# Spearman rank correlation coefficient

The Spearman rank correlation coefficient is a non-parametric measure of the strength and direction of the association between two ranked variables. It assesses how well the relationship between two variables can be described using a monotonic function.
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Updated: Jun 6, 2024

## 3 key takeaways

• The Spearman rank correlation coefficient measures the strength and direction of the relationship between two variables based on their ranked values.
• It is a non-parametric test, meaning it does not assume a normal distribution of the data.
• The coefficient ranges from -1 to 1, where -1 indicates a perfect negative correlation, 0 indicates no correlation, and 1 indicates a perfect positive correlation.

## What is the Spearman rank correlation coefficient?

The Spearman rank correlation coefficient, often denoted by ρ (rho) or r_s, is used to evaluate the relationship between two variables without assuming that the relationship is linear or that the variables are normally distributed. Instead, it relies on the ranks of the data rather than their raw values. This makes it particularly useful for ordinal data or for data that do not meet the assumptions of parametric tests.

### How to calculate the Spearman rank correlation coefficient

The calculation of the Spearman rank correlation coefficient involves the following steps:

1. Rank the data: Assign ranks to the values of each variable. If there are ties (identical values), assign to each tied value the average of the ranks that they would have received had there been no ties.
2. Calculate the differences: Compute the difference between the ranks of each pair of observations.
3. Square the differences: Square each of the rank differences.
4. Sum the squared differences: Sum all the squared differences.
5. Apply the formula: Use the Spearman rank correlation formula to calculate the coefficient.

The formula for the Spearman rank correlation coefficient is:

ρ = 1 – (6 * Σd_i^2) / (n * (n^2 – 1))

where:

• ρ (rho) is the Spearman rank correlation coefficient.
• d_i is the difference between the ranks of each pair of observations.
• n is the number of observations.

### Interpreting the Spearman rank correlation coefficient

The Spearman rank correlation coefficient ranges from -1 to 1:

• +1 indicates a perfect positive correlation: As one variable increases, the other variable also increases in a perfectly monotonic manner.
• 0 indicates no correlation: There is no apparent relationship between the variables.
• -1 indicates a perfect negative correlation: As one variable increases, the other variable decreases in a perfectly monotonic manner.

### Advantages of the Spearman rank correlation coefficient

• Non-parametric: It does not assume a normal distribution of the data, making it suitable for ordinal data and non-linear relationships.
• Robust to outliers: By using ranks instead of raw data, it reduces the influence of outliers on the correlation measure.
• Simplicity: It is straightforward to calculate and interpret.

### Limitations of the Spearman rank correlation coefficient

• Monotonic relationships only: It only measures the strength of monotonic relationships. It may not adequately capture more complex relationships between variables.
• Ties in ranks: The presence of many tied ranks can affect the accuracy of the coefficient, although adjustments can be made.

### Examples of Spearman rank correlation coefficient usage

• Education research: Evaluating the relationship between students’ ranks in different subjects to see if higher performance in one subject correlates with higher performance in another.
• Social sciences: Assessing the association between the ranks of individuals’ socioeconomic status and their health outcomes.
• Finance: Comparing the rankings of companies by different performance metrics to determine if there is a consistent ordering.

The Spearman rank correlation coefficient is a valuable tool in statistics for understanding the relationship between two ranked variables, especially when the assumptions of parametric tests are not met. For further exploration, you might look into related topics such as Pearson correlation coefficient, Kendall’s tau, and non-parametric statistical methods.

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