
10
Understanding Spearman's Rank Correlation and Sample Size
Spearman's rank correlation coefficient, often denoted by \(\rho\) (rho) or \(r_s\), is a non-parametric measure used to assess the strength and direction of the monotonic relationship between two ranked variables. Unlike Pearson correlation, it does not assume the variables are measured on an interval or ratio scale, nor does it require a linear relationship. It works by ranking the data for each variable and then calculating the Pearson correlation coefficient on these ranks.
The formula for Spearman's rank correlation coefficient when there are no tied ranks (or a small number of ties) is:
\(\rho = 1 - \frac{6 \Sigma d_i^2}{n(n^2 - 1)}\)
Where:
- \(\rho\) is the Spearman's rank correlation coefficient.
- \(d_i\) is the difference between the ranks of corresponding values for each pair of observations.
- \(\Sigma d_i^2\) is the sum of the squared differences in ranks.
- \(n\) is the sample size (number of pairs of observations).
Calculating Sample Size 'n' from Spearman's Rho and Sum of Squared Differences
In this problem, we are given the value of the Spearman's rank correlation coefficient and the sum of the squared differences in ranks, and we need to find the sample size 'n'.
Given values:
- Spearman's rank correlation coefficient (\(\rho\)) = 0.7
- Sum of squared differences in ranks (\(\Sigma d_i^2\)) = 49.5
We can substitute these values into the formula:
\(0.7 = 1 - \frac{6 \times 49.5}{n(n^2 - 1)}\)
Now, let's solve for \(n(n^2 - 1)\).
\(0.7 - 1 = - \frac{297}{n(n^2 - 1)}\)
\(-0.3 = - \frac{297}{n(n^2 - 1)}\)
\(0.3 = \frac{297}{n(n^2 - 1)}\)
Rearrange the equation to isolate the term containing 'n':
\(n(n^2 - 1) = \frac{297}{0.3}\)
\(n(n^2 - 1) = 990\)
We need to find an integer value of 'n' that satisfies the equation \(n^3 - n = 990\). We can test the provided options to see which one fits.
Evaluating Options for Sample Size 'n'
Let's check the given options for 'n' in the equation \(n(n^2 - 1) = 990\).
- Option 1: n = 99
\(99(99^2 - 1) = 99(9801 - 1) = 99(9800) = 970200\)
This is not equal to 990.
- Option 2: n = 20
\(20(20^2 - 1) = 20(400 - 1) = 20(399) = 7980\)
This is not equal to 990.
- Option 3: n = 10
\(10(10^2 - 1) = 10(100 - 1) = 10(99) = 990\)
This is equal to 990. This value satisfies the equation derived from the Spearman's rank correlation formula.
- Option 4: n = 990
\(990(990^2 - 1)\)
This is a very large number, clearly not equal to 990.
Based on testing the options, the sample size 'n' that satisfies the equation \(n(n^2 - 1) = 990\) is 10.
Verification
Let's verify the calculation with n=10:
\(\rho = 1 - \frac{6 \Sigma d_i^2}{n(n^2 - 1)} = 1 - \frac{6 \times 49.5}{10(10^2 - 1)} = 1 - \frac{297}{10(100 - 1)} = 1 - \frac{297}{10(99)} = 1 - \frac{297}{990}\)
Now, calculate the fraction:
\(\frac{297}{990} = \frac{297 \div 99}{990 \div 99} = \frac{3}{10} = 0.3\)
So,
\(\rho = 1 - 0.3 = 0.7\)
This matches the given Spearman's rank correlation coefficient.
| Given Value | Symbol | Value |
|---|---|---|
| Spearman's Rho | \(\rho\) | 0.7 |
| Sum of Squared Differences | \(\Sigma d_i^2\) | 49.5 |
| Formula Applied | \(\rho = 1 - \frac{6 \Sigma d_i^2}{n(n^2 - 1)}\) | |
| Equation Derived | \(n(n^2 - 1) = 990\) | |
| Sample Size Found | \(n\) | 10 |
Revision Table: Key Spearman's Rank Correlation Concepts
| Concept | Description | Formula (No ties) |
|---|---|---|
| Spearman's Rho (\(\rho\)) | Measures the strength and direction of monotonic association between two ranked variables. | \(\rho = 1 - \frac{6 \Sigma d_i^2}{n(n^2 - 1)}\) |
| \(d_i\) | Difference between the ranks of the \(i\)-th pair of observations. | |
| \(\Sigma d_i^2\) | Sum of the squares of the differences in ranks. | |
| \(n\) | Sample size (number of pairs of observations). |
Additional Information: Spearman vs. Pearson Correlation
It's helpful to understand how Spearman's rank correlation differs from Pearson correlation.
- Pearson Correlation (r): Measures the strength and direction of the linear relationship between two variables measured on an interval or ratio scale. It assumes the data is normally distributed and the relationship is linear.
- Spearman Correlation (\(\rho\) or \(r_s\)): Measures the strength and direction of the monotonic relationship between two variables. It works on the ranks of the data, making it suitable for ordinal data or interval/ratio data where the relationship might be non-linear but consistently increasing or decreasing (monotonic). It is less sensitive to outliers than Pearson correlation.
A monotonic relationship means that as one variable increases, the other variable either consistently increases (positive monotonic) or consistently decreases (negative monotonic), but not necessarily at a constant rate (which would be a linear relationship).
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