Solved – Mean, Median, Mode:When to use what?

In cases when it is possible to find the mean, median and mode then the usefulness follows from what is important to you.

If we are talking about utility (or money) then the mean is used quite commonly because the expected value (mean) of the money we get is an attractive thing to maximize.

However, the St Petersburg paradox is an example of where the mean is not useful, it involves a gambling game with a 50% chance of losing your money but the mean amount of profit is infinite. This game isn't attractive to most people and thats because they look at the median profit (which is a loss). If you aren't concerned with the absolute value of the profit but you care more about frequently getting above a certain profit then the median is a useful statistic

The mode is useful when you need to be exactly correct. Perhaps you are predicting peoples' ages and they give you a dollar if you are right. Here if you are wrong it doesn't matter if you are above or below the age you guessed, it also doesn't matter if you are wrong by 1 year or by 10 years because those mistakes have equal consequences.

Aside from the usefulness for your context there are a few statistical properties of the central tendencies. The median is a robust way to measure central tendency because outliers don't affect it greatly. On the other hand, the mean is the most efficient estimate of central tendency so the sample mean converges faster than the sample median.