When to use mode over mean ?

Applications of the geometric mean are most common in business and finance, where it is frequently used when dealing with percentages to calculate growth rates and returns on a portfolio of securities
The geometric mean is used in finance to calculate average growth rates and is referred to as the compounded annual growth rate. Consider a stock that grows by 10% in year one, declines by 20% in year two, and then grows by 30% in year three. The geometric mean of the growth rate is calculated as follows:

( (1+0.1) * (1-0.2) * (1+0.3) ) ^ (1/3) = 0.046 or 4.6% annually.

The geometric mean can be good for combining numbers that are expressed in different units. Summing measurements makes a mess. But multiplying them is fine. For example, say I have a list of people's heights and weights, but I want one number that tells me how big each person is. I don't want to discard any data. If I use √height×weight , I get one number for each person, using no free parameters. But if I try to use (height + weight)/2, I'd have to worry about the relative weighting of height and weight, & that leaves me with an extra free parameter. (E.g. if I use weight in grams and height in meters, then height will have almost no effect on my "bigness" result.) I can pick a scale factor that seems reasonable to me, but why am I introducing a parameter here?

F1 score uses harmonic mean of precision and recall

Arithmetic mean is used on values, Geometric mean is usually used on rates, Harmonic mean is usually used on ratios.