The Echoes of the Mind (87-3) Estimation


Many estimates are based upon an initial value that is adjusted to yield a final result. The starting point may be suggested within the problem, or an initial value may arise from partial computation. In either case, adjustments are inadequate. This owes to anchoring bias: different starting points yield divergent estimates based upon the initial value.

The anchoring effect refers to the situation in which an arbitrarily chosen reference point (anchor) significantly influences the decision makers’ value estimates, and the value estimated is insufficiently adjusted away from the reference point toward the true value of the target of estimation ~ Taiwanese information analyst Chin-Shan Wu et al

 Product Estimation

Given only 5 seconds, 2 groups of high school students individually estimated the value of a multiplicative product.

One group estimated:

1 x 2 x 3 x 4 x 5 x 6 x 7 x 8

The other:

8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

To rapidly answer such questions, people perform a few computation steps, and then extrapolate from there. Because adjustment is typically insufficient, this technique should systematically lead to underestimation. Further, as the first product set begins with lower numbers, its estimation should be a lower value.

So it was. The median answer for the ascending sequence was 512, while the average answer for the descending sequence was 2,250. The correct answer: 40,320.


Planning often involves multiple steps or stages. Successful outcomes depend upon accurate estimation, especially when a project is a series of connected (conjunctive) tasks. Estimating sequential events also suffers from anchoring bias.


An experiment tested anchoring bias by type of event. Participants could gamble on 1 of 2 distinct events. 1 of the 2 events was a simple one: drawing from a bag containing marbles, 50% of which were white, 50% red. The other event of choice was compound: a series of elementary events, though of 2 different types: conjunctive and disjunctive.

The conjunctive event was drawing a red marble 7 times in a row, with marble replacement each time, from a bag of 90% red marbles and 10% white marbles.

The disjunctive event was drawing a red marble at least once out of 7 tries from a bag with 10% red marbles and 90% white ones.

The 50–50 simple event had a 50% probability. The connected series (conjunctive) event had a 48% probability. The disconnected (disjunctive) event had a 52% probability.

A significant majority preferred to bet on conjunctive event, which had lower odds than the simple event. Conversely, the strong preference was for the lower-odds, simple event over the disjunctive event.

The simple event is readily understood, and so provides an anchoring for evaluating the compound events via adjustment, which proved inapt. As the results showed, people overestimate the probability of conjunctive events, and underestimate the probability of disjunctive events.


In a conjunctive project, each event in the series must sequentially transpire. The general tendency to overestimate (the probability of achieving) conjunctive events leads to unwarranted optimism of timely completion.

Even when meeting every conjunctive milestone is highly likely, the overall probability of success can be quite low if there are many events. Any estimation error can have a domino effect.

The ‘law’ of sequential choice (or decision) is that if the number of stages in is held constant, the relative overestimation of the chance of guessing correctly at all stages varies directly with a power of the number of alternatives per stage. If, however, the number of alternatives is held constant, the relative overestimation varies exponentially with the number of stages. ~ English psychologist John Cohen et al

Evaluations of risk typically involve disjunction. A complex structure or system will malfunction if any of its essential components fail.

Even when the likelihood of failure for any component is slight, the probability of malfunction increases with the number of working parts. Because of anchoring, there is a strong bias to underestimate the chance of failure in complex systems.

The chain-like structure of conjunctions leads to overestimation, the funnel-like structure of disjunctions leads to underestimation. ~ Amos Tversky & Daniel Kahneman