Measures of Central Tendency: Median and Mode

An analyst examining historical annual returns on a portfolio finds that it has a median value of 9.4%, a mean of 8.7%, and a mode of 5.4%. The sum of all returns is 226.2%. The number of years of returns being examined by the analyst is _closest_ to:

Incorrect. You can get this answer by dividing the sum of annual returns by the median value. Although the median is influenced by the number of returns, it does not depend on the sum of returns.

Incorrect. You can get this answer by dividing the sum of annual returns by the modal value. The mode is the most frequently occurring value, and it is not influenced by either the total number of returns or the sum of returns.

Exactly! The arithmetic mean is the sum of all returns divided by the number of returns. $$\displaystyle \bar{r} = \frac{\sum_{i=1}^n r_i}{n}$$ Given the sum of returns and the mean value, you can calculate the number of returns used in the calculation. $$\displaystyle n = \frac{\sum_{i=1}^n r_i}{\bar{r}} = \frac{226.2\%}{8.7\%} = 26$$ Because these are annual returns, the analyst is examining 26 years of historical data.

24\.

26\.

42\.