Within the small number range, preference for the larger group was statistically significant for all ratios (0.25: t(33) = 3.265, P = 0.003; 0.33: t(33) = 3.260, P = 0.003; 0.50: t(33) = 3.537, P,0.001; 0.67: t(33) = 4.003, P,0.001; 0.75: t(33) = 3.230, P = 0.003); in the large number range, the fish spent significantly more time near the larger shoal for ratios 0.25 (t(33) = 6.922, P,0.001), 0.33 (t(33) = 5.788, P,0.001) and 0.50 (t(33) = 4.466, P,0.001), but not for ratios 0.67 (t(33) = 1.481, P = 0.148) or 0.75 (t(33) = 1.159, P = 0.255). No single ratio significantly differed between the two ranges (two sample t-test, df = 66 all P.0.169). Yet all comparisons had a very low statistical power (0.052#x#0.279) and therefore lack of significance could be due to type II error. Sample size calculation indicates that, for each comparison of our experiment, a minimum of 90 subjects was needed in order to reach a 80% power to detect a 20% difference between treatment groups with a two-sided test and alpha = 0.05 [45]. To test if lack of significance of between-range comparisons was due to inadequate sample size we increased the sample size for one numerical ratio (0.67), by testing 64 additional adult fish, 32 in a small number discrimination (2 vs. 3) and 32 in a large number discrimination (4 vs. 6). The difference between the 2 vs. 3 and the 4 vs. 6 contrasts became statistically significant (independent t-test t(130) = 2.688, P = 0.008). As observed with humans, as discrimination became more difficult, the fish tended to be more accurate in the small number range; they could in fact discriminate between two numbers with a 0.67 or 0.75 ratio in the small number range, whereas they required a 2:1 ratio when tested with large numbers. Discussion Previous studies have reported remarkable similarities in the performance of non-verbal numerical tasks among humans, apes and monkeys, suggesting the existence of the same basic numerical systems among primates [3,29,46,47,48]. Here we provide evidence of a similar correspondence in numerical abilities between humans and teleost fishes. When tested in the same numerical tasks, the students and guppies showed almost identical performance patterns. In both species, the ability to discriminate between large numbers (.4) was approximate and strongly dependent on the ratio between the numerosities. In contrast, in both fish and students, discrimination in the small number range was not dependent on ratio and discriminating 3 from 4 was as easy as discriminating 1 from 4. Likelihood ratio analyses indicate that the lack of ratio effect is 3.68 times more likely than the alternative hypothesis in fish and 3.67 times in students. As a consequence, the discrimination of larger numerical ratios, 0.67 and 0.75, is easy in the small number range for both species, but becomes relatively more difficult (among students) or even impossible (among fish) when confronted with large numbers. It is possible that, in fish, the different discriminative ability in the two numerical ranges is the consequence of a different encounter rate with large or small social groups and thus a different familiarity with large and small numbers. However, the observation that juveniles raised in pairs, and therefore with no previous experience in comparing social groups, showed the same pattern as experienced adults rules this possibility out and indicates that numerical systems are probably innate in fish. The observation that some numerical abilities are exhibited by guppies at birth is certainly remarkable and is line with evidence that both chicks and newborns display some rudimentary numerical skills [49,50]. This reinforces the proposition by some authors of a core Similar Numerical Systems in Humans and Guppies knowledge of number, a system of innate numerical representation shared among the different non-human species [3,51]. Since numerosity normally co-varies with other physical attributes such as the total area occupied by objects, one may argue that in the fish experiment subjects were using these cues instead of numerical information to solve the task. This possibility was not checked in the present study. However in two previous studies we showed that guppies and mosquitofish can discriminate between two schools of fish using the numerosity information only, both within the small number range or outside it [37,52] and that mosquitofish can discriminate between sets of geometric figures in both numerical ranges even after all continuous variables were controlled for. In particular a recent study using a training procedure has shown no difference in the learning rate between fish trained to use numerical information only and fish trained to use continuous quantities only, suggesting that the number per se is not more cognitively demanding than continuous quantities [53]. On the other hand students showed a very similar performance whether continuous variables were controlled or not, thus making the comparisons fully legitimate. One can argue that the guppy and student experiments differ in many respects. In particular students were tested in a sequential presentation, whereas fish saw the simultaneous presentation of the numerosity pairs. However in this respect the two experiments may differ less than may appear at first glance. In our test situation the fish could never see the two stimuli binocularly simultaneously. It was possible for a subject to observe both shoals only when swimming perpendicularly to the main axis, which occurred very rarely during a test. Moreover, in this position, each stimulus was seen with a different eye; in this situation, as a consequence of the lateralization of social recognition [54,55] and reduced inter- hemispheric transfer of information [56,57], fish cannot be expected to guess the larger group. The difference in ratio-dependence suggests the existence in fish, as in humans, of two distinct non-verbal mechanisms of numerical representation, one for numbers 1–4 and one for large quantities. Yet the hypothesis that a precise object-file mechanism does underlie small number discrimination also predicts higher performance in the small number range, a finding not evident in our study. However, the lack of a statistical difference between ranges observed in fish experiment may be due to the limit of the procedure adopted. Previous studies using this procedure have shown that accuracy in selecting the larger shoal never exceed 70% even with very easy numerical ratios (guppies [37,58], mosquitofish [22], topminnows [59], angelfish [60]). Due to large measurement variance combined with small sample size in each numerical contrast, these statistical tests suffered from low power and hence lack of significance may be attributable to a great probability of making a type II error. Sample size calculation indicates that in our experiment a threefold sample size was needed to obtain an adequate statistical power when two treatments had to be compared. As confirmation, a statistical difference emerged between the large and small number range after we increased the sample size in the 0.67 ratio. Regarding this latter evidence, previous work has reported that macaques, mosquitofish, chicks and bees could distinguish two from three items but failed with the same ratio in the large number range [22,29,33,34,61], even though none of these studies could provide a statistical test to support a difference in performance between the two ranges. Our result highlights the possibility that, as in other research fields [62,63,64], many studies are underpowered to detect statistical differences among subgroups. The results reported here differ from those found in a very recent study on angelfish which discriminated 2 vs. 3 but not 3 vs. PLoS ONE | www.plosone.org 6 February 2012 | Volume 7 | Issue 2 | e31923