'Average' is a vague term that usually refers to the arithmetic mean but can also signify the median, mode, geometric mean, or weighted mean. The word stems from a commercial practice of the shipping age. The root aver means to declare, and the shippers of goods would declare the value of their goods. When the goods were sold, a deduction was made from each persons share, based on their declared value, for a portion of the loss or "average."
Average of a function
The RMS is the value obtained by taking the mean of a set of squared values. It provides the value of a periodic quantity and was developed at the beginning of the twentieth century as an accurate measure of voltage and electric current. An RMS error of approximity is also used for equations devised to predict outcomes, such as weather forecasts.