When A and B are each in y and T is in z , T will have a placement on the focus axis that is higher than one when A < B, but equally often it will have a position on the attention axis that is less than 1, when B < A . Additionally, there are six equally probable arrangements when A, B, and T are all in range y . But only two of these six arrangements result in situations where T> one additional infoon the focus axis. Beneath these circumstances, the general chance that T will have a benefit less than one on the attention axis in Fig 1C is .583 . The capacity of the orientation of the attention axis to flip in specific configurations implies that there will be a bias for T to have a price shifted from 1 toward the route of , even even though T was drawn from the same distribution as A.This bias in the consideration axis stems from the overlap amongst distributions used to construct it and noisy estimates of the distribution means. If the estimates A, B and T have been each and every dependent on a large number of samples, or personal responses from a big variety of neurons, they would approximate the true signifies of the distributions, B would almost by no means lie to the correct of A, and the bias would be effectively removed. The volume of bias in the case in point in Fig one can be specifically calculated since the distributions have nicely-outlined properties. To validate the chances in Fig 1C, we simulated the sampling situation ten,000 instances. The indicate likelihood that T < 1 was 0.583 when A and T were randomly drawn from random uniform distributions, while the mean probability that T < A was 0.50 , matching theoretical expectations. Thus, we can account for biased sampling on the attention axis when the statistics of the distributions are known, and potentially correct for it.In practice, however, the attention axis is built on responses from many neurons that have distributions that can only be estimated. It is therefore difficult to know how much bias might enter into attention axis measurements. If the distributions of firing rates of the two response distributions that comprise the attention axis are non-overlapping, then axis inversion is avoided. But because neurons are typically weakly modulated by attention and are often driven by suboptimal sensory stimuli in neurophysiological experiments, overlapping population responses are common.Although the bias described in Fig 1 does not occur if samples are drawn from two non-overlapping distributions, a second bias can occur even when the sample distributions do not overlap and is related to the process of projecting points in multidimensional space. This bias acts in the same direction as that described above, shifting measurements towards 0 on the attention axis. Fig 2 illustrates this other source of bias using a simplified example in which the population responses are based on responses from only two neurons. All responses are noiseless except the response of neuron 2 in the attend-right condition, which is drawn from a uniform distribution.As with the simulations in Fig 1, the attention axis is constructed using only a single sample of the neurons' responses from the attend-right response distribution, A, and a single sample of the neurons' responses from the attend-left response distribution, B .