17.1Independent errors as a special case
Setting aside who may participate does not by itself settle how their estimates should be combined. Consider the stipulated truth 10. Before any exchange, the four estimates [8, 9, 11, 12] average to 10, so the group mean has absolute error 0, while mean individual absolute error is 1.5. Afterward, the estimates [9.5, 10, 11, 11.5] leave mean individual absolute error at 0.75, yet their mean is 10.5, whose absolute error is 0.5. Individual accuracy has improved while aggregate accuracy has worsened.
That arithmetic counterexample makes the aggregation problem visible, but it does not identify a psychological cause. The before-and-after lists are constructed, not observed from an influence experiment. Their point is narrower: individual error and group-mean error are different quantities, and progress in one does not entail progress in the other.
The usual defense of averaging therefore rests on a substantive claim about the errors themselves. Error independence is not produced by having four respondents with four names. It requires assumptions or evidence about how their mistakes arise. Only then can one reasonably expect errors in different directions to cancel rather than accumulate. If the respondents instead share a common source of error, their agreement may reproduce the same mistake at each point in the average. Whether independence obtains is thus something to establish, not something a group roster supplies.
A stipulated positive countercase marks the contrast. If reliable new evidence of the truth is received and all four estimates become 10, both error measures are then 0. That convergence tracks the evidence. It should not be confused with convergence produced by pressure to agree, which can make estimates alike without making any of them better.
The qualification is a matter of what independence licenses: with independent errors, averaging has a principled basis for aggregation. Averaging can still be useful when errors are dependent, but the usual benefits cannot simply be presumed.
Once independence is granted, averaging has a familiar justification: uncorrelated deviations cancel, and the mean estimates the shared truth more reliably than a typical individual does. But the justification depends on what “more reliably” names. The two quantities in E06 separate immediately. With the stipulated truth at 10, the first set of estimates, 8, 9, 11, and 12, yields a mean of exactly 10. Its mean individual absolute error is 1.5, while the absolute error of the group mean is 0. The second set, 9.5, 10, 11, and 11.5, gives a mean of 10.5. Individual accuracy improves, with mean absolute error falling to 0.75, yet the group mean is now off by 0.5.
So the usual benefit of averaging cannot be transferred to dependent errors without argument. Dependence can leave the mean centered while individuals remain dispersed, as in the first set; or it can reduce dispersion while shifting the center, as in the second. Which outcome deserves to be called better depends on the measure of estimation quality one has committed to, and no arithmetic feature of the sets settles that choice. Nor do the numbers diagnose a psychological process: they are invented counterexamples, not observations of how people came to their estimates.
This limits application rather than destroying it. Averaging remains a defensible procedure when one knows enough about the error structure to expect cancellation of the relevant deviations, or when individual error is the quantity of interest. What cannot be presumed is the package: that narrowing individual disagreement tracks improved collective accuracy.
A conditional countercase marks the boundary. If reliable new evidence of the stipulated truth became available and all four estimates moved to 10, both measures would fall to zero together. Improvement through evidence is therefore coherent even where independence fails; the difficulty is identifying such evidence rather than mistaking convergence for it.
17.2What social influence changes
Dependence among errors is one way aggregation loses its ordinary advantages. A common estimate gives a concrete mechanism for that dependence, and exposure need not act on dispersion alone: it can move the center of the responses as well as their spread. Since the two changes bear on different measures of success, contraction and accuracy can come apart.
Take the stipulated truth 10. Before exposure, the four estimates are 8, 9, 11, and 12. Their mean is 10, and the mean absolute individual error is 1.5. After exposure to a common estimate, suppose the responses become 9.5, 10, 11, and 11.5. The spread has narrowed, and mean absolute individual error drops to 0.75: on the typical-response measure, influence helps. The center, however, has shifted to 10.5, so the absolute error of the group mean rises from 0 to 0.5. One and the same contraction therefore improves individual accuracy while degrading the aggregate's estimate. What changed was not only how far apart the answers lie, but where their center sits relative to the truth.
That result is an arithmetic construction, not participant data and not a claim about how often real groups display this pattern; it identifies no psychological cause. Lorenz and colleagues report an estimation-task study examining how social influence can reduce the benefit of aggregating independent estimates (Lorenz, Rauhut, Schweitzer and Helbing, 2011), and the stipulated numbers here illustrate that kind of tension without measuring it. Nor is convergence intrinsically distorting: if the incoming information were reliable evidence of the truth, all four estimates would become 10 and both errors would be 0. The qualification is local to factual estimation, which differs from normative disagreement, and it is no general objection to deliberation. The question after exposure is therefore not simply whether answers converged, but how each of the two error measures moved.
Reduced spread is not yet a verdict on either score. Having seen that exposure can shift both the spread and the center of a set of estimates, consider what the spread alone licenses. In the stipulated E06 sets, the estimates 8, 9, 11, and 12 span four units, while 9.5, 10, 11, and 11.5 span two. That is a real contraction of dispersion, and it is the whole of what the second set's tightness establishes.
Accuracy is a separate question because it requires comparison with the stipulated truth of 10. The positive countercase makes the point cleanly: when reliable new evidence of that truth is supplied, all four estimates become 10, and the responses are both maximally concentrated and correct. Yet concentration does not entail that result. If respondents merely trimmed their estimates toward one another, or toward a single common estimate, the same narrowing could appear without any of the estimates tracking the truth. Tightness is therefore neither sufficient for accuracy nor evidence of it.
The cause of convergence is likewise underdetermined. Pressure and evidence can leave the same distribution. To call a narrow spread coercive would require process information, such as evidence that respondents had abandoned privately held judgments to avoid standing apart, and that supposition must be established rather than read off the numbers. To call it informational would require that the new agreement rests on reasons or evidence the discussion made available, something averaging, which merely combines existing estimates, does not supply by itself.
The inference available here is deliberately modest: a fall in spread records a change in how far responses differ, nothing more. Testing accuracy needs the stipulated truth in view; identifying the mechanism needs evidence about how the estimates were formed. E06 is invented arithmetic built to show that these questions come apart, not participant data, and it supports no claim about how often either pattern occurs in real groups.
17.3Individual and collective outcomes
The relevant comparison is between the accuracy of each person’s estimate and the accuracy of the aggregate formed from those estimates. These targets can move differently even when the values become more alike.
Take the stipulated truth to be 10. In the initial array, the four estimates are 8, 9, 11, and 12. Their mean is exactly 10, so the group mean has zero absolute error. Yet the average individual absolute error is 1.5, because the absolute errors are 2, 1, 1, and 2, whose average is 1.5. After the values are 9.5, 10, 11, and 11.5, the average individual absolute error falls to 0.75: three estimates move closer to 10, the third remains at 11, and the range narrows. But the new mean is 10.5. Its absolute error is 0.5. On the declared measures, average individual accuracy improves while the group estimate becomes less accurate.
This is possible because average individual error rewards proximity to the truth, whereas error of the mean is determined by the sum’s displacement from the truth. A distribution can be tighter around a value that is itself biased, so individual gains and aggregate loss coexist. The example therefore separates two notions of convergence: convergence that brings estimates toward a stipulated truth, and convergence that merely brings them toward one another. Even a lower average individual error does not guarantee a better aggregate: the original positive and negative deviations cancelled in the mean, while the later ones do not.
A separate stipulated countercase makes the contrast clear. If reliable new evidence of the stipulated truth causes all four estimates to become 10, both average individual absolute error and absolute group-mean error are 0. These stipulated numbers cannot tell whether convergence was produced by evidence or by pressure; that attribution requires further information. The present construction is invented arithmetic, not observed participant data, and it establishes only that the two metrics can diverge.
Because the two quantities were declared in advance, their divergence settles one question and leaves another open. Average individual absolute error falls from 1.5 to 0.75, while the error of the group mean rises from 0 to 0.5. That is not a contradiction: the first metric asks how far the typical estimate sits from the stipulated truth of 10, and the second asks where the average estimate itself sits. Compressing the spread of the four numbers can bring individuals closer to the target while simultaneously shifting their mean away from it. A claim of improvement is therefore incomplete until it names the quantity improved.
The inference has a boundary worth marking. These are invented sets of numbers — [8, 9, 11, 12] and [9.5, 10, 11, 11.5] — so they establish only that the two evaluation measures can move in opposite directions. They say nothing about how often that happens in real groups, and they support no conclusion that influence generally degrades collective estimates. Nor do they identify a cause: whether estimates moved through reliable evidence or through social pressure is a separate matter, and the arithmetic is silent between those routes. Convergence through evidence may import genuine information; convergence through pressure merely narrows the distribution.
A stipulated positive countercase shows the limit in the other direction. If reliable new evidence of the stipulated truth is supplied and all four estimates become 10, both error measures reach zero: individuals are perfectly accurate, and the group mean is exact. Nothing in the comparison forbids this outcome.
So the result is narrower than a verdict on deliberation, and narrower too than the point that independence can preserve ignorance. Averaging cannot discover what no estimate contains. The application question is accordingly not whether people heard one another, but what entered the estimates, which metric is at issue, and whether the aggregate's accuracy — not only its members' proximity — improved.
17.4The opposing interpretation
Farrell's response to the social-influence debate challenges an undifferentiated negative interpretation and motivates separating individual from aggregate outcomes (Farrell, 2011). In this study's own numerical construction, the opposing interpretation does not deny the arithmetic; it rejects the question that construction is taken to answer. If the measure of success is how close individuals become to the truth, then the later estimates are plainly better. Their average absolute error falls from 1.5 to 0.75. On that average criterion the later array improves: three estimates become more accurate and one remains unchanged. This says nothing about what caused the changes. A critique organized around individual improvement therefore asks a different question from one organized around aggregate accuracy, and it can accept the second result without conceding that the first has been answered.
The arithmetic shows how the questions come apart. With a stipulated truth of 10, the estimates [8, 9, 11, 12] have mean 10, so the group mean has absolute error 0. Afterward the estimates [9.5, 10, 11, 11.5] have mean 10.5, so the group mean has absolute error 0.5. Average individual error improves while error of the aggregate increases. This is a disagreement about which quantity is being optimized, not a paradox in the numbers.
The inferential step is limited and should be stated narrowly: these two evaluation quantities can move in opposite directions. The construction is invented arithmetic, not observed influence data, and it identifies no psychological cause, no prevalence in real groups, and no general rule that influence degrades aggregation. Consider the stipulated positive countercase: if reliable new evidence of the truth brought all four estimates to 10, both error measures would be zero. Its evidential cause is part of the stipulation, not something that can be inferred from convergence alone. The present example supplies the bare possibility of metric divergence; deciding whether a given episode realizes it requires asking what changed the estimates and what the aggregate is for.
Applied to E06, the opposing interpretation needs a narrower target. The before estimates 8, 9, 11, 12, set against the stipulated truth of 10, average exactly 10, and the average individual absolute error is 1.5. The after estimates 9.5, 10, 11, 11.5 average 10.5; the average individual absolute error falls to 0.75, while the absolute error of the group mean rises from 0 to 0.5. Individuals, taken together, have half the mean absolute error on the first measure, yet the collective answer is slightly worse on the second.
A measurement under which listeners' answers improve does not by itself license a claim about the accuracy of the aggregate. The two quantities are computed differently: one averages distances from the truth across people, the other takes the distance from the truth of an average. In E06 they move in opposite directions. An interpretation that treats improvement in the first as improvement in the second is therefore mistaken; an interpretation confined to individual error remains untouched by this case.
Two limits keep the argument in its place. E06 is invented arithmetic rather than participant data, so it establishes that the reversal is possible and says nothing about how often such reversals occur in real groups. Nor does the arithmetic identify a cause. The tightening of the estimates could reflect shared evidence, mutual pressure, or some mixture of the two; the values alone cannot distinguish these. Whether convergence here is evidential or social has to be settled by something beyond the error counts.
Finally, the aggregate figure is the group mean. Other aggregation rules could give different comparisons, and E06 does not test them. What the case does establish is that a claim that hearing others improved the individual answers and a claim that hearing others improved the group estimate concern different quantities, and can come apart even when both are computed from the same four estimates.
17.5Learning new evidence
The earlier construction left a tension standing. The estimates [8, 9, 11, 12] have a mean of 10 and a mean individual absolute error of 1.5, while the set [9.5, 10, 11, 11.5] improves the mean individual error to 0.75 even as its group mean slips to 10.5, so the error of the mean rises from 0 to 0.5. Individual improvement and aggregate estimation can therefore come apart. What would settle the question is not convergence as such, but what produces it.
Stipulate reliable new evidence bearing on the stipulated truth of 10. All four estimates then become 10. The mean individual absolute error is 0, and the absolute error of the group mean is 0 as well. The inferential step is narrow: in this stipulation the shared input is informative about the quantity at issue and each estimate is revised in light of it, so the resulting agreement is unanimous and accurate at once. Here convergence follows from learning rather than from pressure to resemble one's neighbors; the estimates coincide because they are tracking the same truth. The earlier case shows that improvement in one measure need not improve the other, and this one shows that the conflict is contingent on how the estimates happen to move.
The limit is equally explicit. Reliability is built into the stipulation, and the numbers are invented arithmetic rather than observed influence data; they identify no psychological cause and support no claim about how often real groups receive evidence of this quality. Conditional on shared input being systematically misleading, the four estimates could still move together, toward the same wrong value, yielding unanimity alongside error. What the positive case establishes is what reliable evidence would secure in principle — and that evidence-convergence and pressure-convergence must be kept apart analytically, since a converging group can leave that distinction unresolved.
The positive stipulation completes the contrast rather than replacing it. Here the reliable evidence concerns truth 10, and once it is supplied all four estimates become 10. Average individual absolute error and absolute error of the group mean are then both zero. Set beside the earlier stipulated pair, in which estimates move from 8, 9, 11, 12 to 9.5, 10, 11, 11.5, the point is sharp: individual mean absolute error falls from 1.5 to 0.75 while the mean estimate shifts from 10 to 10.5, raising its error from 0 to 0.5. One stipulation drives both measures to zero; the other lets them move in opposite directions.
That is enough to defeat a blanket opposition to discussion. If an exchange conveys reliable evidence of the stipulated truth, the fact that people revise their estimates is a route to accuracy on both evaluation measures, not a standing threat to estimation. The troublesome stipulation showed only that improvement on one measure is compatible with worsening on the other. It licensed no general rule against hearing from others.
Reliability of evidence is therefore doing explicit premise work. It is stipulated here, not inferred from the resulting agreement. Even perfect convergence on a single number would not by itself establish that the evidence was reliable or that the number was 10; agreement is compatible with influence that carries no information about the truth.
The scope of the claim must remain narrow. Both stipulations are invented arithmetic constructions, not observed influence experiments, and neither identifies a psychological cause nor supports a frequency estimate about real groups. What they establish is structural: the same group-level numbers admit contrary evaluations depending on which error measure is used, and evidence-bearing discussion can be beneficial when reliability is assumed.
That leaves the harder case intact. If participants converge because some copy a confident speaker and no new evidence enters, the arithmetic of agreement is unchanged while its justification is absent.
17.6Convergence without justification
Discussion is worth keeping when it can carry reliable new evidence. The positive countercase makes the point sharply: if new evidence identifies the stipulated truth as 10 and all four estimates become 10, both average individual error and group-mean error fall to zero. But convergence by itself tells a reader nothing about whether this stronger route has occurred. Agreement can also form without anything new being learned.
Take E06’s stipulated estimates. Initially they are [8, 9, 11, 12] around a truth of 10. Their mean is exactly 10, so group-mean error is 0, while average individual absolute error is 1.5. Now suppose, explicitly as a possible mechanism, that the four participants move toward a confident speaker who supplies no further evidence. Their estimates become [9.5, 10, 11, 11.5]. The group mean is now 10.5, so group-mean error has risen to 0.5, even though average individual absolute error has fallen to 0.75. Three estimates move closer to the truth and one is unchanged, yet the aggregate has worsened.
The inferential step is therefore modest but consequential: convergence generated by copying a confident value can tighten the estimates around a center that is not the truth. Evidence-driven convergence may improve both quantities; pressure-driven convergence may improve one and degrade the other. Uniformity after discussion does not distinguish these cases.
The limit matters. These before-and-after estimates are invented arithmetic, not participant observations, and the contrast establishes only that the two evaluation measures can move in opposite directions. It shows no prevalence in real groups and supplies no psychological finding. Whether such copying occurs in a particular case remains open; agreement alone cannot establish it.
Copying a confident speaker without new evidence is one possible mechanism in this stipulated sequence, but it is not established merely by the estimates moving together. Convergence narrows the range of reported values; it does not by itself reveal whether participants revised on a reason, yielded to confidence, or simply happened to agree. The mechanism must therefore be treated as a possible pathway whose presence in a particular group would need separate support.
The scope of the claim follows from E06's arithmetic. Before estimates [8, 9, 11, 12] average to 10 and produce a mean individual absolute error of 1.5 against the stipulated truth 10. Afterwards, the estimates [9.5, 10, 11, 11.5] are closer together, yet their mean is 10.5: mean individual absolute error falls to 0.75 while the absolute error of the group mean rises from 0 to 0.5. The narrower spread and the worse aggregate estimate can therefore occur in the same case. Because these pairs are constructed, the comparison establishes only that the two evaluation quantities can move in opposite directions. It identifies no psychological cause and warrants no claim about how often this pattern occurs in real groups.
The relevant limit separates convergence through evidence from convergence through pressure. If all four estimates became 10 after receiving reliable new evidence about the stipulated truth, both error measures would be 0, and agreement would track accuracy rather than obscure it. Pressure and evidence thus differ in what they contribute to the record, not in whether agreement appears. Discussion can likewise surface information that averaging alone leaves untouched.
The practical application is consequently narrow but useful: convergence alone is an underdetermined signal. It should prompt a question about the route by which agreement arose — changed evidence available to participants, or changed willingness to depart from a speaker — rather than support an inference of motive. That question is the one to make measurable next.
17.7A design that measures both outcomes
Because convergence is compatible with both evidential learning and social pressure, the two outcomes at issue have to be recorded separately rather than read off one another. E06 shows why. Against a stipulated truth of 10, the first set of estimates [8, 9, 11, 12] has a mean of 10 and a mean individual absolute error of 1.5, while the mean estimate is exactly right: its absolute error is 0. The second set [9.5, 10, 11, 11.5] is tighter and individually more accurate, with mean absolute individual error 0.75, yet its mean is 10.5, off by 0.5. Individual improvement and aggregate degradation can thus appear in the same comparison.
A prospective study should therefore record three quantities independently: each participant's absolute error before and after exchange; the absolute error of the group mean before and after exchange; and the evidence each participant actually received during exchange. The third record is what permits the mechanism question to be asked at all. If exchange supplies all four participants with reliable new evidence bearing on the stipulated truth and their estimates all become 10, then individual error and group error both fall to 0. That pattern is consistent with convergence through evidence. If instead estimates tighten while no relevant evidence changes hands, the same movement of individual error would look quite different, and the group mean could drift away from truth as it does in E06.
The limit of this design is that it separates outcomes, not causes. E06 is invented arithmetic rather than observed participant data, so it establishes only that these evaluation quantities can move in opposite directions; it says nothing about how often that happens in real groups, and no psychological motive follows from tightened estimates alone. Attributing convergence to pressure requires evidence about what participants learned and why they moved, not merely evidence that they moved. What these three measurements buy is a comparison in which such attributions can at least be tested rather than assumed.
Separating individual error, group error and changed evidence is only half the design. Each quantity must be anchored to something against which error is defined. E06 supplies that anchor: the stipulated truth of 10 is what makes the before-and-after calculations meaningful at all. Without a specified truth, "improvement" would name nothing more than movement among estimates, and a group that converged tightly on the wrong figure would look no different from one that converged on the right one.
The inferential step is therefore a restriction on scope. A prospective design that computes individual and aggregate errors needs tasks with task-specific ground truth—questions whose answers are verifiable independently of the estimates being scored. Normative disagreement does not meet that condition. There is no single factual answer against which an estimate of, say, what ought to be done can simply be marked right or wrong. Treating convergence in such a case as error reduction would quietly impose one answer as the standard and then celebrate agreement with it as accuracy.
This limit is substantive, not a reason to abandon the study of discussion on evaluative questions. It marks where the error-based measurement applies and where a different assessment would be needed—one that tracks what happened to reasons and evidence rather than distance from a stipulated truth. Conditional on a task supplying verifiable ground truth, the three measures can be computed and compared directly; absent that condition, the same arithmetic would be a category mistake.
The practical upshot is that the design has a defined field of application. It can test whether hearing others improves or degrades aggregate factual estimation, and whether any convergence arrived through new evidence rather than pressure. What it cannot do is generalize those verdicts to normative deliberation, where the very object scored by error does not exist. E06 remains an invented arithmetic case illustrating this division, not a claim about observed frequency in any study.