  We explain how It\^o Stochastic Differential Equations (SDEs) on manifolds
may be defined using 2-jets of smooth functions. We show how this relationship
can be interpreted in terms of a convergent numerical scheme. We show how jets
can be used to derive graphical representations of It\^o SDEs. We show how jets
can be used to derive the differential operators associated with SDEs in a
coordinate free manner. We relate jets to vector flows, giving a geometric
interpretation of the It\^o--Stratonovich transformation. We show how
percentiles can be used to give an alternative coordinate free interpretation
of the coefficients of one dimensional SDEs. We relate this to the jet
approach. This allows us to interpret the coefficients of SDEs in terms of "fan
diagrams". In particular the median of a SDE solution is associated to the
drift of the SDE in Stratonovich form for small times.
