  We consider turbulent advection of a scalar field $T(\B.r)$, passive or
active, and focus on the statistics of gradient fields conditioned on scalar
differences $\Delta T(R)$ across a scale $R$. In particular we focus on two
conditional averages $\langle\nabla^2 T\big|\Delta T(R)\rangle$ and
$\langle|\nabla T|^2\big|\Delta T(R) \rangle$. We find exact relations between
these averages, and with the help of the fusion rules we propose a general
representation for these objects in terms of the probability density function
$P(\Delta T,R)$ of $\Delta T(R)$. These results offer a new way to analyze
experimental data that is presented in this paper. The main question that we
ask is whether the conditional average $\langle\nabla^2 T\big| \Delta
T(R)\rangle$ is linear in $\Delta T$. We show that there exists a dimensionless
parameter which governs the deviation from linearity. The data analysis
indicates that this parameter is very small for passive scalar advection, and
is generally a decreasing function of the Rayleigh number for the convection
data.
