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 $langle abla^2 Tbig|Delta T(R)rangle$ and $langle| abla T|^2big|Delta T(R) rangle$. 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 $langle abla^2 Tbig| Delta T(R)rangle$ 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.
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