Effective Target Width (W_e) in HCI

• Effective target width (W_e) is a measure that adjusts nominal target dimensions by accounting for selection endpoint variability to correct for subject-specific speed–accuracy trade-offs.
• Comparative analysis shows that the univariate method (using σ_x) achieves a stronger model fit (R² ≈ 0.9675) and greater throughput stability than the bivariate alternative.
• Adopting the univariate approach aligns with Fitts’ law’s 1D correction principles, offering a reliable standard for cross-condition and inter-device benchmarking in HCI research.

Effective target width ($W_e$) is a foundational construct in the quantitative assessment of human-computer interaction performance metrics, particularly in Fitts’ law studies of pointing behavior. It serves as a data-driven normalization of the nominal target width by accounting for the observed spread in selection endpoints, thereby correcting for subjective speed–accuracy trade-offs and enabling fair throughput (TP) comparisons across conditions, participant groups, and device modalities.

1. Formal Definition and Role in Fitts’ Law

In classical Fitts’ law, movement time (MT) is modeled as

$\mathrm{MT} = a + b \cdot \mathrm{ID},$

with $\mathrm{ID} = \log_2(A/W + 1)$ where $A$ is the target amplitude and $W$ is the nominal width. However, varying speed–accuracy biases alter error rates (ER), compromising the validity of direct fits between MT and nominal ID. The effective target width remedies this by replacing $W$ with an a posteriori estimate $W_e$ derived from endpoint variability, yielding the effective index of difficulty:

$$\mathrm{ID}_e = \log_2\left( A_e / W_e + 1 \right),$$

where $A_e$ is the empirical mean amplitude. Under the normality assumption, $W_e$ is chosen such that approximately 96% of selection endpoints fall within $\pm W_e/2$, corresponding to an ER near 4%. Throughput is then computed as the mean of $\mathrm{ID}_e/\mathrm{MT}$, offering a performance metric relatively invariant to individual speed–accuracy settings (Yamanaka et al., 4 Feb 2026).

2. Univariate Versus Bivariate Calculation Methods

Two principal methodologies for computing $W_e$ have been established in the literature. The univariate method—endorsed by ISO 9241-411 and Soukoreff & MacKenzie—projects selection coordinates onto the task axis, calculating the standard deviation $\sigma_x$ of these one-dimensional signed distances:

$$\sigma_x = \sqrt{\frac{1}{n-1} \sum_j (x_j - \bar{x})^2 }.$$

The resulting univariate effective width is

$W_{e,\text{univ}} = k \cdot \sigma_x,$

where $k = \sqrt{2\pi e} \approx 4.133$ ensures an ER ≈ 4% under 1D normality.

The bivariate approach, advocated by Wobbrock et al., extends to the two-dimensional dispersion by computing

$$\sigma_{xy} = \sqrt{\frac{1}{n-1} \sum_j \big[ (x_j - \bar{x})^2 + (y_j - \bar{y})^2 \big] },$$

after rotating trials such that the task axis aligns with the $x$ axis. The formula for bivariate effective width uses the same constant:

$W_{e,\text{bi}} = k \cdot \sigma_{xy}.$

Variant	Formula	Theoretical Basis
Univariate	$W_{e,\text{univ}} = 4.133\,\sigma_x$	1D Normal, task axis
Bivariate	$W_{e,\text{bi}} = 4.133\,\sigma_{xy}$	2D spread, all axes equally

The origin of $k$ is rooted in the entropy of a Gaussian distribution, signifying that intervals of length $k\sigma$ contain ≈96.12% of the probability mass under the normal assumption (Yamanaka et al., 4 Feb 2026).

3. Experimental Comparison: Design and Protocol

A large-scale crowdsourced experiment involving 346 mouse-based participants (mean age ≈ 45 years) was conducted via Yahoo! Crowdsourcing to systematically assess the comparative performance of univariate and bivariate $W_e$ calculations under varied speed–accuracy biases. Participants performed ISO-style 2D pointing tasks arrayed around a circular layout, crossing 2 amplitudes (320, 500 px) and 3 widths (20, 45, 100 px). Speed–accuracy bias was manipulated within subjects via task instructions:

• Accurate: “Make as few errors as possible, don’t worry about speed.”
• Neutral: “Be as fast and accurate as possible.”
• Fast: “Be as quick as possible, don’t worry about errors.”

Each block comprised 150 selection trials, with order randomization for biases and amplitude–width pairs. Logged measures included movement time, endpoint coordinates, and first-click error rates. The primary dependent variables for model evaluation included $W_e$ (via $\sigma_x$ or $\sigma_{xy}$), $A_e$, $\mathrm{ID}_e$, and TP (Yamanaka et al., 4 Feb 2026).

4. Empirical Outcomes and Statistical Model Fit

The univariate projection yielded the strongest correspondence between empirical data and the Fitts’ law model. Key results included:

• Nominal ID (no correction): $R^2 \approx 0.8727$ on pooled data.
• Best univariate model ($\sigma_x$, current–previous center axis, nominal $A$): $R^2 \approx 0.9675$.
• All bivariate variants: $R^2 < 0.96$.
• Information criteria (AIC, BIC): Best univariate model favored by $\Delta \mathrm{AIC} > 4$.
• Throughput stability (across speed–accuracy biases):
  • TP$_\text{diff}$ (relative range): 4.74% univariate, ≥5.5% bivariate.
  • TP$_\text{cv}$ (coefficient of variation): 2.77% univariate, ≥3% bivariate.
• Monte Carlo resampling: Univariate model dominated in $R^2$ and TP$_\text{cv}$ for all $N\geq20$; bivariate only sporadically optimal at very low $N$ (N=5).

This establishes the superior empirical validity and stability of univariate $W_e$ calculations, regardless of axis definition or choice between nominal vs. effective amplitude (Yamanaka et al., 4 Feb 2026).

5. Theoretical and Practical Rationale for the Univariate Approach

The principal rationale for univariate $W_e$ is its alignment with the theoretical premise of Fitts’ law: performance is primarily constrained by the one-dimensional accuracy demand along the motion axis, i.e., the requirement to cross target boundaries in that direction. Variability orthogonal to motion does not directly determine “hit” vs. “miss” and incorporating it via bivariate spread dilutes the correlation between observed behaviors (especially biases) and the task’s actual decision boundaries.

The coefficient $k=4.133$ is justified exclusively for 1D normal distributions; its application to bivariate spread lacks analogous theoretical grounding. Empirically, changes in $\sigma_x$ capture the majority of bias-induced variability (increase by ≈54% from “accurate” to “fast” conditions), while $\sigma_y$ rises less (≈35%), supporting the dominance of motion-axis scatter.

Projection onto the task axis is both methodologically simpler and ensures consistency with standards underpinning device and technique benchmarking in HCI. This approach minimizes confounding effects arising from unrelated variability and preserves the centrality of TP as a normalized metric (Yamanaka et al., 4 Feb 2026).

6. Implications and Recommendations for Research and Evaluation

Evidence substantiates the adoption of $W_e = 4.133\sigma_x$ (task axis defined from previous to current target center) as the default standard for cross-condition throughput normalization and Fitts’ law model fitting. Essential reporting practices include explicit identification of calculation variants (univariate or bivariate), axis convention, amplitude type, and model fit indices ($R^2$, AIC, BIC) alongside TP.

Both nominal and effective amplitude $A_e$ are acceptable, with minimal differences; $A_e$ may slightly benefit throughput stability across biases, while nominal $A$ excels in pure model fit, rendering either suitable for most analyses. Recruitment of a minimum of 20 participants is critical to avoid spurious findings favoring inferior bivariate approaches in underpowered studies.

Implications extend to tool development and comparative benchmarking: software implementations should default to the univariate projection to ensure methodological rigor and inter-study comparability. In device, technique, or user group assessments, reliance on univariate $W_e$ yields metrics that authentically reflect pointing efficacy rather than merely risk propensity (Yamanaka et al., 4 Feb 2026).

7. Summary and Impact on Fitts’ Law–based HCI Evaluation

Effective target width, when calculated via the univariate projection onto the motion axis, restores the foundational 1D normal-distribution correction central to ISO-conformant throughput computation. It outperforms bivariate spread measures in fitting empirical selection data across mixed speed–accuracy biases, furnishing more consistent, stable, and interpretable performance metrics. This standardization enhances the validity and replicability of cross-condition and inter-device comparisons in 2D pointing evaluations, reinforcing the primacy of Fitts’ law as an analytic framework in HCI performance assessment (Yamanaka et al., 4 Feb 2026).