Quick Answer: Contingency Coefficient Result

The worked example tested whether school and sex were associated in the student performance dataset. The crosstab contained N = 649 valid students. The observed table showed GP school: 237 F and 186 M, and MS school: 146 F and 80 M. The Pearson chi-square result was χ²(1) = 4.4763, with p = .0344.

The Contingency Coefficient was C = 0.0828. Because C has a maximum below 1 for finite tables, the Excel workbook also calculated the maximum possible value for this 2 × 2 table as C_max = 0.7071 and the adjusted coefficient as adjusted C = 0.1170. Cramer’s V and Phi were both approximately 0.0830. The result is statistically significant at the .05 level, but the association is weak in practical strength.

What Is a Contingency Coefficient?

The Contingency Coefficient, often written as C, measures the strength of association between two categorical variables. It is based on the Pearson chi-square statistic from a contingency table. If the observed counts are very close to the expected counts under independence, C is close to 0. If the observed counts depart more strongly from expected counts, C becomes larger.

A contingency table is a cross-classification of two categorical variables. In this example, the row variable is school with two categories, GP and MS. The column variable is sex with two categories, F and M. The test asks whether the distribution of sex is the same inside both school categories.

The Contingency Coefficient belongs to the same family of categorical association measures as Phi and Cramer’s V. For a 2 × 2 table, Phi and Cramer’s V are often easier to compare because they use a 0 to 1 scale. The unadjusted Contingency Coefficient has a maximum below 1, so the adjusted C is useful when comparing results across tables of different sizes.

Before interpreting C, it helps to understand cross tabulation, frequency distribution, p-value, effect size and null and alternative hypothesis reporting.

Contingency Coefficient Formula

The Contingency Coefficient is calculated from Pearson’s chi-square statistic and the total sample size:

For the worked example, χ² = 4.4763 and N = 649. Substituting those values gives C = √[4.4763 / (4.4763 + 649)] = 0.0828. This is a weak association.

Because the maximum value of C depends on the number of categories in the smaller dimension of the table, an adjusted value can also be calculated:

In a 2 × 2 table, k = 2, so C_max = √(1/2) = 0.7071. The adjusted value is 0.0828 / 0.7071 = 0.1170. The adjusted value is still weak.

The chi-square statistic itself comes from observed and expected counts:

Expected counts are calculated as row total multiplied by column total divided by the grand total. The same logic is used in SPSS, Python, R and Excel.

Null and Alternative Hypotheses for Contingency Coefficient

The Contingency Coefficient itself is an effect-size statistic. The significance decision comes from the chi-square test of independence that produces the chi-square statistic used in the formula.

Dataset and Variables Used

The worked example uses the student performance dataset. The main contingency table compares school and sex. The row variable is school, with categories GP and MS. The column variable is sex, with categories F and M. The analysis uses 649 valid cases.

For categorical data, it is useful to report both the raw counts and row percentages. Raw counts show sample size. Row percentages show how the column distribution changes across row categories. This is why a complete report should include a descriptive statistics section, a crosstab section and an effect size section.

SPSS Output Interpretation for Contingency Coefficient

The SPSS output verifies the same crosstab result using CROSSTABS. SPSS reports the observed counts, expected counts, row percentages, Pearson chi-square test and symmetric measures such as Phi, Cramer’s V and the Contingency Coefficient.

Open the SPSS Contingency Coefficient output PDF

SPSS Observed and Expected Counts

Python Chart-by-Chart Interpretation

The Python charts explain the contingency coefficient from several angles: row percentages, observed counts, standardized residuals, chi-square cell contributions, effect-size summary and observed-versus-expected counts. Together, these charts show not only whether the association is significant, but where it comes from and how strong it is.

Python Chart 1: Row Percentage Profile

This chart converts the school-by-sex table into row percentages. GP is about 56.0% F and 44.0% M, while MS is about 64.6% F and 35.4% M. The blue F section is taller in MS, and the orange M section is taller in GP. That visible shift is the practical pattern behind the significant chi-square result.

The chart also shows why the effect is weak. Both schools still contain large numbers of F and M students; the categories do not separate sharply. The difference is an 8 to 9 percentage-point shift in group composition, not a dramatic change. This supports the conclusion that the association is statistically significant but weak in strength.

Python Chart 2: Observed Count Heatmap

The heatmap displays the four observed cell counts: GP-F = 237, GP-M = 186, MS-F = 146 and MS-M = 80. The largest cell is GP-F, and the smallest cell is MS-M. This is expected partly because GP has more students overall than MS and because F students are more common than M students in the full dataset.

Raw counts alone do not prove association because larger row totals naturally create larger counts. That is why the next charts compare observed counts with expected counts. The heatmap is still important because it gives the reader the actual data structure before the chi-square and contingency coefficient are interpreted.

Python Chart 3: Standardized Residual Map

The residual map explains direction. The GP-F cell is negative (-0.80), meaning fewer F students were observed in GP than expected. The GP-M cell is positive (0.96), meaning more M students were observed in GP than expected. The MS-F cell is positive (1.09), and the MS-M cell is negative (-1.31).

No residual reaches the common ±2 review threshold. This is a key result. The table is significant overall, but no single cell is an extreme departure from independence. The largest standardized residual in absolute value is MS-M at about -1.31, so the evidence is distributed across the 2 × 2 pattern rather than dominated by one extreme cell.

Python Chart 4: Chi-Square Cell Contributions

This chart shows how much each cell contributes to the total chi-square value. The cell contributions are approximately 0.64 for GP-F, 0.92 for GP-M, 1.20 for MS-F and 1.72 for MS-M. The MS-M cell is the largest contributor because the observed count of 80 is meaningfully below its expected count of about 92.63.

The contribution chart is more specific than the p-value. It tells the analyst where the association comes from. In this example, the pattern is mainly that MS has fewer M students and more F students than the independence model would expect, while GP shows the opposite direction.

Python Chart 5: Association Strength Summary

The effect-size chart puts the strength statistics side by side. The unadjusted Contingency Coefficient is 0.0828, normalized C is 0.1170, Cramer’s V is about 0.0830 and Phi is about 0.0830. All values are close to zero.

This chart prevents overclaiming. The p-value is below .05, but the effect size is clearly small. The correct report should say that the school-by-sex association is significant but weak, not that school strongly determines sex distribution.

Python Chart 6: Observed vs Expected Counts

The dashed diagonal represents perfect agreement between observed and expected counts. The GP-F point is below the diagonal because 237 observed is lower than 249.63 expected. The GP-M point is above the diagonal because 186 observed is higher than 173.37 expected. The MS-F point is also above the diagonal, while the MS-M point is below it.

The points are not far away from the diagonal, which again supports a weak association. The plot is useful because it turns the expected-count logic into a simple visual: the closer the points are to the diagonal, the closer the table is to independence.

R Chart-by-Chart Validation

The R charts validate the same contingency coefficient result with a separate workflow. They repeat the row percentage profile, observed count heatmap, standardized residuals, chi-square contributions, effect-size summary and observed-versus-expected counts in a colorful report style. This agreement across Python, R, SPSS and Excel strengthens the interpretation.

R Chart 1: Row Percentage Profile

The R row-percentage chart confirms the Python pattern. MS has a higher F share and lower M share than GP. This is the visible association that the chi-square test detects. The difference is real but not large, which is why the adjusted C remains low.

This chart is the clearest chart for nontechnical readers because percentages are easier to compare than raw frequencies when row totals differ.

R Chart 2: Observed Count Heatmap

The R heatmap confirms the same observed counts. GP-F is the largest cell and MS-M is the smallest cell. This raw table structure matters because every later statistic is built from these counts.

Because row totals are unequal, the raw count chart should not be interpreted alone. It should be read with row percentages, expected counts and residuals.

R Chart 3: Standardized Residual Map

The R residual chart validates the same four-cell direction pattern: GP-F is below expected, GP-M is above expected, MS-F is above expected and MS-M is below expected. The largest absolute residual is still modest, so no cell should be described as extremely unusual.

This is the best chart for explaining the direction of association because positive and negative residuals show which cells drive the deviation from independence.

R Chart 4: Chi-Square Cell Contributions

The R contribution chart again shows that MS-M contributes the most to the chi-square statistic, followed by MS-F, GP-M and GP-F. This confirms the Python conclusion that the MS row contains the stronger departure from expected sex counts.

The chart also shows why the overall chi-square value is not large. The contributions add up to about 4.476, which is just enough to reach significance with one degree of freedom.

R Chart 5: Association Strength Summary

The R effect-size chart confirms the weak-strength conclusion. C, Cramer’s V and Phi are all close to .08, while adjusted C is about .117. These values are far closer to zero than to a moderate or strong association.

This is the chart that should guide the strength language in the final report. The p-value answers whether the variables are independent; this chart answers how strong the association is.

R Chart 6: Observed vs Expected Counts

The R observed-versus-expected plot confirms that all four cells are close to the independence line. Two cells are above expected and two cells are below expected. The departures are coordinated enough to produce a significant test, but not large enough to support a strong practical association.

This chart is helpful for final checking because it shows the complete expected-count logic in one figure.

Excel Results Explained

The uploaded Excel workbook gives a fully worked formula-based calculation for the same school × sex crosstab. It includes a dashboard, raw data, categorical data, observed table, expected counts, chi-square components, main worked calculation, pairwise categorical results and an Excel guide.

Excel Main Worked Result: school × sex

The Excel workbook shows exactly how the result is built. The largest chi-square component is the MS × M cell at approximately 1.72, followed by MS × F at 1.20, GP × M at 0.92 and GP × F at 0.64. This means the MS row contributes slightly more to the association pattern than the GP row.

The workbook’s dashboard interpretation is: Pearson chi-square = 4.4763, df = 1, p-value = 0.034368, C = 0.0828, adjusted C = 0.1170. The decision is significant at .05, but the strength is weak. This is the strongest way to report the result because it combines statistical significance with practical strength.

Excel Pairwise Results: Stronger Categorical Associations

The workbook also calculates many pairwise categorical associations. These are useful because they show how weak the main school-by-sex association is compared with stronger categorical relationships in the same dataset.

The top pair, Dalc × Walc, has adjusted C = 0.7118, which is far stronger than the school-by-sex adjusted C = 0.1170. This comparison helps readers understand effect-size scale. A p-value can say both relationships are significant, but the adjusted coefficient shows that one relationship is far stronger than the other.

Excel Formula Steps Used in the Workbook

SPSS, R, Python and Excel Workflows for Contingency Coefficient

The same Contingency Coefficient workflow can be reproduced in SPSS, R, Python and Excel. The steps are the same in every tool: build the crosstab, compute expected counts, run Pearson chi-square, calculate C and interpret the effect size.

Code Blocks for Contingency Coefficient

SPSS Syntax for Contingency Coefficient

* Contingency Coefficient in SPSS.
* Row variable: school.
* Column variable: sex.

SET PRINTBACK=ON MPRINT=ON.
OUTPUT CLOSE ALL.
OUTPUT NEW NAME=Contingency_Coefficient_Output.

TITLE "Contingency Coefficient: school by sex".

CROSSTABS
  /TABLES=school BY sex
  /FORMAT=AVALUE TABLES
  /STATISTICS=CHISQ PHI CC
  /CELLS=COUNT EXPECTED ROW COLUMN SRESID
  /COUNT ROUND CELL.

* Export the SPSS Viewer output as PDF.
OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Contingency-Coefficient-SPSS-Output.pdf".

Python Code for Contingency Coefficient

import pandas as pd
import numpy as np
from scipy.stats import chi2_contingency

# Load data
df = pd.read_csv("dataset.csv")

# Build observed contingency table
observed = pd.crosstab(df["school"], df["sex"])
chi2, p_value, dof, expected = chi2_contingency(observed, correction=False)

n = observed.to_numpy().sum()
contingency_c = np.sqrt(chi2 / (chi2 + n))
k = min(observed.shape)
c_max = np.sqrt((k - 1) / k)
adjusted_c = contingency_c / c_max
cramers_v = np.sqrt(chi2 / (n * (k - 1)))

expected_table = pd.DataFrame(expected, index=observed.index, columns=observed.columns)
standardized_residuals = (observed - expected_table) / np.sqrt(expected_table)
chi_square_components = ((observed - expected_table) ** 2) / expected_table
row_percentages = observed.div(observed.sum(axis=1), axis=0) * 100

print(observed)
print(expected_table)
print("Chi-square:", chi2)
print("df:", dof)
print("p-value:", p_value)
print("Contingency C:", contingency_c)
print("Adjusted C:", adjusted_c)
print("Cramer's V:", cramers_v)

R Code for Contingency Coefficient

# Contingency Coefficient analysis in R

df <- read.csv("dataset.csv")
observed <- table(df$school, df$sex)

chi_result <- chisq.test(observed, correct = FALSE)
chi_square <- as.numeric(chi_result$statistic)
df_chi <- as.numeric(chi_result$parameter)
p_value <- chi_result$p.value
n <- sum(observed)

contingency_c <- sqrt(chi_square / (chi_square + n))
k <- min(dim(observed))
c_max <- sqrt((k - 1) / k)
adjusted_c <- contingency_c / c_max
cramers_v <- sqrt(chi_square / (n * (k - 1)))

expected <- chi_result$expected
standardized_residuals <- chi_result$stdres
components <- (observed - expected)^2 / expected
row_percentages <- prop.table(observed, margin = 1) * 100

print(observed)
print(expected)
print(row_percentages)
cat("Chi-square =", chi_square, "df =", df_chi, "p =", p_value, "\n")
cat("C =", contingency_c, "Adjusted C =", adjusted_c, "Cramer's V =", cramers_v, "\n")

Excel Formulas for Contingency Coefficient

Observed count for each cell:
=COUNTIFS(row_variable_range,row_category,column_variable_range,column_category)

Expected count:
=(row_total*column_total)/grand_total

Chi-square component:
=(Observed-Expected)^2/Expected

Pearson chi-square:
=SUM(all_chi_square_components)

Degrees of freedom:
=(number_of_rows-1)*(number_of_columns-1)

p-value:
=CHISQ.DIST.RT(chi_square,df)

Contingency Coefficient:
=SQRT(chi_square/(chi_square+N))

Maximum possible C:
=SQRT((MIN(number_of_rows,number_of_columns)-1)/MIN(number_of_rows,number_of_columns))

Adjusted Contingency Coefficient:
=C/Cmax

APA Reporting Wording for Contingency Coefficient

When reporting a Contingency Coefficient, describe the two categorical variables, the crosstab pattern, Pearson chi-square result, p-value, C value, adjusted C value and practical strength. Do not report only that the chi-square test was significant.

APA-Style Full Report

Short APA-Style Version

Excel Worked-Example Wording

Common Mistakes in Contingency Coefficient Interpretation

When to Use Contingency Coefficient

Use Contingency Coefficient when both variables are categorical and you want an effect-size measure based on a chi-square test of independence. It is suitable for nominal variables such as school type, sex, address category, job category, alcohol-use category, internet access or response category.

Use a different method when at least one variable is continuous. For continuous variables, consider Pearson correlation, Spearman correlation or regression. For binary-continuous relationships, consider point-biserial correlation. For group mean differences, consider a two-sample t-test or Welch's t-test. For larger categorical tables, also consider reporting Cramer's V alongside the Contingency Coefficient.

Downloads and Resources for Contingency Coefficient

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