Observed distributions of atmospheric temperature are non-Gaussian. Therefore, the mean and variance are not sufficient in determining likelihood of extreme temperature events. While previous work has sought to estimate  higher-order moments using models and observations, no analytical solution has been derived from first principles. Here, we show an analytical calculation of moments of any order when temperature variability is driven by advection via stochastic meridional wind anomalies. Our results show that nonzero skew and kurtosis arise due to nonlinearity in time-mean temperature, and results are tested using idealized climate model simulations (using ISCA) for the mid-latitudes with specified meridional temperature gradients. Because our method considers large-scale dynamics to calculate statistical moments at the local scale, results can be used to connect large-scale dynamics to local-scale variability and extremes. These results and methods can also serve as a useful diagnostic tool to evaluate the accuracy of temperature output from climate models.