Question

Use the information given in Exercises \(4-7\) to calculate Spearman's rank correlation coefficient, where \(x_{i}\) and \(y_{i}\) are the ranks of the ith pair of observations and \(d_{i}=x_{i}-y_{i} .\) Assume that there are no ties in the ranks. \(S_{x x}=S_{y y}=10 ; S_{x y}=6 ; n=5\)

Short answer

Answer: The Spearman's rank correlation coefficient is 0.6.

Step-by-step solution

Identify the given values

We are given the following values: - \(S_{xx} = 10\) - \(S_{yy} = 10\) - \(S_{xy} = 6\) - \(n = 5\)

Set up the formula for Spearman's rank correlation coefficient

The formula for Spearman's rank correlation coefficient is given by: $$r_s = \frac{S_{xy}}{\sqrt{S_{xx} S_{yy}}}$$

Substitute the given values and calculate the coefficient

Now, we substitute the given values into the formula: $$r_s = \frac{6}{\sqrt{10 \times 10}}$$

Solve the equation

Simplify the expression: $$r_s = \frac{6}{\sqrt{100}} = \frac{6}{10} = 0.6$$

Interpret the result

The Spearman's rank correlation coefficient, \(r_s\), is 0.6. This indicates a moderate positive relationship between the paired observations. When one variable increases, the other variable tends to increase as well, but the relationship is not very strong.

Key concepts

Rank Correlation

In statistics, rank correlation is a method of calculating the relationship or association between two ordinal variables. It is nonparametric, meaning it does not assume a specific statistical distribution. The Spearman's rank correlation coefficient, often denoted by \( r_s \), is a popular measure to assess how well the relationship between two variables can be described using a monotonic function.

Essentially, Spearman's rank correlation examines the ranks assigned to data values rather than their raw numbers. If the ranks of corresponding paired observations resemble each other closely, the Spearman's rank correlation coefficient will be closer to \( +1 \) for positively related ranks, or \( -1 \) for negatively related ranks. A value near zero suggests little or no rank correlation. It's particularly useful because it lessens the influence of outliers and can handle non-linear relationships.

Nonparametric Statistics

Nonparametric statistics refers to statistical methods that do not require population parameters to follow a specific distribution, often making fewer assumptions about the data. This flexibility allows for analysis when parametric assumptions cannot be met, which can be vital for accurately understanding a variety of data types.

They are used when data doesn't lend itself well to standard parametric tests, such as when handling skewed distributions, ordinal data, or populations that are not normally distributed. Spearman's rank correlation coefficient is an example of a nonparametric statistic because it does not rely on any assumptions about the frequency distribution of the variables and is based only on the ranks, therefore being especially robust to data anomalies.

Data Analysis

Data analysis encompasses a broad range of methods to process, visualise, and draw conclusions from data. In the context of Spearman's rank correlation, data analysis involves ranking the variables, assessing their association, and determining the strength and direction of the relationship between them.

The conclusion drawn from a Spearman's rank correlation can influence decision-making in various sectors, including business, healthcare, and social sciences. Data analysis is iterative and exploratory at its core, often benefiting from visualization tools that can help illustrate the strength of a correlation or the pattern of a relationship. An effective data analysis strategy enables researchers to navigate through complex data sets, simplifying the inference of trends and relationships.