Fleiss Kappa Model Overview

Fleiss Kappa is a multi-rater agreement coefficient for categorical ratings. It asks how much the observed agreement exceeds the agreement expected from the overall prevalence of the categories, then scales that excess by the maximum improvement that was still possible. Unlike raw percent agreement, Fleiss Kappa explicitly adjusts for chance agreement produced by unequal category use.

What Question Does Fleiss Kappa Answer?

The method answers: When several classifications are assigned to each item, do the classifications agree more than would be expected from their overall category frequencies? In this worked example, each student is the rated item, and G1, G2 and G3 grade bands act as three classifications of that same student. The analysis therefore evaluates whether the three grade occasions classify students consistently across four bands.

Worked Design at a Glance

There are 649 students and three classifications per student, creating 1,947 ratings. Each grade is converted into one of four categories: Low (<10), Medium (10–13), High (14–16) and Very high (17+). Across all ratings, Medium is the dominant category at 53.78%, while Very high is least common at 5.39%.

How Agreement Is Built

For each student, the method counts how many of the three classifications fall in each category. An item receives agreement P_i = 1 when all three classifications match and P_i = 1/3 when two match and one differs. In this dataset, 413 students have perfect three-way agreement and 236 have a two-against-one pattern; no student has three completely different grade bands.

How Fleiss Kappa Differs from Cohen Kappa

Cohen's Kappa is designed for two raters or two classifications. Fleiss Kappa handles three or more ratings per item by averaging item-level pair agreement and correcting that average for the category prevalences pooled across all ratings. It is not calculated by averaging the three pairwise Cohen kappas.

Primary Worked Result

The average observed item agreement is 0.7575758, expected chance agreement is 0.3754573 and Fleiss Kappa is 0.6118372. The coefficient indicates that approximately 61.18% of the agreement that remained possible beyond chance was achieved. Conventional verbal labels can be supplied as descriptive aids, but the coefficient, study design, observed agreement and prevalence structure should remain the primary report.

Quick Answer: Fleiss Kappa Result

The worked Fleiss Kappa analysis treats G1, G2 and G3 as three classifications of each student's grade band. Every student has exactly three ratings, and all ratings use the same four mutually exclusive categories.

Table of Contents

1. Research question, hypotheses and rating design
2. When to use Fleiss Kappa
3. Fleiss Kappa formula and calculation
4. Variables and data dictionary
5. Complete worked results
6. Fleiss Kappa versus related agreement measures
7. Fleiss Kappa calculator workflow
8. Python chart interpretations
9. R chart pairs and explanations
10. Python, R, SPSS, SAS, Stata and Excel
11. Expandable software code
12. Assumptions, limitations and advanced interpretation
13. APA reporting for Fleiss Kappa
14. Common mistakes and corrections
15. Reports and worked Excel download
16. Related Salar Cafe guides
17. Frequently asked questions
18. Conclusion

Research Question, Hypotheses and Rating Design

Research Question

How consistently do the three grade occasions classify the same students into the four grade bands after accounting for the agreement expected from the overall category distribution?

Null and Alternative Hypotheses

For general guidance on stating statistical hypotheses, see Null and Alternative Hypothesis, P-Value and Type I and Type II Error.

Why G1, G2 and G3 Can Act as Classifications

The usual language describes different human raters, but the calculation only requires multiple categorical classifications per item. Here the three grade occasions classify each student into the same grade-band system. The interpretation is therefore temporal or repeated-classification consistency, not agreement among three independent teachers. This design limitation must be stated clearly.

Nominal Versus Ordinal Treatment

The categories have an obvious order, but the primary workbook uses unweighted Fleiss Kappa and therefore treats every disagreement equally. A one-band disagreement and a three-band disagreement receive the same nominal penalty. A weighted multi-rater coefficient would answer a different question and should be reported separately rather than silently replacing the verified analysis.

When to Use Fleiss Kappa

Simple Test-Selection Logic

Agreement Is Not Association

Two raters can be strongly associated yet disagree systematically. For example, one rater may consistently place cases one category higher than another. A Chi-Square Test of Independence can detect association, but it does not measure exact agreement. Fleiss Kappa specifically rewards category matches and compares them with a chance benchmark.

Why Percent Agreement Is Not Enough

Raw agreement ignores how frequently the categories are used. If one category dominates, several raters may agree often simply because they all select the common category. Fleiss Kappa discounts that expected overlap. See Frequency Distribution and Categorical and Quantitative Variables for foundational context.

Fleiss Kappa Formula and Calculation

The calculation has three stages: calculate item-level agreement, average those item agreements and estimate chance agreement from the pooled category proportions.

Step 1: Build the Item-by-Category Count Matrix

For student i, let n_ij be the number of the three classifications assigned to category j. Each row sums to n = 3. Example rows include [1, 2, 0, 0], [0, 3, 0, 0] and [0, 0, 2, 1].

Step 2: Calculate Item Agreement

For [1, 2, 0, 0], the calculation is:

For [0, 3, 0, 0], all three classifications match:

Step 3: Average Item Agreement

The dataset contains 413 perfect-agreement items and 236 two-against-one items:

Step 4: Calculate Category Proportions

Step 5: Calculate Expected Agreement

Step 6: Calculate Fleiss Kappa

Numerator and Denominator Meaning

The numerator, 0.3821184344, is the observed agreement beyond chance. The denominator, 0.6245426768, is the maximum improvement beyond chance that was available. Dividing the two shows that 61.18% of the available chance-corrected agreement was achieved.

Variables and Data Dictionary

Grade-Band Definitions

Category Counts by Grade Occasion

G3 contains fewer Low classifications and more High or Very high classifications than G1. This prevalence shift affects expected agreement and also explains why agreement is not perfect even when adjacent grade occasions remain strongly related.

Complete Fleiss Kappa Results

Primary Result Table

Item-Level Agreement Pattern

Most Common Rating-Count Patterns

Pairwise Agreement as Supporting Diagnostics

These pairwise percentages help diagnose where disagreement occurs, but they do not replace the multi-classification coefficient. The primary result remains the single Fleiss Kappa calculated from all three classifications simultaneously.

Fleiss Kappa Versus Related Agreement Measures

Fleiss Kappa Versus Cohen Kappa

The phrase Cohen's Kappa vs Fleiss Kappa reflects the most common selection question. Use Cohen Kappa for two raters and Fleiss Kappa for three or more ratings per item. Calculating Cohen Kappa for G1–G2, G1–G3 and G2–G3 answers three pairwise questions; averaging them is not the Fleiss coefficient.

Fleiss Kappa Versus Weighted Kappa

The grade bands are ordinal, so a weighted method can be scientifically useful when a one-band disagreement should receive less penalty than a three-band disagreement. However, weighting changes the estimand. This post preserves the verified unweighted result and treats weighting as an additional analysis, not a correction to the original calculation.

Fleiss Kappa Versus Krippendorff-Type Agreement

Alternative multi-rater measures may handle missing data, unequal numbers of raters, different scale types or custom disagreement functions more naturally. Fleiss Kappa is strongest when the design is balanced and the category set is common across all ratings.

Fleiss Kappa Versus Chi-Square

A Chi-Square Test examines distributional association or fit. Fleiss Kappa examines agreement. A strong association among ratings does not guarantee exact category agreement, so the two methods should not be treated as interchangeable.

Fleiss Kappa Calculator: Step-by-Step Workflow

A reliable Fleiss Kappa calculator should accept an item-by-category count matrix or transform raw rater columns into that matrix. It must verify that every row has the same number of ratings and that category counts are nonnegative integers.

Calculator Inputs for the Worked Example

Calculator Step 1: Check Row Sums

Each item row must sum to three. A row such as [1, 2, 0, 0] is valid; a row summing to two or four indicates missing, duplicated or improperly aggregated ratings.

Calculator Step 2: Calculate Every P_i

Do not estimate observed agreement from the number of rows with perfect agreement alone. The partial-agreement rows contribute one-third each because one of the three rating pairs agrees.

Calculator Step 3: Pool Category Prevalence

Divide each category total by 1,947. The Medium category proportion must be 1,047/1,947 = 0.5377504. Incorrect denominators are a common source of wrong expected agreement.

Calculator Step 4: Confirm the Final Coefficient

The calculator should return P̄ ≈ 0.7575758, P_e ≈ 0.3754573 and κ ≈ 0.6118372. A materially different result usually indicates the wrong matrix orientation, unequal row sums or pairwise Cohen Kappa being used instead.

Calculator Step 5: Interpret the Components

Always report the observed and expected agreement beside κ. Two studies can have similar κ values with very different raw agreement or prevalence structures.

Six Python Fleiss Kappa Chart Interpretations

The Python workflow presents category prevalence, item-level agreement, category relationships, the chance-correction components, common rating patterns and a final result summary. Each chart is interpreted using exact values from the worked analysis.

Chart 1: Category PrevalencePython

Pattern

Medium dominates the rating pool, while Very high is rare.

Key Values

Low 20.65%, Medium 53.78%, High 20.18%, Very high 5.39%.

Interpretation

Unequal prevalence raises the amount of agreement expected by chance, especially through the large squared Medium proportion.

Why It Matters

Expected agreement is driven by these pooled category frequencies, not by a uniform 25% category assumption.

Next step: examine item-level agreement before applying the chance correction.

Chart 2: Item Agreement DistributionPython

Pattern

The distribution has two values: perfect agreement at 1 and partial agreement at one-third.

Key Values

413 students have P_i = 1; 236 have P_i = 0.333333.

Interpretation

Most students remain in the same grade band across all three occasions, while the remainder have one classification that differs.

Why It Matters

The weighted average of these item scores produces P̄ = 0.7575758.

Next step: inspect how category use shifts among G1, G2 and G3.

Chart 3: Category Relationship Across Grade OccasionsPython

Pattern

The occasions are similar but not identical; G3 shifts away from Low and toward High and Very high.

Key Values

Low falls from 157 in G1 to 100 in G3, while Very high rises from 24 to 46.

Interpretation

Systematic distribution movement limits exact agreement even when most differences occur between neighboring grade bands.

Why It Matters

Fleiss Kappa captures exact category consistency, not merely upward or downward trend similarity.

Next step: separate observed agreement from prevalence-based expected agreement.

Chart 4: Kappa ComponentsPython

Pattern

Observed agreement is much higher than expected agreement.

Key Values

P̄ = 0.757576, P_e = 0.375457, κ = 0.611837.

Interpretation

The raw excess agreement is 0.382118, which represents 61.18% of the possible improvement beyond chance.

Why It Matters

This chart prevents κ from being interpreted as the raw percentage of identical ratings.

Next step: examine which item count patterns create perfect and partial agreement.

Chart 5: Agreement PatternsPython

Pattern

All-Medium is the most common perfect-agreement pattern, followed by all-Low and all-High.

Key Values

[0,3,0,0] occurs 235 times; [3,0,0,0] 80 times; [0,0,3,0] 76 times.

Interpretation

The coefficient is supported by many full-agreement items, but the category distribution is not balanced.

Why It Matters

Pattern counts reveal the substantive source of agreement rather than presenting κ as a detached single number.

Next step: consolidate the design, agreement components and final coefficient.

Chart 6: Fleiss Kappa Result SummaryPython

Pattern

The summary confirms a positive chance-corrected agreement coefficient.

Key Values

649 subjects, 3 classifications, P̄ 0.757576, P_e 0.375457 and κ 0.611837.

Interpretation

The classifications are meaningfully consistent but leave substantial room for disagreement reduction.

Why It Matters

This is the most compact chart for presentations, while the earlier figures explain how the coefficient was formed.

Next step: report the coefficient with its design and agreement components.

R Fleiss Kappa Chart Pairs

The R workflow presents six Fleiss Kappa figures in three paired rows. Each pair combines closely related questions so readers can compare prevalence with item agreement, item patterns with occasion-specific distributions, and observed agreement with the final result.

R Pair 1: Overall category proportions and item agreement

Chart 1 Meaning

Prevalence Establishes the Chance Baseline

The Medium band contributes more than half of all ratings. Squaring and summing the four proportions produces expected agreement of 0.3754573.

Check: 402 + 1,047 + 393 + 105 = 1,947 ratings.

Chart 2 Meaning

Most Items Have Full Three-Way Agreement

Perfect agreement occurs for 413 students, while 236 students have a two-one split. The absence of three-category disagreements keeps the mean item agreement relatively high.

Check: 413 + 236 = 649 students.

R Pair 2: Rating-count patterns and grade occasions

Chart 3 Meaning

The Dominant Pattern Is Three Medium Ratings

The [0,3,0,0] pattern is most common. Partial-agreement patterns are mostly adjacent-category splits such as Low/Medium or Medium/High.

The ten most common patterns account for nearly the entire sample.

Chart 4 Meaning

Category Profiles Shift Across Time

G3 has fewer Low and more High or Very high classifications than G1 and G2. This reveals genuine classification movement rather than random rater noise alone.

G3 Very high = 46 versus G1 Very high = 24.

R Pair 3: Observed versus expected agreement and final summary

Chart 5 Meaning

Observed Agreement Clearly Exceeds Chance

The 0.382118 difference between observed and expected agreement is large enough to yield κ = 0.611837 after scaling by the possible improvement.

Observed 0.757576; expected 0.375457.

Chart 6 Meaning

One Coefficient Summarizes All Three Classifications

The final summary reinforces that this is a single multi-classification analysis. Pairwise kappas or raw percentages should be supporting diagnostics only.

Primary report: Fleiss κ = 0.611837 for 649 items and 3 classifications.

Fleiss Kappa in Python, R, SPSS, SAS, Stata and Excel

Expandable Code for Fleiss Kappa

import numpy as np
import pandas as pd
from statsmodels.stats.inter_rater import fleiss_kappa

bands = ["G1_band", "G2_band", "G3_band"]
categories = [1, 2, 3, 4]

# One row per student; one column per category.
counts = np.array([
    [(row[bands] == category).sum() for category in categories]
    for _, row in data.iterrows()
], dtype=int)

assert np.all(counts.sum(axis=1) == 3)
kappa = fleiss_kappa(counts, method="fleiss")
print(kappa)

ratings <- data.frame(G1 = g1_band, G2 = g2_band, G3 = g3_band)
classes <- 1:4
count_matrix <- t(apply(ratings, 1, function(x) {
  sapply(classes, function(k) sum(x == k))
}))
stopifnot(all(rowSums(count_matrix) == 3))
# Run a validated Fleiss kappa function on count_matrix or ratings.

Band 1 count: =COUNTIF($B5,$H$4)+COUNTIF($D5,$H$4)+COUNTIF($F5,$H$4)
Repeat for Bands 2–4.
Item agreement: =(SUMSQ(BandCounts)-Raters)/(Raters*(Raters-1))
Observed agreement: =AVERAGE(ItemAgreementRange)
Category proportion: =CategoryTotal/(Subjects*Raters)
Expected agreement: =SUMSQ(CategoryProportionRange)
Fleiss kappa: =(ObservedAgreement-ExpectedAgreement)/(1-ExpectedAgreement)

* Recode G1, G2 and G3 into the same four grade bands.
* Verify frequencies and pairwise crosstabs.
* Use validated multi-rater syntax or Python/R integration for Fleiss kappa.
* Do not average pairwise Cohen kappa coefficients.

Assumptions, Limitations and Advanced Interpretation

The Prevalence Effect

Medium accounts for 53.78% of all classifications, so chance agreement is not small. This can reduce κ relative to raw agreement. The appropriate response is not to discard the coefficient automatically, but to report P̄, P_e, category proportions and κ together.

The Bias or Marginal-Profile Effect

G1, G2 and G3 do not use categories in exactly the same proportions. G3 shifts toward higher bands. Because Fleiss expected agreement pools all ratings, the coefficient summarizes overall agreement but does not isolate which classifier has the most different marginal profile. The grade-occasion chart provides that diagnostic context.

Interpreting Negative Fleiss Kappa

Negative κ means observed agreement is below the agreement expected from category prevalence. That can occur when classifiers systematically choose different categories. A negative value is not merely “no agreement”; it signals patterned disagreement.

Sample Size and Precision

A larger item count generally supports more stable estimates, but precision also depends on the number of classifications, number of categories and prevalence distribution. When formal inference is required, report a method-specific standard error or confidence interval from a validated implementation and name the method used.

Category-Specific Agreement

The overall coefficient can be supplemented by category-specific positive agreement, confusion patterns, prevalence indices or pairwise tables. These diagnostics are especially useful when one category is clinically or educationally important.

Reliability Versus Validity

Agreement does not prove that the categories are correct. Three classifications can agree consistently and still be systematically wrong. Reliability and validity must be evaluated separately.

APA Reporting for Fleiss Kappa

A strong report names the design, item count, number of classifications, category set, observed agreement, expected agreement and Fleiss Kappa. The worked workbook does not supply a p-value or confidence interval, so the exact worked sentence should remain descriptive.

Copy-Ready Worked Paragraph

Common Fleiss Kappa Mistakes and How to Correct Them

Fleiss Kappa Reports and Worked Excel Download

Related Salar Cafe Guides

Fleiss Kappa Conclusion

Fleiss Kappa provides a transparent way to measure agreement across three or more categorical classifications while adjusting for chance agreement caused by category prevalence. In the worked grade-band analysis, 649 students received three classifications across four categories. Observed agreement was 0.757576, expected agreement was 0.375457 and Fleiss Kappa was 0.611837.

The coefficient should not be reported alone. The strongest interpretation combines κ with the category proportions, item-level agreement pattern and the exact rating design. Here, 413 students had perfect three-way agreement, 236 had a two-against-one pattern, and Medium was the dominant category. These details explain both the strength of the observed consistency and the prevalence-based chance correction.

For the next step, compare this multi-classification approach with Cohen's Kappa for two ratings, the Intraclass Correlation Coefficient for continuous measurements and Cross-Tabulation for detailed categorical disagreement patterns.