Cohen's Kappa Calculator
How much two raters agree beyond chance — the statistic that shows why raw agreement percentages are almost always misleading.
κ = (observed agreement − expected agreement) ÷ (1 − expected agreement).
How the cohen's kappa calculator works
κ = (observed agreement − expected agreement) ÷ (1 − expected agreement). It rescales agreement so that chance is 0 and perfection is 1.
The correction matters enormously when one category dominates. Two raters who both label 95% of cases "negative" will agree about 90% of the time by luck alone, so 92% raw agreement is barely better than guessing — and kappa says so.
Formula: κ = (pₒ − pₑ) / (1 − pₑ)
Worked examples
| Inputs | Cohen's kappa | Note |
|---|---|---|
| Fifty cases, moderate agreement | 0.4 | κ = 0.40 despite 70% raw agreement |
| Perfect agreement | 1 | κ = 1 |
| A rare category | 0.368421 | 94% agreement, κ only 0.36 |
FAQFrequently asked questions
Why not just report percentage agreement?
Because chance agreement is often very high. Two raters labelling a rare condition will agree most of the time whatever they do.
What is a good kappa?
Landis and Koch's labels — slight, fair, moderate, substantial, almost perfect at 0.2, 0.4, 0.6 and 0.8 — are conventional but arbitrary. Most fields want at least 0.6.
Can kappa be negative?
Yes, when agreement is worse than chance. It is rare and usually means the raters understood the categories differently.
What is the kappa paradox?
With a very unbalanced category split, kappa can be low despite near-total agreement, because chance agreement is nearly total too. Report both numbers.
What about more than two raters?
Fleiss' kappa generalises it. For ordered categories, weighted kappa credits near-misses.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.