Sometimes one feels that some data is more important/accurate than other data.
For example, If one wanted to know what the average temperature in July is,
he might feel that temperatures from recent years are more important than
temperatures form less recent years. If the mean temperatures in July were
72 (1996), 69 (1995(, 75 (1994), 73 (1993), and 68 (1992); one could calculate
a weighted mean:

(72x1 + 69x.8 + 75x.6 + 73x.4 + 68x.2)/(1+.8+.6+.4+.2)=71.67

Determining the average size of classes at a university is another problem where weighted means may be appropriate.

(2x5X8)^(1/3)=4.31 or

e^((1/3)(ln(2) + ln(5) + ln(8))) = e^((1/3)(.69+1.61+2.08))=4.31

The geometric mean is the appropriate concept of the mean for average interest or inflation rates. If the inflation rates for three successive years are 3%, 12%, and 5%; the geometric mean of 1.03, 1.12, and 1.05, which is 1.066 gives the annual inflaton rate, 6.6%, which if constant for three years would produce the same increase in prices.

The geometric mean will always be less than or equal to the arithmetic mean.

1/((1/3)((1/2)+(1/5)+(1/8)))=3.64

The canonical problem in which the harmonic mean is employed is that if a car drives 50 miles at 30 mph, 50 miles at 50 mph, and 50 miles at 60 mph, what is its average speed? The total distance is 150 miles, the total time is (50/30)+(50/50)+(50/60)=3.5; hence the average speed is 150/3.5=42.86 mph.

This can be concisely calculated as:

1/((1/3)((1/30)+(1/50)+(1/60)))=42.86

The harmonic mean is always less than or equal to the geometric mean.

A motivation for the coefficient of variation is if one is wondering whether there is more variation in weight among men than mice. Since the heaviest mouse weighs less than the standard deviation of weights of man, the standard deviation of the weights of mice must be less. But the question can be reposed as relative variation, which is what the coefficient of variation measures: the coefficient of variation is the ratio of the standard deviation to the mean. For one of my classes, the mean weight was 152 pounds, with a standard deviation of 31 pounds; the mean height was 69.3 inches, with a standard deviation of 3.86 inches. Hence the coefficients of variation were 31/152=.20 and 3.86/69.3=.056 for weight and height respectively. Note that that the coefficient of variation is independent of what units the data were measured in (pounds or kilograms or stones; inches or feet or metres).