A note on mean absolute deviation

We introduce the idea of Mean Absolute Deviation (MAD). We provide a simple method for calculating the MAD that circumvents the use of the absolute value. We develop this both for the case of continuous and finite valued random variables. We introduce the idea of sub-mean and show how the MAD is related to the difference between the mean and sub-mean. We look at the MAD for random variables with symmetric probability density functions. We consider problem of determining the best estimate of a random variable under various measures of error.