Spearman Rank Correlation: Formula, Interpretation, Python, R, SPSS and Excel Guide

Quick Answer: Spearman Rank Correlation Result

The main Spearman Rank Correlation analysis tests the monotonic association between G1 and G3. G1 is the first-period grade and G3 is the final grade. The analysis uses 649 valid paired cases.

Because the grade variables contain many tied values, the correct Spearman rho is calculated as the Pearson correlation of average ranks. The result is rho = 0.8832876375, with t approximation = 47.9222869, df = 647, and two-tailed p = 4.9873745 × 10^-215. The decision at alpha .05 is to reject H0.

What Is Spearman Rank Correlation?

Spearman Rank Correlation is a nonparametric correlation method that measures whether two variables move together in a monotonic way. A monotonic relationship means that as one variable increases, the other variable tends to increase or tends to decrease, but the pattern does not need to be perfectly linear.

The coefficient is usually written as ρ, rho, or r_s. It ranges from -1 to +1. A value near +1 means the ranks move upward together. A value near -1 means one variable ranks higher when the other ranks lower. A value near 0 means little monotonic association.

In this worked example, G1 and G3 have rho = 0.8833. This means students with higher first-period grades tend to have higher final grades in rank order. The association is very strong and positive.

Spearman correlation is useful when variables are ordinal, not normally distributed, affected by outliers, or monotonic but not strictly linear. It connects naturally with Correlation in Python, Correlation in R, Correlation in SPSS, Correlation in Excel, Correlation Matrix, Correlation Heatmap, Correlation Assumptions, Parametric vs Nonparametric Tests, p-value and effect size.

When Should You Use Spearman Correlation?

Use Spearman correlation when the research question is about rank-order association or monotonic relationship. It is often preferred when Pearson correlation assumptions are questionable.

In the student performance dataset, many variables are ordinal or tied, such as studytime, failures, health, alcohol-use categories and grade scores. That makes Spearman correlation a useful alternative or companion to Pearson correlation.

Spearman vs Pearson Correlation

Spearman and Pearson correlation both measure association, but they answer different questions. Pearson uses raw values and measures linear association. Spearman uses ranks and measures monotonic association.

For G1 and G3, both Pearson and Spearman correlations are high because the raw grade relationship is strong and the rank relationship is also strong. Spearman is especially useful here because tied grade values are common and the method explains the rank-order relationship clearly.

Spearman Rank Correlation Formula

The most reliable general definition of Spearman rho is:

With tied ranks, use average ranks. For this guide:

The classic no-tie shortcut formula is:

Where d is the difference between the two ranks for each case. This shortcut is exact only when there are no tied ranks. In this dataset, tied ranks exist because grade scores repeat many times. Therefore, the preferred rho is 0.8832876, not the shortcut value 0.8847457.

Null and Alternative Hypotheses

The Spearman hypothesis test checks whether the population rank correlation is zero.

Dataset and Variables Used

The main focus pair is G1 and G3. The workbook and reports also calculate pairwise Spearman correlations across numeric and ordinal variables from the student performance dataset.

Many variables have repeated values, which create tied ranks. The workbook therefore uses average ranks rather than assuming every observation has a unique rank.

Verified Spearman Result

The Excel workbook, R report and SPSS output agree on the core result for the focus pair G1 and G3.

Top Pairwise Spearman Correlations

The pairwise matrix ranks all numeric and ordinal variable pairs by absolute Spearman rho. The strongest pair is G2 vs G3, not the worked G1 vs G3 pair. The worked example uses G1 and G3 because it is simple to explain with ranks and rank differences.

The top pairwise results show that grade variables dominate the student performance rank structure. Lifestyle and family-background variables also show meaningful monotonic associations, but they are weaker than grade-to-grade relationships.

Spearman Correlation Matrix Interpretation

The Spearman matrix summarizes rank relationships among all selected variables. Positive values show variables that tend to rank high together. Negative values show that high rank on one variable tends to occur with low rank on the other.

Python Chart-by-Chart Interpretation

The Python charts visualize the focus pair, the rank transformation, the Spearman matrix, the rank-difference diagnostic and the strongest pairwise monotonic associations. The Python PDF link was provided as an asset, but the URL returned a 404 during verification, so the interpretations below are grounded in the uploaded Excel workbook, the accessible R report and the chart file names provided for this post.

Python Chart 1: Original Scatterplot with Monotonic Context

The original scatterplot shows the raw relationship between G1 and G3. The points form a clear upward pattern: students with higher G1 scores generally have higher G3 scores. This supports the very strong positive Spearman result.

The chart is not only about a straight line. Spearman correlation focuses on rank order. Even when many values are tied or repeated, the overall ordering still shows that higher G1 positions are associated with higher G3 positions.

This chart is useful as the first visual because it connects the reader’s intuition to the later rank transformation. Before looking at ranks, the raw scores already suggest a strong increasing relationship.

Python Chart 2: Rank Scatterplot with Equal Rank Line

The rank scatterplot is the most important Spearman visual. The x-axis shows average ranks of G1 and the y-axis shows average ranks of G3. The equal-rank reference line helps readers see whether cases keep a similar rank position across the two variables.

Because the points cluster around an upward diagonal pattern, the Spearman rho is very high. The verified result is rho = 0.8833. Cases far from the diagonal represent larger rank disagreement between G1 and G3.

This chart also explains why average ranks matter. Grade values repeat, so tied observations receive averaged rank positions. Spearman rho is then calculated as the Pearson correlation between those average rank columns.

Python Chart 3: Spearman Correlation Matrix Heatmap

The matrix heatmap shows all rank correlations at once. The strongest positive area appears among G1, G2 and G3. These grade variables rank together strongly, especially G2 and G3 with rho = 0.9445.

The heatmap also shows meaningful positive monotonic relationships between Medu and Fedu and between Dalc and Walc. Negative cells show variables that move in opposite rank directions, such as failures with grade variables.

This chart is helpful for identifying clusters. It shows that academic performance variables form the strongest block, while lifestyle and family variables form weaker but still interpretable patterns.

Python Chart 4: Rank Differences by Case

The rank-difference chart shows the difference between each student’s G1 rank and G3 rank. Values near zero mean the student has similar rank positions on both variables. Large positive or negative values show rank disagreement.

The workbook reports Σd² = 5,250,964. This value is used in the classic no-tie formula and is also useful for teaching how rank gaps reduce perfect rank agreement.

This chart gives the most concrete explanation of why rho is below 1.00. The association is very strong, but not perfect, because some students rank much higher in G1 than G3 or much higher in G3 than G1.

Python Chart 5: Top Pairwise Spearman Correlations

The top-pair chart ranks the strongest monotonic associations in the dataset. The top three are G2 vs G3, G1 vs G2 and G1 vs G3. This confirms that grade variables dominate the rank-order structure.

The chart also includes positive non-grade relationships such as Medu vs Fedu and Dalc vs Walc, plus negative academic-risk relationships involving failures and grades.

This chart is useful for a report because it reduces the large matrix into a ranked summary. Instead of reading every cell, readers can immediately see which monotonic relationships are strongest.

R Chart-by-Chart Interpretation

The R report validates the same result and includes colorful versions of the Spearman charts. The R output confirms the focus pair G1 vs G3, n = 649, rho = 0.8832876, t approximation = 47.92229, df = 647, and p = 4.987375e-215.

R Chart 1: Colorful Original Scatterplot

The R original scatterplot confirms the same increasing pattern between G1 and G3. Higher first-period grades tend to align with higher final grades.

This chart supports the monotonic interpretation visually before ranks are introduced. It shows the same story as the Python original scatterplot using a separate software workflow.

R Chart 2: Colorful Rank Scatterplot

The R rank scatterplot displays the actual ranked values used in the Spearman calculation. The upward diagonal pattern confirms a very strong positive rank relationship.

This is the best R chart for explaining the coefficient. Spearman rho is not mysterious: it is simply the correlation of these rank columns when tied values are handled with average ranks.

R Chart 3: Colorful Spearman Matrix Heatmap

The R heatmap validates the same rank-correlation matrix. Grade variables show the strongest positive cluster, failures show negative grade relationships, and Dalc–Walc plus Medu–Fedu show strong positive monotonic associations.

Using both Python and R matrices is useful because it shows that the findings are not software-specific. The same rank relationships appear across workflows.

R Chart 4: Colorful Rank Differences by Case

The R rank-difference chart shows where G1 and G3 ranks disagree. Large positive or negative rank gaps reduce the coefficient, while near-zero gaps support high rank agreement.

Because rho is still very high, most rank positions are broadly aligned. However, the diagnostic chart makes clear that the relationship is not perfect.

R Chart 5: Colorful Top Pairwise Spearman Correlations

The R top-pairs chart highlights the same strongest monotonic relationships: G2–G3, G1–G2, G1–G3, Medu–Fedu and Dalc–Walc.

The negative failure-grade relationships are also important. They show that students with higher ranks on past failures tend to rank lower on grade outcomes.

SPSS Output Interpretation

The SPSS output confirms the focus-pair result using SPSS nonparametric correlations. SPSS reports the G1–G3 Spearman coefficient as approximately .883, with Sig. (2-tailed) = .000 and N = 649. In report writing, SPSS .000 should be written as p < .001, not p = .000.

Open the SPSS Spearman Rank Correlation output PDF

The SPSS output is useful for students because it shows both the nonparametric correlation table and a manual rank-construction section. This proves the method: rank G1, rank G3, use average ranks for ties, and correlate the rank columns.

Excel Worked Formula Explanation

The uploaded Excel workbook provides a fully worked Spearman Rank Correlation analysis. It includes the dataset, rank working sheet, final report, pairwise matrix and top pairwise results.

Excel Formula Method

The workbook uses RANK.AVG for tied ranks and CORREL to calculate Spearman rho as the correlation of average ranks.

In the workbook, the first few working rows show how ranks can differ sharply. For example, one case has G1 = 0 and G3 = 11, producing rank G1 = 1 and rank G3 = 249.5, so the rank difference is -248.5. These large rank gaps explain why rho is less than 1, even though the association is very strong.

Python, R, SPSS and Excel Workflows

Spearman Rank Correlation can be calculated in Python, R, SPSS and Excel. The key is to use the same variables, missing-value rule and tied-rank method.

Code Blocks and Excel Formulas

Python Code for Spearman Rank Correlation

import pandas as pd
from scipy import stats

df = pd.read_csv("dataset.csv")

x_var = "G1"
y_var = "G3"

work = df[[x_var, y_var]].dropna().copy()
work[x_var] = pd.to_numeric(work[x_var], errors="coerce")
work[y_var] = pd.to_numeric(work[y_var], errors="coerce")
work = work.dropna()

rho, p_value = stats.spearmanr(work[x_var], work[y_var])
n = len(work)

t_approx = rho * ((n - 2) / (1 - rho**2)) ** 0.5
dfree = n - 2

print("X variable:", x_var)
print("Y variable:", y_var)
print("N:", n)
print("Spearman rho:", rho)
print("t approximation:", t_approx)
print("df:", dfree)
print("p-value:", p_value)

Python Code for Manual Average-Rank Check

import pandas as pd
from scipy import stats

df = pd.read_csv("dataset.csv")
work = df[["G1", "G3"]].dropna().copy()

work["rank_G1"] = work["G1"].rank(method="average", ascending=True)
work["rank_G3"] = work["G3"].rank(method="average", ascending=True)
work["d"] = work["rank_G1"] - work["rank_G3"]
work["d_squared"] = work["d"] ** 2

rho_from_ranks, p_from_ranks = stats.pearsonr(work["rank_G1"], work["rank_G3"])

n = len(work)
classic_no_tie_rho = 1 - (6 * work["d_squared"].sum()) / (n * (n**2 - 1))

print("Spearman rho as Pearson correlation of average ranks:", rho_from_ranks)
print("Sum d squared:", work["d_squared"].sum())
print("Classic no-tie formula rho:", classic_no_tie_rho)
print("Note: no-tie formula is exact only when there are no tied ranks.")

R Code for Spearman Rank Correlation

df <- read.csv("dataset.csv", stringsAsFactors = FALSE)

work <- na.omit(df[, c("G1", "G3")])
work$G1 <- as.numeric(work$G1)
work$G3 <- as.numeric(work$G3)

test <- cor.test(work$G1, work$G3, method = "spearman", exact = FALSE)

print(test)

rank_G1 <- rank(work$G1, ties.method = "average")
rank_G3 <- rank(work$G3, ties.method = "average")

rho_from_ranks <- cor(rank_G1, rank_G3, method = "pearson")
d <- rank_G1 - rank_G3
sum_d_squared <- sum(d^2)

n <- nrow(work)
classic_no_tie_rho <- 1 - (6 * sum_d_squared) / (n * (n^2 - 1))

cat("rho from average ranks:", rho_from_ranks, "\n")
cat("sum d squared:", sum_d_squared, "\n")
cat("classic no-tie formula rho:", classic_no_tie_rho, "\n")

SPSS Syntax for Spearman Rank Correlation

* Spearman Rank Correlation in SPSS.
OUTPUT CLOSE ALL.
OUTPUT NEW NAME=Spearman_Rank_Correlation_Output.

DESCRIPTIVES VARIABLES=G1 G3
  /STATISTICS=MEAN STDDEV MIN MAX.

NONPAR CORR
  /VARIABLES=G1 G3
  /PRINT=SPEARMAN TWOTAIL
  /MISSING=PAIRWISE.

* Manual rank construction for fully worked output.
RANK VARIABLES=G1 G3 (A)
  /RANK
  /PRINT=YES
  /TIES=MEAN.

COMPUTE rank_difference = RG1 - RG3.
COMPUTE rank_difference_squared = rank_difference ** 2.
EXECUTE.

CORRELATIONS
  /VARIABLES=RG1 RG3
  /PRINT=TWOTAIL
  /MISSING=PAIRWISE.

GRAPH
  /SCATTERPLOT(BIVAR)=G1 WITH G3
  /TITLE='Spearman Rank Correlation: Original Values'.

GRAPH
  /SCATTERPLOT(BIVAR)=RG1 WITH RG3
  /TITLE='Spearman Rank Correlation: Average Ranks'.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE='Spearman-Rank-Correlation-SPSS-Output.pdf'.

Excel Formula Patterns

Assume:
X_range = G1 values
Y_range = G3 values

Rank G1 with average ranks:
=RANK.AVG(B9,$B$9:$B$657,1)

Rank G3 with average ranks:
=RANK.AVG(C9,$C$9:$C$657,1)

Rank difference:
=rank_G1-rank_G3

Squared rank difference:
=rank_difference^2

Preferred Spearman rho with tied ranks:
=CORREL(rank_G1_range,rank_G3_range)

Classic no-tie shortcut:
=1-(6*SUM(d_squared_range))/(n*(n^2-1))

t approximation:
=rho*SQRT((n-2)/(1-rho^2))

Degrees of freedom:
=n-2

Two-tailed p-value:
=T.DIST.2T(ABS(t),df)

Decision:
=IF(p_value<0.05,"Reject H0","Fail to reject H0")

Assumptions and Diagnostics

Spearman correlation has fewer assumptions than Pearson correlation, but it still needs thoughtful checking.

Spearman is not a free replacement for every correlation problem. It measures monotonic rank association. If the relationship is random or non-monotonic, Spearman rho can still be small even when a complex nonlinear pattern exists.

How to Report Spearman Rank Correlation

A complete Spearman report should include the variables, sample size, rho, p-value, direction, strength and tie handling if relevant.

Common Mistakes in Spearman Correlation Interpretation

Downloads and Resources

External References

For additional learning, review statistical documentation on Spearman’s rho, rank-order correlation, monotonic association, tied ranks, nonparametric correlation tests and correlation matrix interpretation.

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