Gwet's AC1 Agreement Coefficient Calculator

Gwet's AC1 answers the same question as Cohen's kappa — how much do two raters agree beyond chance? — but it estimates the “beyond chance” part differently so it does not misbehave when one category dominates. You give the tool a q×q agreement matrix where nₖₗ is the number of items Rater A put in category k and Rater B put in category l. The diagonal holds the agreements; everything off it is a disagreement.

From the grand total N, the row marginals and the column marginals, the calculation follows Gwet (2008):

• Observed agreement: pₐ = (Σ nₖₖ) / N
• Averaged marginal: πₖ = (pₖ₊ + p₊ₖ) / 2
• Chance agreement: pₑ = Σ πₖ(1 − πₖ) / (q − 1)
• Gwet's AC1: AC1 = (pₐ − pₑ) / (1 − pₑ)

The one line that matters is the chance term. Cohen's kappa uses pₑ(κ) = Σ pₖ₊·p₊ₖ, the product of the two raters' marginals. When 95% of items are one category, that product is roughly 0.90, so κ's correction eats almost all of the observed agreement. Gwet's Σ πₖ(1 − πₖ)/(q − 1) uses each category's own prevalence in a way that shrinks — not grows — as a category becomes dominant. The result is that AC1 keeps reporting the raters' real agreement where kappa collapses. This tool shows both numbers together and flags “kappa-paradox territory” whenever they diverge by 0.15 or more.

A useful property falls out of that definition: because Σ πₖ² is at least 1/q, the chance term pₑ can never exceed 1/q, so the denominator 1 − pₑnever reaches zero. AC1 is therefore always defined — even at 100% agreement on a single category, the exact case where Cohen's kappa becomes an undefined 0/0. Finally the AC1 value is mapped to a strength-of-agreement band on your chosen scale — Landis & Koch (1977), Altman (1991), or Fleiss (1981) — for a one-line verdict. All arithmetic is exact and runs in your browser; nothing is uploaded.

Scope note: this version computes the unweighted, two-rater AC1 for complete data. The Gwet (2008) standard-error and confidence-interval estimator is a planned follow-up — we ship the verified point estimates rather than an unverified interval.

• Gwet, K. L. (2008). Computing inter-rater reliability and its variance in the presence of high agreement. British Journal of Mathematical and Statistical Psychology, 61(1), 29–48 — the AC1 definition and chance-agreement formula.
• Gwet, K. L. (2014). Handbook of Inter-Rater Reliability, 4th ed., Advanced Analytics LLC — canonical AC1 notation and benchmarking guidance.
• Wongpakaran, N. et al. (2013). A comparison of Cohen's Kappa and Gwet's AC1 when calculating inter-rater reliability coefficients. BMC Medical Research Methodology, 13:61 — documents the kappa paradox AC1 corrects.
• Cohen, J. (1960). A Coefficient of Agreement for Nominal Scales. Educational and Psychological Measurement, 20(1), 37–46 — the Cohen's κ chance term shown for comparison.
• Landis, J. R., & Koch, G. G. (1977). The Measurement of Observer Agreement for Categorical Data. Biometrics, 33(1), 159–174 — the strength-of-agreement bands.

Every formula on this page was cross-checked against these sources on 2026-07-18, and the AC1 chance term is verified inside the tool against its algebraically-equivalent sum-of-squares form. Your agreement matrix never leaves your browser.