We determine the orbital eccentricities of individual small Kepler planets , through a combination of asteroseismology and transit light-curve analysis . We are able to constrain the eccentricities of 51 systems with a single transiting planet , which supplement our previous measurements of 66 planets in multi-planet systems . Through a Bayesian hierarchical analysis , we find evidence that systems with only one detected transiting planet have a different eccentricity distribution than systems with multiple detected transiting planets . The eccentricity distribution of the single-transiting systems is well described by the positive half of a zero-mean Gaussian distribution with a dispersion \sigma _ { e } = 0.32 \pm 0.06 , while the multiple-transit systems are consistent with \sigma _ { e } = 0.083 ^ { +0.015 } _ { -0.020 } . A mixture model suggests a fraction of 0.76 ^ { +0.21 } _ { -0.12 } of single-transiting systems have a moderate eccentricity , represented by a Rayleigh distribution that peaks at 0.26 ^ { +0.04 } _ { -0.06 } . This finding may reflect differences in the formation pathways of systems with different numbers of transiting planets . We investigate the possibility that eccentricities are “ self-excited ” in closely packed planetary systems , as well as the influence of long-period giant companion planets . We find that both mechanisms can qualitatively explain the observations . We do not find any evidence for a correlation between eccentricity and stellar metallicity , as has been seen for giant planets . Neither do we find any evidence that orbital eccentricity is linked to the detection of a companion star . Along with this paper we make available all of the parameters and uncertainties in the eccentricity distributions , as well as the properties of individual systems , for use in future studies .