Enter the difference between the ranks of each observation, the sum of the squares of the rank differences, and the number of observations into the calculator to determine Spearman’s Rho. This calculator can also evaluate any of the variables given the others are known.

Spearman’s Rho Formula

The following formula is used to calculate Spearman’s Rho. 

ρ = 1 - (6 * Σd^2) / (n * (n^2 - 1))

Variables:

  • ρ is Spearman’s rank correlation coefficient
  • d is the difference between the ranks of each observation
  • Σd^2 is the sum of the squares of the rank differences
  • n is the number of observations

To calculate Spearman’s Rho, first find the difference between the ranks of each observation. Square these differences and sum them up. Multiply this sum by 6. Then, multiply the number of observations by the difference between the square of the number of observations and 1. Finally, subtract the first result from 1 and divide by the second result.

What is a Spearman’s Rho?

Spearman’s Rho, also known as Spearman’s rank correlation coefficient, is a non-parametric measure of statistical dependence between two variables. It assesses how well the relationship between two variables can be described using a monotonic function. In other words, it measures the strength and direction of the association between two ranked variables. It is denoted by the Greek letter rho (ρ) and can take values from +1 to -1, where +1 indicates a perfect positive correlation, -1 a perfect negative correlation, and 0 no correlation.

How to Calculate Spearman’s Rho?

The following steps outline how to calculate Spearman’s Rho:

  1. First, calculate the difference between the ranks of each observation (d).
  2. Next, square each difference (d^2).
  3. Next, sum up all the squared differences (Σd^2).
  4. Next, calculate n * (n^2 – 1), where n is the number of observations.
  5. Next, multiply Σd^2 by 6.
  6. Finally, calculate Spearman’s Rho using the formula ρ = 1 – (6 * Σd^2) / (n * (n^2 – 1)).
  7. After inserting the variables and calculating the result, check your answer with the calculator above.

Example Problem:

Use the following variables as an example problem to test your knowledge:

Number of observations (n) = 10

Difference between ranks (d): 2, 4, 1, 3, 5, 0, -1, -3, -2, -4

Sum of squared differences (Σd^2) = 70

Calculate Spearman’s Rho using the formula ρ = 1 – (6 * Σd^2) / (n * (n^2 – 1)).