The arithmetic average is the sum of returns in each period divided by the number of periods. This is the most commonly used average and is generally just referred to as the "average".
In the use of investments the arithmetic average can be a very poor and deceptively inaccurate measure. The reason is that the arithmetic average assumes that each return is independent of each other. The arithmetic average does not take into account the change in the amount of capital available to investment as a result of a previous gain or loss.
If one was to invest $100 for two years. Assume after the first year the investment loss 50% of it's value. The next year the investment gained 50% of it's value by the end of years end. If the arithmetic average were used to determine the rate of return for both years the investor should have broke even at 0% leaving his value at $100. But in reality the investor suffered a loss of 25% and is left with a value of $75. This is because after the first year the capital available to investment was only $50. So an increase of $50 by 50% would take the value to $75 at the end of 2 years.