Quick Answer: Kendall’s Tau Result

The main analysis tested the monotonic association between G1 and G3. G1 is the first-period grade, and G3 is the final grade. Kendall’s tau-b was used because grade data contain many tied values.

The result was Kendall’s tau-b = 0.766198, with N = 649 complete pairs and p = 2.108694e-155. The direction is positive, the practical strength label is strong, and the decision is Reject H0: significant monotonic association.

What Is Kendall’s Tau?

Kendall’s Tau is a rank-based measure of association between two variables. Instead of using raw distances between values, it studies the ordering of observations. If higher values of one variable usually match higher values of the other variable, tau is positive. If higher values of one variable usually match lower values of the other variable, tau is negative.

Kendall’s Tau is useful for ordinal data, ranked data, non-normal data and data with many repeated values. In this guide, G1 and G3 are numeric grade variables, but the repeated grade values create many ties. That is why Kendall’s tau-b is used instead of a tie-free tau-a version.

Kendall’s Tau is related to Correlation in Python, Correlation in R, Correlation Matrix, Correlation Assumptions, p-value, effect size and descriptive statistics explained.

Kendall’s Tau Formula

The simple tie-free Kendall tau idea is:

Here, C is the number of concordant pairs, D is the number of discordant pairs, and n(n − 1) / 2 is the total number of observation pairs.

Because real datasets often contain ties, Kendall’s tau-b uses a tie-corrected denominator:

The large number of concordant pairs compared with discordant pairs explains why the coefficient is strongly positive.

Null and Alternative Hypotheses

The Kendall’s Tau hypothesis test checks whether the population rank association is zero.

Dataset and Variables Used

The analysis used the student performance dataset. The main variables were G1 and G3. A wider Kendall’s tau-b matrix also included age, Medu, Fedu, traveltime, studytime, failures, famrel, freetime, goout, Dalc, Walc, health, absences, G1, G2 and G3.

Before interpreting Kendall’s Tau, check whether the variables are ordered, whether rows are independent, whether the relationship is monotonic, and whether ties are present. In this dataset, ties are common, so tau-b is the appropriate version.

Verified Kendall’s Tau Results

The main result and top pairwise results are shown below. Pairs are ranked by absolute tau-b strength.

The strongest overall pair is G2 vs G3, while the primary teaching example is G1 vs G3. The failures variable has the strongest negative grade relationships.

Python Chart-by-Chart Interpretation

The Python report includes six charts: rank scatterplot, pair classification counts, Kendall matrix heatmap, p-value heatmap, top tau pairs and primary variable distributions.

Python Chart 1: Kendall’s Tau Rank Scatterplot

The rank scatterplot shows that higher G1 ranks generally align with higher G3 ranks. The pattern is upward and monotonic, which supports the positive Kendall’s tau-b value of 0.766198.

This chart is useful because Kendall’s Tau is a rank-based method. The chart does not require a perfect straight line; it only needs a consistent ordered pattern. The strong upward rank pattern supports the final conclusion.

Python Chart 2: Pair Classification Counts

This chart explains the tau-b calculation directly. The analysis found 158,466 concordant pairs and 13,791 discordant pairs. Concordant pairs dominate the result, which is why tau-b is strongly positive.

The chart also shows ties in X only, ties in Y only and ties on both variables. These ties are expected because grade scores repeat across many students. Kendall’s tau-b corrects for those ties.

Python Chart 3: Kendall’s Tau Matrix Heatmap

The matrix heatmap shows ordered associations across all selected numeric and ordinal variables. The grade block stands out clearly: G1, G2 and G3 have strong positive tau-b relationships.

The heatmap also shows meaningful negative grade relationships involving failures. This means students with more previous failures tend to have lower ordered grade positions.

Python Chart 4: Kendall’s Tau p-value Heatmap

The p-value heatmap separates statistical evidence from practical strength. Strong grade relationships have extremely small p-values. Some weaker associations may also be significant because the sample size is large.

This chart should be interpreted together with the tau-b matrix. The tau-b value tells strength and direction, while the p-value tells whether the association is statistically detectable.

Python Chart 5: Top Kendall’s Tau Pairs

The top-pairs chart ranks relationships by absolute tau-b value. The strongest pair is G2 vs G3, followed by G1 vs G2 and G1 vs G3. This shows that the grade variables form the strongest ordered cluster.

Negative relationships involving failures also appear in the ranking. These should be described as negative monotonic relationships, not causal effects.

Python Chart 6: Primary Variable Distributions

The distribution chart shows that G1 and G3 contain repeated grade values. These repeated values create tied ranks. Because ties are present, tau-b is better than tau-a.

This chart supports the method choice. Kendall’s tau-b is designed for ordered data with ties, making it appropriate for grade variables with repeated scores.

R Chart-by-Chart Interpretation

The R report validates the same result using a separate workflow. It includes colorful charts for rank scatterplot, pair classification counts, tau matrix, top tau pairs and primary variable distributions.

R Chart 1: Colorful Kendall’s Tau Rank Scatterplot

The R rank scatterplot confirms the same upward ordered pattern. Higher G1 ranks generally match higher G3 ranks. This visual pattern agrees with tau-b = 0.766198.

R Chart 2: Colorful Pair Classification Counts

The R pair-count chart confirms that concordant pairs are far more common than discordant pairs. This explains the strong positive coefficient. The tie categories again show why tau-b is reported.

R Chart 3: Colorful Kendall’s Tau Matrix Heatmap

The R tau matrix confirms the same strongest ordered relationships. The G1, G2 and G3 block is strongly positive, while failures have negative relationships with grade variables.

R Chart 4: Colorful Top Kendall’s Tau Pairs

The top-pairs chart confirms the ranking: G2 vs G3 is strongest, G1 vs G2 is second, and G1 vs G3 is third. The repeated agreement across Python, R and Excel strengthens the result.

R Chart 5: Colorful Primary Variable Distributions

The R distribution chart again shows many tied values in G1 and G3. This supports the choice of Kendall’s tau-b and helps explain why tie correction is required.

SPSS Output Interpretation

The SPSS output confirms the primary nonparametric correlation between G1 and G3. SPSS reports Kendall’s tau-b correlation coefficient of approximately .766, Sig. (2-tailed) = .000, and N = 649.

Open the SPSS Kendall’s Tau Output PDF

The SPSS result is shorter than the Python and R reports, but it gives the formal nonparametric correlation table many students need for assignments, theses and research reports.

Excel Worked File Explanation

The Excel workbook gives a fully worked Kendall’s Tau analysis. It includes the original dataset, primary G1 and G3 data, a first-30-observation pairwise demo, a full Kendall matrix, top pairwise results and an assumptions sheet.

Download the Kendall’s Tau Fully Worked Excel File

The Excel dashboard confirms the primary result: Kendall’s tau-b = 0.7661978205, p = 2.1086936598e-155, and decision Reject H0: significant monotonic association. The pairwise demo also shows how pair classification works for the first 30 observations.

This workbook is especially useful for teaching because it shows the logic behind the coefficient instead of only presenting the final value.

Python, R, SPSS and Excel Workflows

Code Blocks and Excel Formulas

Python Code for Kendall’s Tau

import pandas as pd
from scipy.stats import kendalltau

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

x_var = "G1"
y_var = "G3"

work = df[[x_var, y_var]].dropna()
tau, p_value = kendalltau(work[x_var], work[y_var])

print("Complete pairs:", len(work))
print("Kendall tau-b:", tau)
print("Two-tailed p-value:", p_value)

if p_value < 0.05:
    print("Reject H0: significant monotonic association")
else:
    print("Fail to reject H0")

Python Code for Pair Classification

import numpy as np

x = work[x_var].to_numpy()
y = work[y_var].to_numpy()

C = D = Tx = Ty = Tboth = 0

for i in range(len(x)):
    for j in range(i + 1, len(x)):
        dx = np.sign(x[j] - x[i])
        dy = np.sign(y[j] - y[i])

        if dx == 0 and dy == 0:
            Tboth += 1
        elif dx == 0:
            Tx += 1
        elif dy == 0:
            Ty += 1
        elif dx == dy:
            C += 1
        else:
            D += 1

tau_b_manual = (C - D) / np.sqrt((C + D + Tx) * (C + D + Ty))

print("Concordant:", C)
print("Discordant:", D)
print("X ties only:", Tx)
print("Y ties only:", Ty)
print("Ties on both:", Tboth)
print("Manual tau-b:", tau_b_manual)

R Code for Kendall's Tau

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

x_var <- "G1"
y_var <- "G3"

work <- na.omit(df[, c(x_var, y_var)])

test <- cor.test(
  work[[x_var]],
  work[[y_var]],
  method = "kendall",
  exact = FALSE
)

print(test)

cat("Complete pairs:", nrow(work), "\n")
cat("Kendall tau-b:", unname(test$estimate), "\n")
cat("p-value:", test$p.value, "\n")

SPSS Syntax for Kendall's Tau

* Kendall's Tau in SPSS.
OUTPUT CLOSE ALL.
OUTPUT NEW NAME=Kendalls_Tau_Output.

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

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

NONPAR CORR
  /VARIABLES=age Medu Fedu traveltime studytime failures famrel freetime goout Dalc Walc health absences G1 G2 G3
  /PRINT=KENDALL TWOTAIL
  /MISSING=PAIRWISE.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE='Kendalls-Tau-SPSS-Output.pdf'.

Excel Formula Patterns for Kendall's Tau

For each observation pair i < j:

Difference in X:
=X_j-X_i

Difference in Y:
=Y_j-Y_i

Sign of X difference:
=SIGN(X_j-X_i)

Sign of Y difference:
=SIGN(Y_j-Y_i)

Pair class:
=IF(AND(sign_dx=0,sign_dy=0),"Both tied",
 IF(sign_dx=0,"Tie X only",
 IF(sign_dy=0,"Tie Y only",
 IF(sign_dx=sign_dy,"Concordant","Discordant"))))

Concordant count:
=COUNTIF(pair_class_range,"Concordant")

Discordant count:
=COUNTIF(pair_class_range,"Discordant")

X ties only:
=COUNTIF(pair_class_range,"Tie X only")

Y ties only:
=COUNTIF(pair_class_range,"Tie Y only")

Kendall tau-b:
=(C-D)/SQRT((C+D+Tx)*(C+D+Ty))

How to Report Kendall's Tau

A good Kendall's Tau report should name the variables, sample size, tau-b value, p-value, direction, practical strength and tie handling.

Common Mistakes in Kendall's Tau Interpretation

Downloads and Resources

External References

For software documentation, see SciPy documentation for kendalltau, R documentation for cor.test, IBM SPSS documentation for nonparametric correlations and Microsoft Excel documentation for ranking and COUNT formulas.

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