In a distribution that is skewed by a few extreme outliers, what would be the best choice for a measure of central tendency?

Question

According to my Professor, the answer to this question is the MEDIAN. However, wouldn't MODE be a better measure of central tendency? The mode is almost never affected by extreme outliers..please help explain this. THANKS

Answer 1

If your sample consists of 1000 distinct points in $[0,1]$ and two more samples equal to $100$, I would say $100$ is the outlier, but it is also the mode. The mode in this case is clearly NOT a good measure of central tendency, right?

Answer 2

No. The mode is very sensitive to the observed frequencies. For example, the data

$$\{ 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 7, 10, 100 \}$$

has a mode of 1, but the similar shaped distribution

$$\{ 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 7, 10, 100 \}$$

has a mode of 4.