The main underlying assumption made in retrievals, direct radiance assimilation and in the 3 and 4 dimensional variational (VAR) data assimilation (DA) schemes is that the variables and/or observational errors are Normally (Gaussian) distributed. However, there is evidence that some of these errors are not Normal and are often lognormal. There is also a state variable, humidity, which when used in retrieval and 3D-4D VAR is transformed to its logarithm to be compatible with the Normal framework which implies a lognormal structure. There are commonly two approaches to overcome this problem of non-Normal errors. The first is to apply the logarithm, as we have just mentioned, and then apply the current Normal framework to assimilate. After the minimum state has been found for this variable, the transform is inverted back to the model state. The second approach is to simply apply the Normal framework directly to the non-Normal variable. Both approaches introduce a bias into the analysis state which has to be bias corrected is some form. In this paper we quantify these biases by identifying the impact of applying a transform (retrievals) or assuming a Normal framework (radiances) instead of using the correct probability model. We also present a method which uses the correct probability model and show the difference between the two Normal methods and the correct probability model.