When Ensembles Beat Items: Conditional Aggregate Dominance in Ensemble Perception
A group’s average can be easier to report than any of its members. Try a one-second test, then explore the paper’s worked example of when a pooled estimate beats a reconstruction from individual items.
A one-second test
Twelve circles will appear for one second, followed by a mask. Then a single circle appears. Decide whether it was one of the twelve.
What you saw
The twelve circles from the test, with the test circle on the right. It was not in the set. Its diameter is exactly the average of the twelve.
The test circle was never shown. The twelve circles come in four diameters, three of each, and the test circle’s diameter is exactly their mean. After briefly seeing an array of circles, an observer may judge whether a probe circle is larger than the array’s mean more accurately than report the size of a particular member (Ariely, 2001). With faces, observers can judge a crowd’s average emotion while struggling to report which expression belonged to which face.
Such tasks raise the question of how a group summary can remain accessible when its members are poorly reportable. This paper asks when a pooled summary should beat a reconstruction from the items themselves.
Two routes to the same average
The paper compares two noisy estimates of one quantity, the mean of the display actually shown. The experimenter can calculate it exactly. The observer has no noiseless access to it. The pooled route forms a group estimate directly. The item-mediated route builds the estimate from item representations, here from two circles.
The worked example in Section 2.4 fixes every parameter before set size varies. Display variance is held at 16. Item noise has variance 4 with correlation 0.25. The pool takes in up to eight effective items, with pooling noise 8 and a comparison floor of 1. The values are in squared feature units, chosen for calculation rather than fitted to observer data.
Which route has the smaller error?
- Effective items pooled
- 8at most eight in this example
- Item error minus pooled error
- +6.5positive when pooling wins
- Smaller error
- Pooled routeconditional error variance
8 circles shown. The pool takes in 8 effective items. Pooling is exhaustive at this set size, so both readings give the same pooled error. The pooled route has the smaller error, by 6.5.
Computed from Eqs. (4), (6) and (7) of the preprint with the Section 2.4 parameters. With comparison noise at 1, the values match the paper’s table. The paper also reports the eight-circle reversal when comparison noise rises to 8. The other combinations apply the same equations to the same parameters.
| Circles shown | Effective items pooled | Item route | Pooled, only some items get in | Pooled, every item gets in |
|---|---|---|---|---|
| 4 | 4 | 6.5 | 3.0 | 3.0 |
| 8 | 8 | 8.5 | 2.0 | 2.0 |
| 16 | 8 | 9.5 | 3.0 | 2.0 |
At eight circles, raising only the pooled route’s comparison variance from 1 to 8 gives a pooled error of 9, above the item route’s 8.5, so the ordering reverses.
The equations behind the bars
Eq. (4) gives the item route’s error. The first term is the error from averaging only part of the display. The second is sensory error, whose correlated part sets a floor that averaging cannot remove.
VI = (1 − k/n) sx²/k + σI² [ρ + (1 − ρ)/k]
Eq. (6) gives the pooled route’s error when only some items get in, with pooling noise and a comparison or readout floor.
VA = (1 − keff/n) sx²/keff + σP²/keff + σR²
Eq. (7) applies when every item gets in but overlap limits the effective count. The two agree while pooling is exhaustive, because the first term of Eq. (6) vanishes at keff = n.
VA† = σP²/keff + σR²
The worked example sets sx² = 16, k = 2, σI² = 4, ρ = 0.25, keff = min(n, 8), σP² = 8 and σR² = 1. At sixteen circles the item error is (1 − 2/16) × 16/2 + 4 (0.25 + 0.75/2) = 9.5, and the pooled error when only some items get in is (1 − 8/16) × 16/8 + 8/8 + 1 = 3.
Pooling wins when enough independent information gets in
If the pooled route’s conditional error variance is no larger than the item route’s, the pooled route has no greater mean-squared error. For Gaussian mean-shift discrimination it also has no lower detection power (Proposition 1). That ordering is classical decision theory, in the line from Rao’s work on estimation. The contribution is the decomposition.
Each route’s error splits into parts that experiments can change separately. These are effective sample size, finite-population subset error, correlated sensory noise, pooling noise, attentional selection, and comparison or readout noise. Pooled dominance then requires enough effectively independent pooled information to overcome pooling and downstream noise, not merely more displayed items (Corollary 1).
Once the pool is full, the two readings separate. If only some items get in, adding items raises pooled error by sx²(1/n₁ − 1/n₂). That amount is set by the display, with nothing fitted. If every item gets in but they overlap, pooled error stays flat (Proposition 2). The ideal 1/√n law is therefore a limiting case, not a rule for every ensemble task.
A sharper test does not assume that noise stays fixed across set sizes. At each set size, regress the squared error of the mean report on the display’s own variance. If only some items get in, the slope is 1/keff − 1/n, and once the pool is full it changes between set sizes by exactly 1/n₁ − 1/n₂, provided pooling and readout noise do not depend on the display’s variance. If every item gets in, and under resource-limited integration of every item with independent noise, the slope is zero. Unequal weights spread over every item give a positive slope that falls roughly as 1/n. A slope that rises with set size therefore requires a saturated pool.
Attention can act at selection, pooling, comparison or readout. Two tasks with the same stimuli can therefore show opposite attention effects, which the paper treats as more diagnostic than asking whether ensembles need attention.
The paper also specifies a direct test of the two routes. Present and mask an array, then cue either its mean or a randomly chosen subset of locations. On mean trials the observer adjusts a probe to report the mean. On item trials the observer reports the size at each cued location, and the item estimate is built from those reports afterwards. Repeated presentations of the same arrays then estimate each route’s bias and variance against the display actually shown.
What the paper does not claim
The worked example’s values are chosen for calculation, not fitted to observer data. The paper calls it an illustration, not validation. A predictive test would calibrate effective counts and noise, fix them and the pooling interpretation, and then predict held-out set sizes and interventions without refitting.
The result commits neither to arithmetic-mean computation by the brain nor to pre-attentive pooling. The two routes are an information-flow model, not a claim about anatomical implementation. Loss of item identity is treated as a separate compression hypothesis.
A null set-size benefit alone does not falsify the account, and a flat set-size profile alone does not identify a unique mechanism.
The one-second test on this page demonstrates a known effect. It records nothing, and it is not data from the paper.
Related research
The paper relates its comparison to my work on allocation and aggregate verification, which applies the classical comparison-of-experiments argument when an individual record is a state-independent randomisation of an aggregate record. That assumption does not establish a relation between perceptual routes, so here the error ordering must be measured.
It is part of my research on cognition and artificial intelligence.
Paper and citation
J. Armstrong, “When Ensembles Beat Items: Conditional Aggregate Dominance in Ensemble Perception,” PsyArXiv preprint, 2026. DOI 10.31234/osf.io/uzfam_v1. Version 1, posted on 21 October 2026.