0.6
0.3
0.5
0.08
7.5
0.4432
0.7568
2
0.6
0.3
0.5
0.08
7.5
0.4432
0.7568
2
Fleiss' Kappa is a statistical measure used to assess the reliability of agreement between multiple raters when assigning categorical ratings. Unlike Cohen's Kappa, which is limited to exactly two raters, Fleiss' Kappa generalizes the concept to any fixed number of raters, making it an indispensable tool in inter-rater reliability studies across medicine, psychology, and social sciences.
This calculator provides a simplified approach by accepting the observed agreement proportion (Pₒ) and expected agreement proportion (Pₑ) directly, allowing you to compute the kappa statistic quickly and interpret the degree of agreement among your raters.
Fleiss' Kappa is calculated using the fundamental formula:
$$\kappa = \frac{P_o - P_e}{1 - P_e}$$
Where:
The numerator (Pₒ − Pₑ) represents the agreement beyond chance, while the denominator (1 − Pₑ) normalizes this value against the maximum possible agreement beyond chance. A kappa of 1.0 indicates perfect agreement, 0 indicates agreement no better than chance, and negative values suggest systematic disagreement.
To compute Pₒ in a full Fleiss' Kappa analysis, one sums the proportion of agreeing rater pairs across all subjects:
$$P_o = \frac{1}{N \cdot n \cdot (n-1)} \sum_{i=1}^{N} \sum_{j=1}^{k} n_{ij}(n_{ij} - 1)$$
Where N is the number of subjects, n is the number of raters, and nᵢⱼ is the count of raters who assigned subject i to category j. The expected agreement Pₑ is computed as the sum of squared marginal proportions for each category.
The standard interpretation scale proposed by Landis and Koch (1977) classifies agreement strength from 'poor' (κ < 0) through 'slight', 'fair', 'moderate', 'substantial', to 'almost perfect' (κ ≥ 0.81).
Interpreting Fleiss' Kappa requires consideration of both the numerical value and the study context. The Landis-Koch scale provides general guidance:
However, these benchmarks are not absolute. In clinical diagnosis, a kappa above 0.60 may be considered adequate, while in high-stakes classification tasks, values above 0.80 are often required. The standard error helps assess whether the observed kappa is statistically significant when compared to zero (no agreement beyond chance).
Inputs
Results
Three doctors rating 50 patient scans into 'normal' vs 'abnormal'. With 80% observed agreement and 50% expected by chance, κ = 0.60, indicating moderate agreement.
Inputs
Results
Five raters classifying 100 articles into 3 categories. With high observed agreement (92%) against 33% chance agreement, κ = 0.88, showing almost perfect agreement.
Cohen's Kappa measures agreement between exactly two raters, while Fleiss' Kappa extends to any number of raters (≥2). Fleiss' Kappa assumes that all subjects are rated by the same number of raters, but the specific raters may vary across subjects. Cohen's Kappa, by contrast, requires the same two raters for every subject.
For Pₒ, create a matrix where each row is a subject and each column is a category, with cells containing the count of raters who assigned that category. Then compute the pairwise agreement proportion across all subjects. For Pₑ, compute the marginal proportion for each category (total assignments to that category divided by total assignments), then sum the squares of these proportions.
Yes. A negative kappa indicates that the observed agreement is less than what would be expected by chance alone, suggesting systematic disagreement among raters. This may indicate that raters are applying contradictory criteria or that the rating scheme is poorly defined.
Generally, a minimum of 30 subjects is recommended for stable kappa estimates. Larger samples (50–100+) provide more precise estimates. The number of raters also matters — more raters per subject can improve the stability of the estimate, though 3–5 raters is common in practice.
Yes. When the distribution of categories is highly imbalanced (one category dominates), the expected agreement Pₑ becomes large, which can result in lower kappa values even when observed agreement is high. This is known as the 'kappa paradox'. Consider reporting both kappa and the raw agreement proportions.
Under the null hypothesis of no agreement beyond chance (κ = 0), the test statistic z = κ / SE follows approximately a standard normal distribution. If |z| > 1.96, the kappa is statistically significant at the 0.05 level. This calculator provides an approximate standard error for this purpose.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
Random Number Generator
Advanced & Specialized Statistical Tools
Central Limit Theorem Calculator
Advanced & Specialized Statistical Tools
Empirical Rule Calculator
Advanced & Specialized Statistical Tools
Chebyshev's Theorem Calculator
Advanced & Specialized Statistical Tools
Monte Carlo Estimation Calculator
Advanced & Specialized Statistical Tools
Power Analysis Calculator
Advanced & Specialized Statistical Tools
