Quick Answer: Kendall’s Tau-c Result

The primary ordinal pair in this report is studytime by G3. Studytime has 4 ordered levels, while G3 has 17 observed grade levels. Because the table is rectangular and ordinal, Kendall’s Tau-c is useful for summarizing the direction and strength of association between the two ordered variables.

The primary result is Kendall’s Tau-c = 0.225039, based on N = 649 students. The calculation uses 79,088 concordant pairs and 43,543 discordant pairs. The approximate p-value is 2.51457e-12, so the association is statistically significant at α = .05.

What Is Kendall’s Tau-c?

Kendall’s Tau-c is an ordinal association coefficient used when both variables can be arranged from low to high. It is especially useful for contingency tables where the number of categories differs between the two variables. In this post, studytime has four ordered categories and G3 has seventeen observed final-grade categories, so the table is clearly rectangular. Kendall’s Tau-c is designed to handle this kind of ordered table more directly than a simple two-variable linear correlation.

The logic behind Tau-c is based on comparing pairs of observations. If one student has a higher studytime category and also a higher G3 category than another student, that pair is concordant. If one student has a higher studytime category but a lower G3 category, that pair is discordant. Tau-c summarizes whether concordant ordering is more common than discordant ordering after adjusting for the smaller number of categories in the table.

A positive Tau-c means that higher categories of the first variable tend to occur with higher categories of the second variable. A negative Tau-c means higher categories of one variable tend to occur with lower categories of the other variable. A Tau-c value near zero means little ordered association.

Kendall’s Tau-c is related to Correlation in Python, Correlation in R, Correlation in Excel, Correlation Matrix, Correlation Assumptions, Cross Tabulation, Effect Size, p-value and Parametric vs Nonparametric Tests.

When Should You Use Kendall’s Tau-c?

Use Kendall’s Tau-c when your variables are ordered categories and you want a single effect-size coefficient for the ordinal association. It is useful when both variables have meaningful order but are not necessarily continuous in the Pearson correlation sense. It is also helpful when the table is rectangular, because Tau-c contains an adjustment for the smaller number of row or column categories.

For the student performance example, studytime is naturally ordered because categories represent increasing study duration. G3 is also ordered because a higher final grade indicates a higher performance level. Therefore, the coefficient has a meaningful direction: positive values indicate that higher studytime levels tend to align with higher final-grade categories.

Kendall’s Tau-b vs Kendall’s Tau-c

Kendall’s Tau-b and Kendall’s Tau-c are both ordinal association measures, but they are not identical. Tau-b is often used when ties are important in a square or nearly square ordinal setting. Tau-c is especially useful for rectangular contingency tables where the number of row categories and column categories differ.

Kendall’s Tau-c Formula

The Kendall’s Tau-c formula used in the Python, R and Excel reports is:

In this formula, C is the number of concordant pairs, D is the number of discordant pairs, n is the total number of observations, and m is the smaller number of row or column categories. The value of m matters because Tau-c adjusts the maximum possible association for rectangular tables.

The positive value comes from the fact that concordant pairs outnumber discordant pairs. However, the coefficient is not close to 1, so the relationship is not strong. It is best interpreted as a statistically significant weak positive ordinal association.

Null and Alternative Hypotheses

The hypothesis test asks whether the ordinal association is zero in the population. Although Tau-c itself is an ordinal effect-size coefficient, the report uses an approximate Kendall rank-test p-value as significance context.

Dataset and Variables Used

The analysis used the student performance dataset with 649 valid cases. The main pair was studytime by G3. The full matrix also included age, Medu, Fedu, traveltime, studytime, failures, famrel, freetime, goout, Dalc, Walc, health, absences, G1, G2 and G3.

This dataset is useful for Tau-c because it contains a mixture of ordinal variables, grade variables, repeated values and rectangular tables. It also shows the difference between a modest ordinal relationship such as studytime by G3 and a very strong relationship such as G2 by G3.

Primary Contingency Table: Studytime by G3

The primary contingency table counts students by studytime category and final grade category. Studytime has four ordered levels, while G3 has seventeen observed ordered values. This makes the table rectangular, which is one reason Tau-c is appropriate.

The table shows that studytime level 2 contains the largest group of students, followed by studytime level 1. Higher studytime levels have fewer students, so visual interpretation should be cautious. The positive Tau-c does not mean every higher-studytime student earned a higher final grade. It means that, across all ordered pairs, the higher-studytime/higher-grade ordering occurs more often than the higher-studytime/lower-grade ordering.

Verified Kendall’s Tau-c Results

The primary pair and the strongest matrix relationships are shown below. The primary pair explains the method, while the top-pair table shows where the strongest ordinal relationships appear in the full dataset.

The strongest positive associations are among G1, G2 and G3. This is expected because grade variables measure related academic performance across time. The failures variable shows the most meaningful negative grade relationships, meaning higher failure counts tend to align with lower grade categories.

Python Chart-by-Chart Interpretation

The Python report includes five charts: Tau-c heatmap, primary contingency heatmap, top Tau-c relationships, p-value context and the ordinal pattern for the primary pair. These charts are designed to explain both the full matrix and the worked studytime by G3 example.

Python Chart 1: Kendall’s Tau-c Heatmap

The Tau-c heatmap summarizes the direction and strength of ordinal association across the full variable set. Positive values show that higher categories of one variable tend to align with higher categories of another variable. Negative values show that higher categories of one variable tend to align with lower categories of another variable.

The strongest positive block is among G1, G2 and G3. These grade variables have very high positive associations because students who rank higher in one grade period usually rank higher in another. The heatmap also shows negative grade relationships for failures, indicating that students with more past failures tend to be ordered lower on grade outcomes.

For interpretation, this chart should not be read as a prediction model. It is an association map. It tells which ordered variables move together, which move in opposite directions and which relationships are weak or near zero.

Python Chart 2: Primary Contingency Heatmap

The primary contingency heatmap displays the cell counts for studytime by G3. Studytime levels appear on one axis and G3 final-grade categories appear on the other. Darker or more intense cells represent larger counts.

The table is not evenly distributed. Most students fall in studytime categories 1 and 2, while categories 3 and 4 contain fewer observations. This matters because a positive association may be statistically significant but still modest if the higher studytime categories are small and the grade distributions overlap.

The heatmap supports the final conclusion: the relationship is positive but weak. Higher studytime categories show some tendency toward higher G3 values, but the pattern is not strong enough to call the association large.

Python Chart 3: Top Tau-c Relationships

The top-relationships chart ranks pairs by absolute Tau-c value. The strongest overall pair is G2 by G3, followed by G1 by G2 and G1 by G3. These are very strong positive ordinal associations.

The chart also shows that Medu by Fedu is strong, meaning mother and father education levels tend to move together. Dalc by Walc is moderate, showing that weekday and weekend alcohol-use categories are positively ordered together.

The chart is important because it places the primary studytime by G3 result in context. Studytime by G3 is statistically significant, but it is not among the very strongest relationships in the full dataset. The strongest relationships are the grade-to-grade pairs.

Python Chart 4: Tau-c p-value Context

The p-value context chart shows the approximate statistical evidence for the top Tau-c relationships. The strongest grade relationships have extremely small p-values, but the studytime by G3 result is also statistically significant with p much smaller than .05.

This chart helps avoid a common mistake: confusing statistical significance with practical strength. A relationship can be statistically significant because the sample is large, even if the effect size is weak. Therefore, the primary result should be reported as statistically significant and weak positive, not as a strong association.

Python Chart 5: Primary Pair Ordinal Pattern

The ordinal pattern chart shows studytime categories against G3 final grades with small jitter so overlapping points are easier to see. The plot shows that higher studytime categories include many students with medium-to-high G3 scores, but there is still considerable overlap between studytime categories.

This overlap explains why the coefficient is only 0.225039. If studytime strongly separated G3 outcomes, the higher studytime categories would cluster almost entirely at higher G3 values. Instead, students across studytime categories still share many similar grades.

The correct visual interpretation is that there is a positive tendency, not a deterministic pattern. Studytime is associated with G3 in the expected direction, but other factors clearly also influence final grades.

R Chart-by-Chart Interpretation

The R report validates the Python output using a separate workflow and colorful charts. It confirms the same primary pair, the same Tau-c formula, the same top relationships and the same interpretation pattern.

R Chart 1: Colorful Kendall’s Tau-c Heatmap

The R heatmap confirms the same matrix pattern as Python. The grade variables form the strongest positive block, and failures has negative associations with the grade variables. This software-to-software agreement strengthens confidence in the result.

The heatmap also helps students understand that Tau-c is not restricted to one pair. A full matrix can reveal which ordinal relationships are strongest, which are weak and which move in opposite directions.

R Chart 2: Colorful Primary Contingency Heatmap

The R contingency heatmap displays the same studytime by G3 count table with a different color style. It confirms that most observations are concentrated in studytime levels 1 and 2, with fewer students in levels 3 and 4.

This chart is important because Tau-c is based on the contingency table, not only on a raw scatterplot. The distribution of counts across ordered cells determines the concordant and discordant pair totals.

R Chart 3: Colorful Top Tau-c Relationships

The R top-relationships chart confirms the strongest pairs: G2 by G3, G1 by G2 and G1 by G3. These grade relationships are much stronger than the primary studytime by G3 example.

This chart is useful for writing a balanced report. It allows the article to say that studytime by G3 is significant, while also showing that grade-to-grade consistency is the dominant association pattern in the dataset.

R Chart 4: Colorful Tau-c p-value Context

The R p-value context chart confirms that the top relationships are statistically significant. The grade-pair relationships have extremely small p-values, and the primary studytime by G3 pair is also significant.

However, this chart should not be used alone. A p-value does not measure strength. Always combine it with the Tau-c coefficient and the contingency table interpretation.

R Chart 5: Colorful Primary Pair Ordinal Pattern

The R ordinal pattern chart confirms the weak positive pattern between studytime and G3. Higher studytime categories show some tendency toward higher final grades, but there is substantial overlap among categories.

This chart is the best visual support for the final wording. It shows why the association is statistically significant but weak: the general direction is positive, yet studytime categories do not sharply separate final-grade outcomes.

Excel Worked File Explanation

The Excel workbook is a fully worked Kendall’s Tau-c analysis file. It includes the raw data, selected variables, primary contingency table, worked Tau-c formula calculation, pairwise Tau-c results, a coefficient heatmap sheet, p-value matrix and top findings. This makes it useful for students who want to understand how Tau-c is calculated from a contingency table instead of only reading software output.

Download the Kendall’s Tau-c Fully Worked Excel File

Excel Primary Result

The Excel workbook reports the same primary result: studytime by G3, Kendall’s Tau-c = 0.2250390352, and approximate p-value 2.514568793e-12. It also identifies the strongest pair overall as G2 by G3, with Tau-c = 0.8310255674.

Excel Contingency Table Logic

The workbook builds a contingency table first, then calculates concordant and discordant pair contributions. For each cell, concordant contributions come from cells that are lower-left or upper-right relative to that cell. Discordant contributions come from cells that are upper-left or lower-right relative to that cell. The workbook then divides by 2 because cross-cell pair contributions are counted twice when every cell is evaluated.

Python, R and Excel Workflows

Kendall’s Tau-c can be calculated in different software tools, but the central logic is the same: create an ordered contingency table, count concordant and discordant pairs, identify the smaller table dimension, and apply the Tau-c formula.

Python and R are best for full automated matrices. Excel is best for showing learners how the formula works cell by cell. For a blog post, using all three is ideal because it provides reproducibility, validation and teaching value.

Code Blocks and Excel Formulas

Python Code for Kendall’s Tau-c

import pandas as pd
import numpy as np
from scipy.stats import kendalltau

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

x_var = "studytime"
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()

# Ordered contingency table
tab = pd.crosstab(work[x_var], work[y_var]).sort_index(axis=0).sort_index(axis=1)

def kendalls_tau_c_from_table(table):
    arr = table.to_numpy()
    r, c = arr.shape
    n = arr.sum()
    m = min(r, c)

    concordant = 0
    discordant = 0

    for i in range(r):
        for j in range(c):
            count = arr[i, j]
            if count == 0:
                continue

            # Concordant cells: lower-right and upper-left
            lower_right = arr[i+1:, j+1:].sum()
            upper_left = arr[:i, :j].sum()

            # Discordant cells: lower-left and upper-right
            lower_left = arr[i+1:, :j].sum()
            upper_right = arr[:i, j+1:].sum()

            concordant += count * (lower_right + upper_left)
            discordant += count * (lower_left + upper_right)

    # Divide by 2 because cross-cell pairs are counted twice
    concordant = concordant / 2
    discordant = discordant / 2

    tau_c = (2 * m * (concordant - discordant)) / (n**2 * (m - 1))
    return tau_c, concordant, discordant, n, m

tau_c, C, D, n, m = kendalls_tau_c_from_table(tab)

# Approximate p-value context using Kendall rank test
tau_context, p_value = kendalltau(work[x_var], work[y_var])

print("Primary pair:", x_var, "by", y_var)
print("N:", n)
print("m:", m)
print("Concordant pairs:", C)
print("Discordant pairs:", D)
print("Kendall's Tau-c:", tau_c)
print("Approximate p-value context:", p_value)
print(tab)

R Code for Kendall’s Tau-c

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

x_var <- "studytime"
y_var <- "G3"

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

tab <- table(work[[x_var]], work[[y_var]])

kendalls_tau_c_from_table <- function(tab){
  arr <- as.matrix(tab)
  r <- nrow(arr)
  c <- ncol(arr)
  n <- sum(arr)
  m <- min(r, c)

  C <- 0
  D <- 0

  for(i in 1:r){
    for(j in 1:c){
      count <- arr[i, j]
      if(count == 0) next

      lower_right <- if(i < r && j < c) sum(arr[(i+1):r, (j+1):c, drop = FALSE]) else 0
      upper_left <- if(i > 1 && j > 1) sum(arr[1:(i-1), 1:(j-1), drop = FALSE]) else 0

      lower_left <- if(i < r && j > 1) sum(arr[(i+1):r, 1:(j-1), drop = FALSE]) else 0
      upper_right <- if(i > 1 && j < c) sum(arr[1:(i-1), (j+1):c, drop = FALSE]) else 0

      C <- C + count * (lower_right + upper_left)
      D <- D + count * (lower_left + upper_right)
    }
  }

  C <- C / 2
  D <- D / 2

  tau_c <- (2 * m * (C - D)) / (n^2 * (m - 1))
  list(tau_c = tau_c, concordant = C, discordant = D, n = n, m = m)
}

result <- kendalls_tau_c_from_table(tab)
print(result)

# Approximate p-value context
test_context <- cor.test(work[[x_var]], work[[y_var]], method = "kendall", exact = FALSE)
print(test_context)

Excel Formula Patterns for Kendall's Tau-c

Primary formula:
tau-c = [2*m*(C-D)] / [n^2*(m-1)]

Where:
C = concordant pairs
D = discordant pairs
n = total valid observations
m = MIN(number_of_row_categories, number_of_column_categories)

Contingency table count:
=COUNTIFS(studytime_range, row_category, G3_range, column_category)

Row total:
=SUM(row_cells)

Column total:
=SUM(column_cells)

Total N:
=SUM(all_contingency_cells)

m:
=MIN(number_of_row_levels, number_of_column_levels)

Cell concordant contribution:
=cell_count*(SUM(lower_right_cells)+SUM(upper_left_cells))

Cell discordant contribution:
=cell_count*(SUM(lower_left_cells)+SUM(upper_right_cells))

Total C:
=SUM(all_concordant_contributions)/2

Total D:
=SUM(all_discordant_contributions)/2

Kendall's Tau-c:
=(2*m*(C-D))/(N^2*(m-1))

Decision:
=IF(p_value<0.05,"Significant ordinal association","Not significant")

Assumptions and Data Checks

Kendall's Tau-c is nonparametric, so it does not require normality in the same way that Pearson correlation does. However, it still has important requirements. The variables must have meaningful order, the observations should be independent, and the coefficient should be interpreted as ordinal association rather than causal effect.

These checks help keep the interpretation accurate. The coefficient is meaningful because both variables are ordered, but the weak size of the coefficient and the overlap in the ordinal pattern mean the result should not be overstated.

How to Report Kendall's Tau-c

A complete Kendall's Tau-c report should include the variables, sample size, row and column categories, coefficient, p-value, direction, strength and a plain-language interpretation. Since Tau-c is often used for ordinal contingency tables, it is also helpful to mention the number of concordant and discordant pairs.

Common Mistakes in Kendall's Tau-c Interpretation

Downloads and Resources

External References

For additional learning, check documentation and references for Kendall rank correlation, ordinal association, contingency-table analysis, Python statistical workflows, R correlation workflows and Excel formulas for count tables. When comparing this method with other measures, also review nominal association measures such as Cramer's V and rank measures such as Kendall's Tau-b and Spearman correlation.

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