*Crowding* is the failure to recognize an object due to surrounding clutter. Our visual crowding survey measured 13 crowding distances (or “critical spacings”) twice in each of 50 observers. The survey includes three eccentricities (0, 5, and 10 deg), four cardinal meridians, two orientations (radial and tangential), and two fonts (Sloan and Pelli). The survey also tested foveal acuity, twice. Remarkably, fitting a two-parameter model—the well-known Bouma law, where crowding distance grows linearly with eccentricity—explains 82% of the variance for all 13 × 50 measured log crowding distances, cross-validated. An enhanced Bouma law, with factors for meridian, crowding orientation, target kind, and observer, explains 94% of the variance, again cross-validated. These additional factors reveal several asymmetries, consistent with previous reports, which can be expressed as crowding-distance ratios: 0.62 horizontal:vertical, 0.79 lower:upper, 0.78 right:left, 0.55 tangential:radial, and 0.78 Sloan-font:Pelli-font. Across our observers, peripheral crowding is independent of foveal crowding and acuity. Evaluation of the Bouma factor, *b* (the slope of the Bouma law), as a biomarker of visual health would be easier if there were a way to compare results across crowding studies that use different methods. We define a *standardized Bouma factor b*′ that corrects for differences from Bouma's 25 choice alternatives, 75% threshold criterion, and linearly symmetric flanker placement. For radial crowding on the right meridian, the standardized Bouma factor *b*′ is 0.24 for this study, 0.35 for Bouma (1970), and 0.30 for the geometric mean across five representative modern studies, including this one, showing good agreement across labs, including Bouma's. Simulations, confirmed by data, show that peeking can skew estimates of crowding (e.g., greatly decreasing the mean or doubling the *SD* of log *b*). Using gaze tracking to prevent peeking, individual differences are robust, as evidenced by the much larger 0.08 *SD* of log *b* across observers than the mere 0.03 test–retest SD of log *b* measured in half an hour. The ease of measurement of crowding enhances its promise as a biomarker for dyslexia and visual health.

*flankers*, and is characterized by the

*crowding distance*(or “critical spacing”), which is the center-to-center distance from target to flanker at which recognition attains a criterion level of performance. Crowding distance increases linearly with eccentricity (Bouma, 1970; Kooi, Toet, Tripathy, & Levi, 1994; Levi & Carney, 2009; Pelli et al., 2004; Toet & Levi, 1992), and increases with target-flanker similarity (Andriessen & Bouma, 1976; Chastain, 1982; Kooi et al., 1994; Leat, Li, & Epp, 1999; Nazir, 1992; Pelli et al., 2004), as well as the number of distractors (Grainger, Tydgat, & Issele, 2010; Strasburger et al., 1991). Crowding also occurs for moving stimuli (Bex & Dakin, 2005; Bex, Dakin, & Simmers, 2003). For a review of the crowding literature, see Herzog, Sayim, Chicherov, and Manassi (2015), Levi (2008), Pelli and Tillman (2008), Strasburger (2020), Strasburger, Rentschler, and Juttner (2011), and Whitney and Levi (2011). Among the normally sighted, crowding was first reported in the periphery and, after some debate, has now been convincingly demonstrated in the fovea (Atkinson, Pimm-Smith, Evans, Harding, & Braddick, 1986; Coates, Levi, Touch, & Sabesan, 2018; Flom, Heath, et al., 1963; Liu & Arditi, 2000; Malania et al., 2007; Pelli et al., 2016; Siderov, Waugh, & Bedell, 2013; Toet & Levi, 1992).

*E*

_{2}value for acuity is more than 5 times larger than the

*E*

_{2}value for crowding (

*E*

_{2}= 2.72 for acuity;

*E*

_{2}= 0.45 for crowding) (Latham & Whitaker, 1996; Petrov & Meleshkevich, 2011; Rosenholtz, 2016; Song, Levi, & Pelli, 2014; Strasburger, 2020). This seems inconsistent with a common cause. Here, we use our data from 50 observers to study the relationship between acuity and crowding in the fovea.

*b*was 0.12, with a 0.31 standard deviation (

*SD*) of log

*b*. Using awaited fixation, geometric mean

*b*was higher (0.20), with a lower

*SD*of log

*b*(0.18). The awaited-fixation histogram (red) is compact. The unmonitored fixation histogram (green) is much broader, extending to much lower values of

*b*. Our interpretation of the broader histogram and lower geometric mean

*b*in unmonitored fixation is that observers occasionally “peek”—that is, fixate near an anticipated location of the target instead of the fixation cross as instructed. Indeed, at the end of the Results section, we present a quantitative peeking model showing that peeking reduces geometric mean

*b*and broadens its distribution, consistent with the observed results.

*b*(slope of crowding distance vs. eccentricity) estimated from Equation 10, to minimize error in fitting log

*ŝ*. Here, crowding distance

*ŝ*is the required center-to-center spacing (in deg) for 70% correct report of the middle letter in a triplet.

^{2}. The white background never changed throughout the experiment; the black crosshair and letters were drawn on it. The observer viewed the screen binocularly at one of several different viewing distances. The software required a special keypress by the experimenter at the beginning of every block with a new observer or a new viewing distance to affirm that the new viewing distance (eye to screen) was correct as measured with a tape measure and that the screen center was orthogonal to the observer's line of sight. To measure crowding and acuity in the fovea, the viewing distance was 200 cm. For ±5 and ±10 deg eccentricity the distance was 40 cm, and for ±20 and ±30 deg eccentricity it was 20 cm. The long viewing distance gives good rendering of small stimuli; the short viewing distance results in a wide angular subtense of the display, which allows presentation of peripheral targets on either side of a central fixation. Stimuli were rendered using CriticalSpacing.m software (Pelli et al., 2016) implemented in MATLAB 2021 (MathWorks, Natick, MA) using the Psychtoolbox (Brainard, 1997; Pelli, 1997). Every Sloan letter was at least 8 pixels wide, and every Pelli digit was at least 4 pixels wide.

*SD*of repeated measurements of threshold spacing

*s*is roughly proportional to the mean spacing, but the

*SD*of log spacing

*S*is independent of mean spacing. Therefore, our fitting minimizes the root-mean-square (RMS) error in log spacing

*S*= log

_{10}

*s*. The fitting is nonlinear (using the MATLAB fmincon function) because we minimize error in

*S*, whereas each model is linear in

*s*, not

*S*. We estimated the participant, meridional, crowding orientation, and font factors by solving several models (see model equations in Table 4).

*SSE*):

*S*is the

_{i}*i*-th log crowding distance, and \({\hat S_i}\) is the

*i*-th predicted log crowding distance. The variance explained by each model is

*is the mean log crowding distance.*

_{i}S_{i}*F*test was used for pairwise model comparison. The model with fewer parameters is referred to as “simple,” and the model with more parameters is referred to as “full.” After calculating the sum of squared errors

*SSE*

_{simple}and

*SSE*

_{full}for each model (Equation 1), we calculated the

*F*-statistic:

*n*

_{full}is the number of parameters in the full model,

*n*

_{simple}is the number of parameters in the simple model, and

*N*is the number of observations. The

*p*value is estimated using the

*F*distribution. A

*p*value less than 0.05 indicates that the model with more parameters provides a significantly better explanation of the data.

*R*is calculated by Equation 2.

^{2}*b*′ as the slope of crowding distance versus radial eccentricity multiplied by a correction factor that accounts for methodological differences from Bouma's number of choices (25), threshold criterion (75% correct), and linear spacing (vs. log).

*b*across studies that used various numbers of response choices (e.g., nine Sloan letters or two orientations of a tumbling T), various threshold criteria (e.g., 70% or 75% correct), and linear or log flanker spacing. Including the present one, we know of five studies that have compared crowding distance across meridians. We have taken Bouma (1970) as the standard for this standardized way of reporting the strength of crowding.

*∆S*is illustrated in Figure 3B. Using the Coates et al. (2021) reanalysis of Bouma's (1970) data, we estimated the Bouma factor for a 75% threshold criterion applied to Bouma's (1970) results (see the Bouma factor paragraph in the Discussion section).

*P*is a Weibull function of log spacing

*S*:

*n*, and using our own estimate of the steepness parameter β = 2.3.

*n*, and thus guessing rates γ = 1/

*n*, we corrected for guessing:

*P*gives us the corresponding “true” proportion correct criterion

*P*to apply after correction for guessing. Similarly, applying correction for guessing to the psychometric function (Equation 4) gives us the “true” proportion correct:

^{*}*S*at its threshold criterion

*P*.

^{*}*S*relative to the Bouma standard:

*P*

^{*}and \(P^*_{{\rm{Bouma}}}\) are the “true” proportion correct threshold criteria computed by Equation 5 from the study's criterion

*P*and Bouma's

*P*

_{Bouma}= 0.75 (Andriessen & Bouma, 1976).

*B*= log

*b*were assessed using an analysis of variance (ANOVA) with

*B*as the dependent variable. Two sample comparisons were made with the Wilcoxon rank-sum test. We report Pearson's

*r*correlation coefficient for test–retest reliability and correlations of crowding distance.

*b*(see Equation 10 below). Foveal crowding and acuity are presented as crowding distance (deg) and acuity as letter size (deg). Figure 5 plots a scatter diagram of estimates from first versus second session for each combination of font and task. The second session improved over the first only for the Pelli font (Figure 5B), with a ratio of geometric mean retest:retest = 0.88. This training benefit was much smaller (and insignificant) for the Sloan font (0.95), presumably because Sloan is more similar (than Pelli) to familiar fonts and thus benefits less from learning. In general, each threshold is derived from a QUEST staircase with 35 trials, which takes about 3.5 minutes and has very good reproducibility. The analyses performed in the following sections are based on the geometric mean threshold across both sessions.

*SD*for radial Sloan with two meridians (with awaited fixation) in Figure 1 corresponds to the 0.17 total

*SD*with four meridians (also with awaited fixation) in Table 3. The 0.08

*SD*for Sloan radial crowding in Figure 8 corresponds to the 0.08

*SD*across observers in Table 3. Meridian contributed the most variance.

*ŝ*is crowding distance (in deg), φ is radial eccentricity (in deg), and φ

_{0}(in deg) and

*b*(dimensionless) are positive fitted constants (Bouma, 1970; Rosen, Chakravarthi, & Pelli, 2014). The dimensionless slope

*b*is the Bouma factor. The horizontal intercept is –φ

_{0}, and the vertical intercept is φ

_{0}

*b*(Liu & Arditi, 2000; Strasburger et al., 2011; Toet & Levi, 1992).

*E*

_{2}value that is the eccentricity at which threshold reaches twice its foveal value (Levi et al., 1985). In the Bouma law (Equation 10),

*E*

_{2}= φ

_{0.}

*R*

^{2}= 82.45%) by the two-parameter linear Bouma law (Equation 10), showing that most of variation in crowding in our data is explained by eccentricity. Just two degrees of freedom,

*b*and φ

_{0}, suffice to fit all 650 data points (13 thresholds measured in each of 50 observers). The estimated slope

*b*was 0.23, just over half of Bouma's 0.4. (We return to this apparent discrepancy below in Discussion: Standardized Bouma factor.) Our database consists of measurements at five locations with radial and tangential flankers and two fonts. To capture the effect of these parameters on the Bouma factor we propose an extended version of the Bouma law.

*b*

_{θ}, which allows

*b*to depend on the meridian θ (right, left, up, or down):

*b*from Equation 10 now represents the

*geometric mean*of

*b*

_{θ},

*b*= \(10\hat{\;}({\rm mean}(\log b_{\theta}\!))\). Note that the meridian is undefined at the fovea.

*o*depends on the observer:

_{i}*R*

^{2}= 89%) and crowding orientation (

*R*

^{2}= 93%). Adding the target-kind factor explains hardly any more variance, with an increase from 92.54% to 92.63%. Finally, the most enhanced model, with an observer factor, explains 94% of variance. The models are all cross-validated, so the additional variance explained is not a necessary consequence of the increase in parameters. If the additional parameters were overfitting the training data, then we would find less variance accounted for in the left-out test data.

*F*(3,196) = 92.76,

*p*< 0.001, and post hoc analysis showed that the Bouma estimates at each meridian were significantly different from each other (all

*p*< 0.001, corrected for multiple comparisons). For each meridian, we also estimated the eccentricity φ

_{0}at which the crowding distance reached twice its foveal value; φ

_{0}was 0.37 ± 0.02 deg for the right, 0.29 ± 0.02 deg for the left, 0.22 ± 0.01 deg for the lower, and 0.17 ± 0.01 deg for the upper meridians.

*z*= −0.73,

*p*= 0.49). The standardized (see section on corrected Bouma factor below) tangential Bouma factor was small: 0.13 on the right and 0.14 on the left meridian. The tangential:radial ratio was 0.60 in the right meridian and 0.50 in the left.

*z*= 3.58,

*p*< 0.001). The model performance is slightly improved by adding the target-kind factor (Equation 13). This factor contributed little to the overall variance explained by the model because most of the data came from trials with the same target kind (Sloan letters). So, even though excluding target kind as a factor from the model caused inaccurate predictions for the Pelli font, the reduction in variance explained was negligible because nearly all of the dataset is based on the Sloan font. We anticipate that the target-kind factor will account for more variance in datasets that focus on comparing target kinds.

*SD*of the log of the threshold. The radial Bouma factor for the Sloan font varied approximately twofold across observers (

*SD*of log

*b*= 0.08). The variance was very similar for tangential flankers (

*SD*of log

*b*= 0.08) and larger for the Pelli font (

*SD*of log

*b*= 0.11). Foveal acuity

*a*and foveal crowding distance

*s*also varied twofold. For crowding, the φ

_{0}values also varied twofold and ranged between 0.17 and 0.37 (Song et al., 2014). We also report the SD between test and retest for the log Bouma factor estimated with radial flankers and the Sloan font (Figure 8B). For each observer, we fit one log Bouma factor for the test session and one log Bouma factor for the retest. Differences across observers were much larger than those of test–retest. The 0.08

*SD*of the log Bouma factor across observers is nearly three times larger than the 0.03

*SD*of test and retest, showing that one such Bouma factor estimate, measured in half an hour, is enough to distinguish individual differences. That measurement of log

*b*consists of eight thresholds (2 eccentricities × 4 meridians) and 280 trials (8 thresholds × 35 trials/threshold).

*F*(1,398) = 42.3,

*p*< 0.001, and there was no interaction between eccentricity and meridian,

*F*(3,392) = 0.513,

*p*= 0.674. This shows that the growth of crowding distance with eccentricity is actually more than linear. Indeed, the Coates et al. (2021) reanalysis of Bouma (1970) shows a similar supralinearity. Motivated by this finding, we invited 10 observers already in the main dataset (0, 5, and 10 deg) to also measure crowding distance at 20 and 30 deg eccentricity.

_{0}(in deg),

*b*, and

*c*are degrees of freedom.

*r*between two measurements. Rows are sorted so that the average correlation decreases from top to bottom. We found that log crowding distance measured on the right meridian at 10 deg eccentricity with the Sloan font and radial flankers yielded the highest average correlation with other log crowding distances (

*r*= 0.39 with all, and

*r*= 0.41 when fovea is excluded). Foveal log crowding distance measured with Pelli font yielded the smallest average correlation with the rest of the log crowding distances.

*r*= 0.54. On the other hand, when we changed the location only and kept the stimulus properties (e.g., radial flankers, Sloan font, 5 deg) we obtained a much lower correlation of

*r*= 0.32. This indicates that, when correlating crowding distances, location matters more than any other stimulus property.

*b*and intercept φ

_{0}

_{0}and

*b*, which are anticorrelated,

*r*= −0.51 (geometric mean across meridians).

*b*and the negative intercept φ

_{0}. Estimating two parameters requires two measurements, but many crowding studies report crowding distance at only one eccentricity. In the complete case, we have thresholds

*s*

_{0}and

*s*at eccentricities 0 and φ, and we use the definition of the Bouma factor

*b*as the slope

*b*= (

*s*–

*s*

_{0})/(φ – 0). In the incomplete case, we have only threshold

*s*at eccentricity φ. One might try to estimate the missing foveal threshold

*s*

_{0}or negative intercept φ

_{0}, but the simplest thing to do is to neglect φ

_{0}(pretend it is zero), and estimate \(\hat b = s/\varphi \). The estimate has fractional error \( \in\, =\, ( {\hat b - b} )/b = {\varphi _0}/\varphi \). Thus, neglecting φ

_{0}, possibly because the foveal threshold is unknown, leads to a fractional error φ

_{0}/φ. The studies in Table 2 used eccentricities φ ≥ 2 deg. Our measurements estimated φ

_{0}= 0.24 deg. Thus, at 2 deg or beyond, the fractional error in the estimated Bouma factor will be at most 0.24/2 = 12%. The fractional error drops to 5% at 5 deg and to 2% at 10 deg.

*b*′ for each study (Table 6). Figure 11 compares crowding across studies by plotting the standardized Bouma factor vs. meridian.

*t*

_{kind}is meant to account for in Equation 13. The tumbling clocks may be more like each other than the nine Sloan letters are and therefore produce larger crowding distance. Almost all other studies (rows 3, 6, 9, and 10) cluster above the Sloan font and show the same dependence on meridian. In general, we found that Courier New letters (row 8) produced the smallest radial standardized Bouma factor (0.23 on the right meridian) and tumbling Ts (row 10) produced the largest radial standardized Bouma factor (0.39 on the right meridian).

*r*= 0.64). (It may be mere coincidence, since they are different measures made with different fonts, but we were struck by the near equality of geo. mean foveal size and spacing thresholds: 0.07 deg acuity and 0.08 deg crowding distance, Figure 8). Thus, foveal acuity predicts foveal crowding but not peripheral crowding. If peripheral crowding is of interest (e.g., as a possible limit to reading speed), then it should be measured, as it is not predicted by foveal acuity.

*p*of the trials, and that the peeking eye movement travels only a fraction

*k*of the distance from the crosshair to the possible target location, with a Gaussian error (0.5 deg

*SD*in

*x*and in

*y*). The peeking distribution has a mode corresponding to each possible target location, but at a fraction

*k*of the possible target eccentricity. Gaze position is randomly sampled from the peeking distribution on a proportion

*p*of trials and otherwise from the awaited-fixation distribution.

*r*of target-flanker spacing to actual target eccentricity, where

*b*

_{true}is the true Bouma factor and steepness β is 2.3. For simplicity, the model omits threshold criterion and finger-error probability delta. Bouma factor

*b*is estimated by 35 trials of QUEST, assuming the true psychometric function, with a prior guess = 0.11 of

*r*and an assumed

*SD*= 2 of log

*r.*

*-*fixation distribution

*-*fixation distribution was 3500 actual gaze positions at stimulus onset (35 trials × 50 participants × 2 sessions) measured with an EyeLink eye tracker in our awaited

*-*fixation dataset. Recall that the stimulus was presented only when the gaze had been within 1.5 deg of the crosshair center for 250 ms.

*b*was lowest (0.03) for peeking with one possible location and highest with four (0.37), given a true Bouma factor

*b*= 0.30. With no peeking (

*p*= 0), gaze position was from the awaited

*-*fixation distribution, and the model estimate of Bouma factor was 0.28, very close to the true value of 0.30. The

*SD*of log Bouma factor

*b*was highest (0.40) with two possible locations and lowest (0.22) with four.

*b*= 0.3; see arrow on the vertical axis in Figure 14D) in estimating Bouma factor grows with the proportion

*p*and fraction

*k*of peeking (Figure 14D). Observing that the error of the estimated Bouma factor grows proportionally with

*k*and that its

*SD*grows proportionally with

*k*

^{2}(note the parabolic shape in Figure 14E), we produced new plots (Figures 14F, 14G) showing that the geometric mean of

*b*is roughly linear with

*p*×

*k*and the

*SD*of log

*b*is roughly linear with \(\sqrt {p{\rm{\;}} \times {\rm{\;}}k} \).

*p*,

*k*, and the true Bouma factor, our peeking-observer model predicts the estimated Bouma factor

*b*. We estimated

*p*and

*k*by fitting the model using maximum likelihood optimization. The higher the log-likelihood, the better the fit. We wanted to determine how often (

*p*) and how far (

*k*) the observer peeks and the estimated Bouma factor

*b.*In Figures 15A and 15B, the scatter and breadth of the log-likelihood distribution result in a broad confidence interval for the product

*p*×

*k*. To estimate the error of the fit, we bootstrapped it by removing 25% of data at each iteration (

*n*= 100). For unmonitored fixation (Figure 15A), the bootstrapped parameters were consistent with high peeking (0.5 <

*p*×

*k*≤ 1) and rejected no peeking (

*p*×

*k*= 0). For awaited fixation (Figure 15B), the bootstrapped parameters were consistent with low peeking (

*p*×

*k <*0.5) and rejected high peeking (

*p*×

*k*= 1). For each dataset, Figures 15E and 15F show that the human data are well matched by the simulated histogram. The two geometric means of

*b*match, as do the standard deviations of log

*b*. Our peeking model of eye position and crowding predicts the estimated Bouma factor in both cases, showing that the unmonitored fixation results are well fit by high peeking, and the awaited fixation results are well fit by low peeking. One simple model of eye position and crowding fits all our data.

*b*′ as the reported Bouma factor

*b*(ratio of crowding distance to radial eccentricity) multiplied by a correction factor that account for differences in task from Bouma's 25 choice alternatives, 75% threshold criterion, and linear flanker symmetry. Bouma reported a “roughly” 0.5 slope for radial letter crowding versus eccentricity (Bouma, 1970). Andriessen and Bouma (1976) later reported a slope of 0.4 for crowding of lines. Coates et al. (2021) reanalyzed Bouma's original data with various threshold criteria so we interpolated between the 70% and 80% thresholds to estimate the 75% threshold. Estimating Bouma factor from Bouma's original data using this criterion yielded a Bouma factor of 0.35, in line with modern estimates of 0.3 (Table 6 and Supplementary Table S2). Figure 11A shows that the corrected Bouma factor

*b*′ ranges from 0.23 for Courier New letters to 0.39 for tumbling T measured with radial flankers on the right meridian. That residual difference may be due to target kind (Coates et al., 2021; Grainger et al., 2010). This is further supported by our finding that Bouma factor was 0.78 lower for the Sloan font than for the Pelli font.

*b*measured with the Sloan font and radial flankers for test–retest is much lower than the standard deviation of log Bouma factor across observers (0.03 vs. 0.08).

*o*, in the enhanced model. Adding the observer factor improved the explained variance from 92.6% to 94% (Table 4). Although this effect may seem negligible at first, we found that the Bouma factor varied twofold across observers, ranging from 0.20 to 0.38 (Figure 8A). A similar, twofold variation was observed for all other thresholds that we estimated (Figure 8A).

_{i}*r*= 0.39 averaged across all peripheral locations) and hardly any between fovea and periphery (

*r*= 0.11). We also found that crowding measured with radial flankers correlated highly with crowding measured with tangential flankers at the same location (

*r*= 0.53 for the right meridian,

*r*= 0.50 for the left meridian). The threshold measurement that best predicted all other peripheral thresholds (excluding the fovea), with a correlation

*r*= 0.41, was radial Sloan crowding at 10 deg in the right meridian.

*r*= 0.54) when the location was the same and the stimulus configuration was changed, than (

*r*= 0.32) when the stimulus configuration was the same and the location changed. Correlation of crowding distance depends more on location than configuration. Paralleling our result, Poggel and Strasburger (2004) found only a weak correlation across meridians for visual reaction times. Surprisingly little is known about the spatial correlations of basic measures such acuity and contrast sensitivity.

*b*from 0.12 to 0.20 and nearly halved the

*SD*of log

*b*from 0.31 to 0.18. Histograms are shown in Figure 1. This peeking-observer model assumes, first, that performance on each trial depends solely on target eccentricity (relative to gaze position); second, that the observer peeks on a fraction

*p*of the trials and fixates near the crosshair on the rest of the trials; and, third, that the location of the peek is a fraction

*k*of the distance from the fixation mark to the anticipated target location.

*p*. In awaited fixation, peeking is prevented by using gaze-contingent display and discarding any trials where gaze left the fixation cross while the target was present. Suppose there are two possible target locations. The Bouma factor distribution is unimodal for low values of

*p*and becomes bimodal for high values for

*p*. Our unmonitored

*b*histogram is bimodal and is best fit with a peeking probability of 50%. Our awaited

*-*fixation

*b*histogram is unimodal and best fit by peeking restricted to the 1.5 deg from the crosshair allowed by the gaze tracker. Upgrading from the bimodal to the unimodal

*b*distribution raised the mean

*b*from 0.12 to 0.27 and nearly halved the standard deviation of log

*b*from 0.31 to 0.19.

- 1 The well-known Bouma law—crowding distance depends linearly on radial eccentricity—explains 82% of the variance of log crowding distance, cross-validated. Our enhanced Bouma law, with factors for observer, meridian, and target kind, explains 94% of the variance, cross-validated. The very good fit states the central accomplishment of the paper and shows how well the linear Bouma law fits human data.
- 2 The Bouma factor varies twofold across observers, meridians, and crowding orientations.
- 3 Consistent with past reports, five asymmetries each confer an advantage expressed as a ratio of Bouma factors: 0.62 horizontal:vertical, 0.79 lower:upper, 0.78 right:left, 0.55 tangential:radial, and 0.78 Sloan font:Pelli font.
- 4 As noted above, the Bouma factor varies twofold across observers. Differences across observers are much larger than those of test–retest. The 0.08
*SD*of log Bouma factor across observers is nearly triple the 0.03*SD*of test–retest when log*b*is measured in half an hour (2 eccentricities × 4 meridians = 8 thresholds and 8 × 35 deg = 280 trials). - 5 The growth of crowding distance with eccentricity is supralinear, which becomes obvious when measurements extend out to 30 deg eccentricity. The linear fit is adequate for most purposes.
- 6 Crowding distance measured at 10 deg eccentricity along the right meridian is the best predictor of average crowding distance elsewhere (average
*r*= 0.39). - 7 Peripheral crowding is independent of foveal crowding and foveal acuity.
- 8 Simulations and data show that peeking can skew estimates of crowding (e.g., greatly decrease the mean or double the SD of log
*b*). Thus it is important to minimize peeking, e.g. by using awaited fixation (with gaze tracking) or manual tracking of a moving crosshair (without gaze tracking).

**Authors contributed:**Jan Kurzawski collected data with awaited fixation, analyzed data with awaited and unmonitored fixation, and wrote the paper; Augustin Burchell performed the initial analysis of the unmonitored fixation data. Darshan Thapa supervised testing for data with unmonitored fixation; Najib Majaj discussed, designed, analyzed, and wrote; Jonathan Winawer discussed, and helped design, analyze, and write; Denis Pelli directed the project, including design, software, data collection, analysis, and writing. Jan Kurzawski, Najib Majaj, and Denis Pelli met most days for a year to do the revisions.

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