Reaction time distributions are universally positively skewed: there is a sharp rise on the left (fast responses) and a long tail on the right (slow responses). One natural explanation is that RT arises from the multiplicative combination of many processing stages, just as the normal distribution arises from additive combination. By the multiplicative central limit theorem, the product of many independent positive random variables converges to a log-normal distribution.
Parameterization
The log-normal distribution for RT has two parameters:
σ = standard deviation of ln(RT)
E[RT] = exp(μ + σ²/2)
Var[RT] = [exp(σ²) − 1] × exp(2μ + σ²)
Note that μ and σ are the parameters of the log-transformed variable, not of RT itself. The mean and variance of the actual RT are nonlinear functions of both parameters. A three-parameter version adds a shift parameter θ, giving the shifted log-normal f(t − θ) for t > θ, which accounts for a minimum possible response time.
Empirical Performance
Ulrich and Miller (1993) conducted a systematic comparison of distributional models and found that the log-normal often provides fits comparable to or better than the ex-Gaussian. The log-normal tends to fit the body of the RT distribution well, though it can underpredict the extreme right tail in some tasks. Van Zandt (2000) showed that no single parametric family dominates across all experimental conditions, and the choice between log-normal and ex-Gaussian should be guided by theoretical commitments and specific data characteristics.
The log-normal arises when RT is the product of many independent factors: RT = X₁ × X₂ × ... × Xₙ, so ln(RT) = ln(X₁) + ln(X₂) + ... + ln(Xₙ). This is plausible if processing stages have durations that scale proportionally rather than additively — e.g., if later stages take longer when earlier stages are slow, as might happen with cascaded activation or resource sharing.
Analysis Considerations
A practical advantage of the log-normal model is that standard statistical methods (t-tests, ANOVA, linear regression) become applicable after log-transforming the data. This approach has been widely adopted, though it has been criticized because effects on the log scale do not straightforwardly map to effects on the original RT scale. Specifically, if experimental effects are additive on the original scale, they become non-additive on the log scale, and vice versa.
Despite these concerns, the log-normal remains one of the most commonly used distributional models of reaction time, valued both for its theoretical motivation and its analytical convenience.